niels bohr hydrogen experiment

Atomic Physics

Bohr’s theory of the hydrogen atom, learning objectives.

By the end of this section, you will be able to:

• Describe the mysteries of atomic spectra.
• Explain Bohr’s theory of the hydrogen atom.
• Explain Bohr’s planetary model of the atom.
• Illustrate energy state using the energy-level diagram.
• Describe the triumphs and limits of Bohr’s theory.

The great Danish physicist Niels Bohr (1885–1962) made immediate use of Rutherford’s planetary model of the atom. (Figure 1). Bohr became convinced of its validity and spent part of 1912 at Rutherford’s laboratory. In 1913, after returning to Copenhagen, he began publishing his theory of the simplest atom, hydrogen, based on the planetary model of the atom. For decades, many questions had been asked about atomic characteristics. From their sizes to their spectra, much was known about atoms, but little had been explained in terms of the laws of physics. Bohr’s theory explained the atomic spectrum of hydrogen and established new and broadly applicable principles in quantum mechanics.

Figure 1. Niels Bohr, Danish physicist, used the planetary model of the atom to explain the atomic spectrum and size of the hydrogen atom. His many contributions to the development of atomic physics and quantum mechanics, his personal influence on many students and colleagues, and his personal integrity, especially in the face of Nazi oppression, earned him a prominent place in history. (credit: Unknown Author, via Wikimedia Commons)

Mysteries of Atomic Spectra

As noted in Quantization of Energy , the energies of some small systems are quantized. Atomic and molecular emission and absorption spectra have been known for over a century to be discrete (or quantized). (See Figure 2.) Maxwell and others had realized that there must be a connection between the spectrum of an atom and its structure, something like the resonant frequencies of musical instruments. But, in spite of years of efforts by many great minds, no one had a workable theory. (It was a running joke that any theory of atomic and molecular spectra could be destroyed by throwing a book of data at it, so complex were the spectra.) Following Einstein’s proposal of photons with quantized energies directly proportional to their wavelengths, it became even more evident that electrons in atoms can exist only in discrete orbits.

Figure 2. Part (a) shows, from left to right, a discharge tube, slit, and diffraction grating producing a line spectrum. Part (b) shows the emission line spectrum for iron. The discrete lines imply quantized energy states for the atoms that produce them. The line spectrum for each element is unique, providing a powerful and much used analytical tool, and many line spectra were well known for many years before they could be explained with physics. (credit for (b): Yttrium91, Wikimedia Commons)

In some cases, it had been possible to devise formulas that described the emission spectra. As you might expect, the simplest atom—hydrogen, with its single electron—has a relatively simple spectrum. The hydrogen spectrum had been observed in the infrared (IR), visible, and ultraviolet (UV), and several series of spectral lines had been observed. (See Figure 3.) These series are named after early researchers who studied them in particular depth.

The observed hydrogen-spectrum wavelengths can be calculated using the following formula:

[latex]\displaystyle\frac{1}{\lambda}=R\left(\frac{1}{n_{\text{f}}^2}-\frac{1}{n_{\text{i}}^2}\right)\\[/latex],

where λ is the wavelength of the emitted EM radiation and R is the Rydberg constant , determined by the experiment to be  R = 1.097 × 10 7  / m (or m −1 ).

The constant n f is a positive integer associated with a specific series. For the Lyman series, n f  = 1; for the Balmer series, n f  = 2; for the Paschen series, n f  = 3; and so on. The Lyman series is entirely in the UV, while part of the Balmer series is visible with the remainder UV. The Paschen series and all the rest are entirely IR. There are apparently an unlimited number of series, although they lie progressively farther into the infrared and become difficult to observe as n f increases. The constant n i is a positive integer, but it must be greater than n f . Thus, for the Balmer series, n f  = 2 and n i  = 3, 4, 5, 6, …. Note that n i can approach infinity. While the formula in the wavelengths equation was just a recipe designed to fit data and was not based on physical principles, it did imply a deeper meaning. Balmer first devised the formula for his series alone, and it was later found to describe all the other series by using different values of n f . Bohr was the first to comprehend the deeper meaning. Again, we see the interplay between experiment and theory in physics. Experimentally, the spectra were well established, an equation was found to fit the experimental data, but the theoretical foundation was missing.

Figure 3. A schematic of the hydrogen spectrum shows several series named for those who contributed most to their determination. Part of the Balmer series is in the visible spectrum, while the Lyman series is entirely in the UV, and the Paschen series and others are in the IR. Values of n f and n i are shown for some of the lines.

Example 1. Calculating Wave Interference of a Hydrogen Line

What is the distance between the slits of a grating that produces a first-order maximum for the second Balmer line at an angle of 15º?

Strategy and Concept

For an Integrated Concept problem, we must first identify the physical principles involved. In this example, we need to know two things:

• the wavelength of light
• the conditions for an interference maximum for the pattern from a double slit

Part 1 deals with a topic of the present chapter, while Part 2 considers the wave interference material of Wave Optics.

Solution for Part 1

Hydrogen spectrum wavelength . The Balmer series requires that n f  = 2. The first line in the series is taken to be for n i  = 3, and so the second would have n i  = 4.

The calculation is a straightforward application of the wavelength equation. Entering the determined values for n f and n i yields

[latex]\begin{array}{lll}\frac{1}{\lambda}&=&R\left(\frac{1}{n_{\text{f}}^2}-\frac{1}{n_{\text{i}}^2}\right)\\\text{ }&=&\left(1.097\times10^7\text{ m}^-1\right)\left(\frac{1}{2^2}-\frac{1}{4^2}\right)\\\text{ }&=&2.057\times10^6\text{ m}^{-1}\end{array}\\[/latex]

Inverting to find λ gives

[latex]\begin{array}{lll}\lambda&=&\frac{1}{2.057\times10^6\text{ m}^-1}=486\times10^{-9}\text{ m}\\\text{ }&=&486\text{ nm}\end{array}\\[/latex]

Discussion for Part 1

This is indeed the experimentally observed wavelength, corresponding to the second (blue-green) line in the Balmer series. More impressive is the fact that the same simple recipe predicts all of the hydrogen spectrum lines, including new ones observed in subsequent experiments. What is nature telling us?

Solution for Part 2

Double-slit interference (Wave Optics). To obtain constructive interference for a double slit, the path length difference from two slits must be an integral multiple of the wavelength. This condition was expressed by the equation  d sin θ = mλ , where d is the distance between slits and θ is the angle from the original direction of the beam. The number m is the order of the interference; m =1 in this example. Solving for d and entering known values yields

[latex]\displaystyle{d}=\frac{\left(1\right)\left(486\text{ nm}\right)}{\sin15^{\circ}}=1.88\times10^{-6}\text{ m}\\[/latex]

Discussion for Part 2

This number is similar to those used in the interference examples of Introduction to Quantum Physics (and is close to the spacing between slits in commonly used diffraction glasses).

Bohr’s Solution for Hydrogen

Bohr was able to derive the formula for the hydrogen spectrum using basic physics, the planetary model of the atom, and some very important new proposals. His first proposal is that only certain orbits are allowed: we say that the orbits of electrons in atoms are quantized . Each orbit has a different energy, and electrons can move to a higher orbit by absorbing energy and drop to a lower orbit by emitting energy. If the orbits are quantized, the amount of energy absorbed or emitted is also quantized, producing discrete spectra. Photon absorption and emission are among the primary methods of transferring energy into and out of atoms. The energies of the photons are quantized, and their energy is explained as being equal to the change in energy of the electron when it moves from one orbit to another. In equation form, this is Δ E  = hf =  E i  −  E f .

Figure 4. The planetary model of the atom, as modified by Bohr, has the orbits of the electrons quantized. Only certain orbits are allowed, explaining why atomic spectra are discrete (quantized). The energy carried away from an atom by a photon comes from the electron dropping from one allowed orbit to another and is thus quantized. This is likewise true for atomic absorption of photons.

Here, Δ E is the change in energy between the initial and final orbits, and hf is the energy of the absorbed or emitted photon. It is quite logical (that is, expected from our everyday experience) that energy is involved in changing orbits. A blast of energy is required for the space shuttle, for example, to climb to a higher orbit. What is not expected is that atomic orbits should be quantized. This is not observed for satellites or planets, which can have any orbit given the proper energy. (See Figure 4.)

Figure 5 shows an energy-level diagram , a convenient way to display energy states. In the present discussion, we take these to be the allowed energy levels of the electron. Energy is plotted vertically with the lowest or ground state at the bottom and with excited states above. Given the energies of the lines in an atomic spectrum, it is possible (although sometimes very difficult) to determine the energy levels of an atom. Energy-level diagrams are used for many systems, including molecules and nuclei. A theory of the atom or any other system must predict its energies based on the physics of the system.

Figure 5. An energy-level diagram plots energy vertically and is useful in visualizing the energy states of a system and the transitions between them. This diagram is for the hydrogen-atom electrons, showing a transition between two orbits having energies E 4 and E 2 .

Bohr was clever enough to find a way to calculate the electron orbital energies in hydrogen. This was an important first step that has been improved upon, but it is well worth repeating here, because it does correctly describe many characteristics of hydrogen. Assuming circular orbits, Bohr proposed that the angular momentum L of an electron in its orbit is quantized , that is, it has only specific, discrete values. The value for L is given by the formula [latex]L=m_{e}vr_{n}=n\frac{h}{2\pi}\left(n=1,2,3,\dots\right)\\[/latex], where L is the angular momentum, m e is the electron’s mass, r n is the radius of the n th orbit, and h is Planck’s constant. Note that angular momentum is L  =  Iω . For a small object at a radius r ,  I =  mr 2  and [latex]\omega=\frac{v}{r}\\[/latex], so that [latex]L=\left(mr^2\right)\frac{v}{r}=mvr\\[/latex]. Quantization says that this value of mvr can only be equal to [latex]\frac{h}{2},\frac{2h}{2},\frac{3h}{2}\\[/latex], etc. At the time, Bohr himself did not know why angular momentum should be quantized, but using this assumption he was able to calculate the energies in the hydrogen spectrum, something no one else had done at the time.

From Bohr’s assumptions, we will now derive a number of important properties of the hydrogen atom from the classical physics we have covered in the text. We start by noting the centripetal force causing the electron to follow a circular path is supplied by the Coulomb force. To be more general, we note that this analysis is valid for any single-electron atom. So, if a nucleus has Z protons ( Z  = 1 for hydrogen, 2 for helium, etc.) and only one electron, that atom is called a hydrogen-like atom . The spectra of hydrogen-like ions are similar to hydrogen, but shifted to higher energy by the greater attractive force between the electron and nucleus. The magnitude of the centripetal force is [latex]\frac{m_{e}v^2}{r_n}\\[/latex], while the Coulomb force is [latex]k\frac{\left(Zq_{e}\right)\left(q_e\right)}{r_n^2}\\[/latex]. The tacit assumption here is that the nucleus is more massive than the stationary electron, and the electron orbits about it. This is consistent with the planetary model of the atom. Equating these,

[latex]k\frac{Zq_{e}^2}{r_n^2}=\frac{m_{e}v^2}{r_n}\text{ (Coulomb = centripetal)}\\[/latex].

Angular momentum quantization is stated in an earlier equation. We solve that equation for v , substitute it into the above, and rearrange the expression to obtain the radius of the orbit. This yields:

[latex]\displaystyle{r}_{n}=\frac{n^2}{Z}a_{\text{B}},\text{ for allowed orbits }\left(n=1,2,3\dots\right)\\[/latex],

where a B is defined to be the Bohr radius , since for the lowest orbit ( n  = 1) and for hydrogen ( Z  = 1), r 1  =  a B . It is left for this chapter’s Problems and Exercises to show that the Bohr radius is

[latex]\displaystyle{a}_{\text{B}}=\frac{h^2}{4\pi^2m_{e}kq_{e}^{2}}=0.529\times10^{-10}\text{ m}\\[/latex].

These last two equations can be used to calculate the radii of the allowed (quantized) electron orbits in any hydrogen-like atom . It is impressive that the formula gives the correct size of hydrogen, which is measured experimentally to be very close to the Bohr radius. The earlier equation also tells us that the orbital radius is proportional to n 2 , as illustrated in Figure 6.

Figure 6. The allowed electron orbits in hydrogen have the radii shown. These radii were first calculated by Bohr and are given by the equation [latex]r_n=\frac{n^2}{Z}a_{\text{B}}\\[/latex]. The lowest orbit has the experimentally verified diameter of a hydrogen atom.

To get the electron orbital energies, we start by noting that the electron energy is the sum of its kinetic and potential energy:  E n   = KE + PE.

Kinetic energy is the familiar [latex]KE=\frac{1}{2}m_{e}v^2\\[/latex], assuming the electron is not moving at relativistic speeds. Potential energy for the electron is electrical, or PE =  q e V , where V is the potential due to the nucleus, which looks like a point charge. The nucleus has a positive charge Zq e  ; thus, [latex]V=\frac{kZq_e}{r_n}\\[/latex], recalling an earlier equation for the potential due to a point charge. Since the electron’s charge is negative, we see that [latex]PE=-\frac{kZq_e}{r_n}\\[/latex]. Entering the expressions for KE  and PE , we find

[latex]\displaystyle{E}_{n}=\frac{1}{2}m_{e}v^2-k\frac{Zq_{e}^{2}}{r_{n}}\\[/latex].

Now we substitute r n and v from earlier equations into the above expression for energy. Algebraic manipulation yields

[latex]\displaystyle{E}_{n}=-\frac{Z^2}{n^2}E_0\left(n=1,2,3,\dots\right)\\[/latex]

for the orbital energies of hydrogen-like atoms . Here, E 0 is the ground-state energy ( n  = 1) for hydrogen ( Z  = 1) and is given by

[latex]\displaystyle{E}_{0}=\frac{2\pi{q}_{e}^{4}m_{e}k^{2}}{h^2}=13.6\text{ eV}\\[/latex]

Thus, for hydrogen,

[latex]\displaystyle{E}_n=-\frac{13.6\text{ eV}}{n^2}\left(n=1,2,3\dots\right)\\[/latex]

Figure 7. Energy-level diagram for hydrogen showing the Lyman, Balmer, and Paschen series of transitions. The orbital energies are calculated using the above equation, first derived by Bohr.

Figure 7 shows an energy-level diagram for hydrogen that also illustrates how the various spectral series for hydrogen are related to transitions between energy levels.

Electron total energies are negative, since the electron is bound to the nucleus, analogous to being in a hole without enough kinetic energy to escape. As n approaches infinity, the total energy becomes zero. This corresponds to a free electron with no kinetic energy, since r n gets very large for large n , and the electric potential energy thus becomes zero. Thus, 13.6 eV is needed to ionize hydrogen (to go from –13.6 eV to 0, or unbound), an experimentally verified number. Given more energy, the electron becomes unbound with some kinetic energy. For example, giving 15.0 eV to an electron in the ground state of hydrogen strips it from the atom and leaves it with 1.4 eV of kinetic energy.

Finally, let us consider the energy of a photon emitted in a downward transition, given by the equation to be ∆ E = hf =  E i  − E f .

Substituting E n   = (–13.6 eV/ n 2 ), we see that

[latex]\displaystyle{hf}=\left(13.6\text{ eV}\right)\left(\frac{1}{n_{\text{f}}^2}-\frac{1}{n_{\text{i}}^2}\right)\\[/latex]

Dividing both sides of this equation by hc  gives an expression for [latex]\frac{1}{\lambda}\\[/latex]:

[latex]\displaystyle\frac{hf}{hc}=\frac{f}{c}=\frac{1}{\lambda}=\frac{\left(13.6\text{ eV}\right)}{hc}\left(\frac{1}{n_{\text{f}}^2}-\frac{1}{n_{\text{i}}^2}\right)\\[/latex]

It can be shown that

[latex]\displaystyle\left(\frac{13.6\text{ eV}}{hc}\right)=\frac{\left(13.6\text{ eV}\right)\left(1.602\times10^{-19}\text{ J/eV}\right)}{\left(6.626\times10^{-34}\text{ J }\cdot\text{ s}\right)\left(2.998\times10^{8}\text{ m/s}\right)}=1.097\times10^7\text{ m}^{-1}=R\\[/latex]

is the Rydberg constant . Thus, we have used Bohr’s assumptions to derive the formula first proposed by Balmer years earlier as a recipe to fit experimental data.

[latex]\displaystyle\frac{1}{\lambda}=R\left(\frac{1}{n_{\text{f}}^2}-\frac{1}{n_{\text{i}}^2}\right)\\[/latex]

We see that Bohr’s theory of the hydrogen atom answers the question as to why this previously known formula describes the hydrogen spectrum. It is because the energy levels are proportional to [latex]\frac{1}{n^2}\\[/latex], where n is a non-negative integer. A downward transition releases energy, and so n i must be greater than n f . The various series are those where the transitions end on a certain level. For the Lyman series, n f  = 1—that is, all the transitions end in the ground state (see also Figure 7). For the Balmer series, n f  = 2, or all the transitions end in the first excited state; and so on. What was once a recipe is now based in physics, and something new is emerging—angular momentum is quantized.

Triumphs and Limits of the Bohr Theory

Bohr did what no one had been able to do before. Not only did he explain the spectrum of hydrogen, he correctly calculated the size of the atom from basic physics. Some of his ideas are broadly applicable. Electron orbital energies are quantized in all atoms and molecules. Angular momentum is quantized. The electrons do not spiral into the nucleus, as expected classically (accelerated charges radiate, so that the electron orbits classically would decay quickly, and the electrons would sit on the nucleus—matter would collapse). These are major triumphs.

But there are limits to Bohr’s theory. It cannot be applied to multielectron atoms, even one as simple as a two-electron helium atom. Bohr’s model is what we call semiclassical . The orbits are quantized (nonclassical) but are assumed to be simple circular paths (classical). As quantum mechanics was developed, it became clear that there are no well-defined orbits; rather, there are clouds of probability. Bohr’s theory also did not explain that some spectral lines are doublets (split into two) when examined closely. We shall examine many of these aspects of quantum mechanics in more detail, but it should be kept in mind that Bohr did not fail. Rather, he made very important steps along the path to greater knowledge and laid the foundation for all of atomic physics that has since evolved.

PhET Explorations: Models of the Hydrogen Atom

How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental results.

Click to download the simulation. Run using Java.

Section Summary

• The planetary model of the atom pictures electrons orbiting the nucleus in the way that planets orbit the sun. Bohr used the planetary model to develop the first reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, with hydrogen spectrum wavelengths given by the formula [latex]\frac{1}{\lambda }=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right)\\[/latex], where λ  is the wavelength of the emitted EM radiation and R  is the Rydberg constant, which has the value  R  = 1.097 × 10 7 m −1 .
• The constants n i and n f are positive integers, and n i must be greater than n f .
• Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by ∆ E  = hf = E i − E f , where ∆ E is the change in energy between the initial and final orbits and hf  is the energy of an absorbed or emitted photon. It is useful to plot orbital energies on a vertical graph called an energy-level diagram.
• Bohr proposed that the allowed orbits are circular and must have quantized orbital angular momentum given by [latex]L={m}_{e}{\text{vr}}_{n}=n\frac{h}{2\pi }\left(n=1, 2, 3 \dots \right)\\[/latex], where L  is the angular momentum, r n  is the radius of the n th orbit, and h  is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is given by [latex]{r}_{n}=\frac{{n}^{2}}{Z}{a}_{\text{B}}\left(\text{allowed orbits }n=1, 2, 3, …\right)\\[/latex],  Z  is the atomic number of an element (the number of electrons is has when neutral) and a B is defined to be the Bohr radius, which is [latex]{a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\text{0.529}\times {\text{10}}^{-\text{10}}\text{ m}\\[/latex].
• Furthermore, the energies of hydrogen-like atoms are given by [latex]{E}_{n}=-\frac{{Z}^{2}}{{n}^{2}}{E}_{0}\left(n=1, 2, 3 …\right)\\[/latex], where E 0 is the ground-state energy and is given by [latex]{E}_{0}=\frac{{2\pi }^{2}{q}_{e}^{4}{m}_{e}{k}^{2}}{{h}^{2}}=\text{13.6 eV}\\[/latex]. Thus, for hydrogen, [latex]{E}_{n}=-\frac{\text{13.6 eV}}{{n}^{2}}\left(n,=,1, 2, 3 …\right)\\[/latex].
• The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.

Conceptual Questions

• How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.
• Explain how Bohr’s rule for the quantization of electron orbital angular momentum differs from the actual rule.
• What is a hydrogen-like atom, and how are the energies and radii of its electron orbits related to those in hydrogen?

Problems & Exercises

• By calculating its wavelength, show that the first line in the Lyman series is UV radiation.
• Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation.
• Look up the values of the quantities in [latex]{a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}\\[/latex] , and verify that the Bohr radius a B is 0.529 × 10 −10 m.
• Verify that the ground state energy E 0 is 13.6 eV by using [latex]{E}_{0}=\frac{{2\pi }^{2}{q}_{e}^{4}{m}_{e}{k}^{2}}{{h}^{2}}\\[/latex].
• If a hydrogen atom has its electron in the n  = 4 state, how much energy in eV is needed to ionize it?
• A hydrogen atom in an excited state can be ionized with less energy than when it is in its ground state. What is n  for a hydrogen atom if 0.850 eV of energy can ionize it?
• Find the radius of a hydrogen atom in the n  = 2 state according to Bohr’s theory.
• Show that [latex]\frac{\left(13.6 \text{eV}\right)}{hc}=1.097\times10^{7}\text{ m}=R\\[/latex] (Rydberg’s constant), as discussed in the text.
• What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?
• Show that the entire Paschen series is in the infrared part of the spectrum. To do this, you only need to calculate the shortest wavelength in the series.
• Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.
• (a) Which line in the Balmer series is the first one in the UV part of the spectrum? (b) How many Balmer series lines are in the visible part of the spectrum? (c) How many are in the UV?
• A wavelength of 4.653 µm is observed in a hydrogen spectrum for a transition that ends in the n f = 5 level. What was n i for the initial level of the electron?
• A singly ionized helium ion has only one electron and is denoted He + . What is the ion’s radius in the ground state compared to the Bohr radius of hydrogen atom?
• A beryllium ion with a single electron (denoted Be 3+ ) is in an excited state with radius the same as that of the ground state of hydrogen. (a) What is n  for the Be 3+ ion? (b) How much energy in eV is needed to ionize the ion from this excited state?
• Atoms can be ionized by thermal collisions, such as at the high temperatures found in the solar corona. One such ion is C +5 , a carbon atom with only a single electron. (a) By what factor are the energies of its hydrogen-like levels greater than those of hydrogen? (b) What is the wavelength of the first line in this ion’s Paschen series? (c) What type of EM radiation is this?
• Verify Equations [latex]{r}_{n}=\frac{{n}^{2}}{Z}{a}_{\text{B}}\\[/latex] and [latex]{a}_{B}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{kq}_{e}^{2}}=0.529\times{10}^{-10}\text{ m}\\[/latex] using the approach stated in the text. That is, equate the Coulomb and centripetal forces and then insert an expression for velocity from the condition for angular momentum quantization.
• The wavelength of the four Balmer series lines for hydrogen are found to be 410.3, 434.2, 486.3, and 656.5 nm. What average percentage difference is found between these wavelength numbers and those predicted by [latex]\frac{1}{\lambda}=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right)\\[/latex]? It is amazing how well a simple formula (disconnected originally from theory) could duplicate this phenomenon.

hydrogen spectrum wavelengths:  the wavelengths of visible light from hydrogen; can be calculated by

[latex]\displaystyle\frac{1}{\lambda }=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right)\\[/latex]

Rydberg constant:  a physical constant related to the atomic spectra with an established value of 1.097 × 10 7 m −1

double-slit interference:  an experiment in which waves or particles from a single source impinge upon two slits so that the resulting interference pattern may be observed

energy-level diagram:  a diagram used to analyze the energy level of electrons in the orbits of an atom

Bohr radius:  the mean radius of the orbit of an electron around the nucleus of a hydrogen atom in its ground state

hydrogen-like atom:  any atom with only a single electron

energies of hydrogen-like atoms:  Bohr formula for energies of electron states in hydrogen-like atoms: [latex]{E}_{n}=-\frac{{Z}^{2}}{{n}^{2}}{E}_{0}\left(n=\text{1, 2, 3,}\dots \right)\\[/latex]

Selected Solutions to Problems & Exercises

1. [latex]\displaystyle\frac{1}{\lambda}=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right)\Rightarrow \lambda =\frac{1}{R}\left[\frac{\left({n}_{\text{i}}\cdot{n}_{\text{f}}\right)^{2}}{{n}_{\text{i}}^{2}-{n}_{\text{f}}^{2}}\right];{n}_{\text{i}}=2,{n}_{\text{f}}=1\\[/latex], so that

[latex]\displaystyle\lambda =\left(\frac{m}{1.097\times {\text{10}}^{7}}\right)\left[\frac{\left(2\times1\right)^{2}}{{2}^{2}-{1}^{2}}\right]=1\text{.}\text{22}\times {\text{10}}^{-7}\text{m}=\text{122 nm}\\[/latex] , which is UV radiation.

3. [latex]\begin{array}{lll}{a}_{\text{B}}&=&\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kZq}}_{e}^{2}}\\\text{ }&=&\frac{\left(\text{6.626}\times {\text{10}}^{-\text{34}}\text{J }\cdot\text{ s}\right)^{2}}{{4\pi }^{2}\left(9.109\times {\text{10}}^{-\text{31}}\text{kg}\right)\left(8.988\times {\text{10}}^{9}\text{N}\cdot{\text{m}}^{2}/{C}^{2}\right)\left(1\right)\left(1.602\times {\text{10}}^{-\text{19}}\text{C}\right)^{2}}\\\text{ }&=&\text{0.529}\times {\text{10}}^{-\text{10}}\text{m}\end{array}\\[/latex]

5. 0.850 eV

7. 2.12 × 10 −10 m

9. 365 nm; it is in the ultraviolet.

11. No overlap; 365 nm; 122 nm

15. (a) 2; (b) 54.4 eV

17. [latex]\displaystyle\frac{{\text{kZq}}_{e}^{2}}{{r}_{n}^{2}}=\frac{{m}_{e}{V}^{2}}{{r}_{n}}\\[/latex], so that [latex]\displaystyle{r}_{n}=\frac{{\text{kZq}}_{e}^{2}}{{m}_{e}{V}^{2}}=\frac{{\text{kZq}}_{e}^{2}}{{m}_{e}}\frac{1}{{V}^{2}}\\[/latex]. From the equation [latex]\displaystyle{m}_{e}{vr}_{n}=n\frac{h}{2\pi}\\[/latex], we can substitute for the velocity, giving:

[latex]\displaystyle{r}_{n}=\frac{{\text{kZq}}_{e}^{2}}{{m}_{e}}\cdot \frac{{4\pi }^{2}{m}_{e}^{2}{r}_{n}^{2}}{{n}^{2}{h}^{2}}\\[/latex]

[latex]\displaystyle{r}_{n}=\frac{{n}^{2}}{Z}\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\frac{{n}^{2}}{Z}{a}_{\text{B}}\\[/latex],

[latex]\displaystyle{a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}\\[/latex].

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Bohr Model of the Atom Explained

Planetary Model of the Hydrogen Atom

ThoughtCo / Evan Polenghi

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The Bohr Model has an atom consisting of a small, positively charged nucleus orbited by negatively charged electrons. Here's a closer look at the Bohr Model, which is sometimes called the Rutherford-Bohr Model.

Overview of the Bohr Model

Niels Bohr proposed the Bohr Model of the Atom in 1915. Because the Bohr Model is a modification of the earlier Rutherford Model, some people call Bohr's Model the Rutherford-Bohr Model. The modern model of the atom is based on quantum mechanics. The Bohr Model contains some errors, but it is important because it describes most of the accepted features of atomic theory without all of the high-level math of the modern version. Unlike earlier models, the Bohr Model explains the Rydberg formula for the spectral emission lines of atomic hydrogen .

The Bohr Model is a planetary model in which the negatively charged electrons orbit a small, positively charged nucleus similar to the planets orbiting the sun (except that the orbits are not planar). The gravitational force of the solar system is mathematically akin to the Coulomb (electrical) force between the positively charged nucleus and the negatively charged electrons.

Main Points of the Bohr Model

• Electrons orbit the nucleus in orbits that have a set size and energy.
• The energy of the orbit is related to its size. The lowest energy is found in the smallest orbit.
• Radiation is absorbed or emitted when an electron moves from one orbit to another.

Bohr Model of Hydrogen

The simplest example of the Bohr Model is for the hydrogen atom (Z = 1) or for a hydrogen-like ion (Z > 1), in which a negatively charged electron orbits a small positively charged nucleus. Electromagnetic energy will be absorbed or emitted if an electron moves from one orbit to another. Only certain electron orbits are permitted. The radius of the possible orbits increases as n 2 , where n is the principal quantum number . The 3 → 2 transition produces the first line of the Balmer series . For hydrogen (Z = 1) this produces a photon having wavelength 656 nm (red light).

Bohr Model for Heavier Atoms

Heavier atoms contain more protons in the nucleus than the hydrogen atom. More electrons were required to cancel out the positive charge of all of these protons. Bohr believed each electron orbit could only hold a set number of electrons. Once the level was full, additional electrons would be bumped up to the next level. Thus, the Bohr model for heavier atoms described electron shells. The model explained some of the atomic properties of heavier atoms, which had never been reproduced before. For example, the shell model explained why atoms got smaller moving across a period (row) of the periodic table, even though they had more protons and electrons. It also explained why the noble gases were inert and why atoms on the left side of the periodic table attract electrons, while those on the right side lose them. However, the model assumed electrons in the shells didn't interact with each other and couldn't explain why electrons seemed to stack in an irregular manner.

Problems With the Bohr Model

• It violates the Heisenberg Uncertainty Principle because it considers electrons to have both a known radius and orbit.
• The Bohr Model provides an incorrect value for the ground state orbital angular momentum .
• It makes poor predictions regarding the spectra of larger atoms.
• It does not predict the relative intensities of spectral lines.
• The Bohr Model does not explain fine structure and hyperfine structure in spectral lines.
• It does not explain the Zeeman Effect.

Refinements and Improvements to the Bohr Model

The most prominent refinement to the Bohr model was the Sommerfeld model, which is sometimes called the Bohr-Sommerfeld model. In this model, electrons travel in elliptical orbits around the nucleus rather than in circular orbits. The Sommerfeld model was better at explaining atomic spectral effects, such the Stark effect in spectral line splitting. However, the model couldn't accommodate the magnetic quantum number.

Ultimately, the Bohr model and models based upon it were replaced Wolfgang Pauli's model based on quantum mechanics in 1925. That model was improved to produce the modern model, introduced by Erwin Schrodinger in 1926. Today, the behavior of the hydrogen atom is explained using wave mechanics to describe atomic orbitals.

• Lakhtakia, Akhlesh; Salpeter, Edwin E. (1996). "Models and Modelers of Hydrogen". American Journal of Physics . 65 (9): 933. Bibcode:1997AmJPh..65..933L. doi: 10.1119/1.18691
• Linus Carl Pauling (1970). "Chapter 5-1".  General Chemistry  (3rd ed.). San Francisco: W.H. Freeman & Co. ISBN 0-486-65622-5.
• Niels Bohr (1913). "On the Constitution of Atoms and Molecules, Part I" (PDF). Philosophical Magazine . 26 (151): 1–24. doi: 10.1080/14786441308634955
• Niels Bohr (1914). "The spectra of helium and hydrogen". Nature . 92 (2295): 231–232. doi:10.1038/092231d0
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Bohr's Hydrogen Atom

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Niels Bohr introduced the atomic Hydrogen model in 1913. He described it as a positively charged nucleus, comprised of protons and neutrons, surrounded by a negatively charged electron cloud. In the model, electrons orbit the nucleus in atomic shells. The atom is held together by electrostatic forces between the positive nucleus and negative surroundings.

Hydrogen Energy Levels

The Bohr model is used to describe the structure of hydrogen energy levels. The image below represents shell structure, where each shell is associated with principal quantum number n . The energy levels presented correspond with each shell. The amount of energy in each level is reported in eV, and the maxiumum energy is the ionization energy of 13.598eV.

F igure 1: Some of the orbital shells of a Hydrogen atom. The energy levels of the orbitals are shown to the right.

Hydrogen Spectrum

The movement of electrons between these energy levels produces a spectrum. The Balmer equation is used to describe the four different wavelengths of Hydrogen which are present in the visible light spectrum. These wavelengths are at 656, 486, 434, and 410nm. These correspond to the emission of photons as an electron in an excited state transitions down to energy level n=2. The Rydberg formula, below, generalizes the Balmer series for all energy level transitions. To get the Balmer lines, the Rydberg formula is used with an n f of 2.

Rydberg Formula

The Rydberg formula explains the different energies of transition that occur between energy levels. When an electron moves from a higher energy level to a lower one, a photon is emitted. The Hydrogen atom can emit different wavelengths of light depending on the initial and final energy levels of the transition. It emits a photon with energy equal to the difference of square of the final (\(n_f\)) and initial (\(n_i\)) energy levels.

\[\text{Energy}=R\left(\dfrac{1}{n^2_f}-\dfrac{1}{n^2_i}\right) \label{1}\]

The energy of a photon is equal to Planck’s constant, h=6.626*10 - 34 m 2 kg/s, times the speed of light in a vacuum, divided by the wavelength of emission.

\[E=\dfrac{hc}{\lambda} \label{2}\]

Combining these two equations produces the Rydberg Formula.

\[\dfrac{1}{\lambda}=R\left(\dfrac{1}{n^2_f}-\dfrac{1}{n^2_i}\right) \label{3}\]

The Rydberg Constant (R) = \(10,973,731.6\; m^{-1}\) or \(1.097 \times 10^7\; m^{-1}\).

Limitations of the Bohr Model

The Bohr Model was an important step in the development of atomic theory. However, it has several limitations.

• It is in violation of the Heisenberg Uncertainty Principle. The Bohr Model considers electrons to have both a known radius and orbit, which is impossible according to Heisenberg.
• The Bohr Model is very limited in terms of size. Poor spectral predictions are obtained when larger atoms are in question.
• It cannot predict the relative intensities of spectral lines.
• It does not explain the Zeeman Effect, when the spectral line is split into several components in the presence of a magnetic field.
• The Bohr Model does not account for the fact that accelerating electrons do not emit electromagnetic radiation.
• Bohr, Niels. "On the Constitution of Atoms and Molecules, Part I." Philosophical Magazine 26 (1913): 1-24. < http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf >
• Bohr, Niels. "On the Constitution of Atoms and Molecules, Part II." Philosophical Magazine 26 (1913): 476-502. < http://web.ihep.su/dbserv/compas/src/bohr13b/eng.pdf >
• Turner, J. E. Atoms, Radiation, and Radiation Protection . Weinheim: Wiley-VCH, 2007. Print.

1. An emission spectrum gives one of the lines in the Balmer series of the hydrogen atom at 410 nm. This wavelength results from a transition from an upper energy level to n=2. What is the principal quantum number of the upper level?

2. The Bohr model of the atom was able to explain the Balmer series because:

• larger orbits required electrons to have more negative energy in order to match the angular momentum.
• differences between the energy levels of the orbits matched the difference between energy levels of the line spectra.
• electrons were allowed to exist only in allowed orbits and nowhere else.
• none of the above

3. One reason the Bohr model of the atom failed was because it did not explain why

• accelerating electrons do not emit electromagnetic radiation.
• moving electrons have a greater mass.
• electrons in the orbits of an atom have negative energies.
• electrons in greater orbits of an atom have greater velocities.

1. (1/λ) = R*[ 1/(2 2 ) - 1/(n 2 ) ] , R=1.097x10 7 m -1 , λ=410nm

(1/410nm) = (1.097x10 7 m -1 ) * [ 1/(2 2 ) - 1/(n 2 ) ]

[ (1/4.10x10 -7 m) / (1.097x10 7 m -1 ) ] - [ (1/4) ] = [ -1/(n 2 ) ]

-1/-0.02778 = n 2

36 = n 2 , n=6 --> The emission resulted from a transition from energy level 6 to energy level 2.

2. (B) differences between the energy levels of the orbits matched the difference between energy levels of the line spectra.

3. (A) accelerating electrons do not emit electromagnetic radiation.

Contributors and Attributions

• Michelle Faust

Niels Bohr (1885-1962)

On the spectrum of hydrogen.

Hydrogen possesses not only the smallest atomic weight of all the elements, but it also occupies a peculiar position both with regard to its physical and its chemical properties. One of the points where this becomes particularly apparent is the hydrogen line spectrum.

The spectrum of hydrogen observed in an ordinary Geissler tube consists of a series of lines, the strongest of which lies at the red end of the spectrum, while the others extend out into the ultra-violet, the distance between the various lines, as well as their intensities, constantly decreasing. In the ultraviolet the series converges to a limit. ...

We shall now consider the second part of the foundation on which we shall build, namely, the conclusions arrived at from experiments with the rays emitted by radioactive substances. I have previously here in the Physical Society had the opportunity of speaking of the scattering of α rays in passing through thin plates, and to mention how Rutherford (1911) has proposed a theory for the structure of the atom in order to explain the remarkable and unexpected results of these experiments. I shall, therefore, only remind you that the characteristic feature of Rutherford's theory is the assumption of the existence of a positively charged nucleus inside the atom. A number of electrons are supposed to revolve in closed orbits around the nucleus, the number of these electrons being sufficient to neutralize the positive charge of the nucleus. The dimensions of the nucleus are supposed to be very small in comparison with the dimensions of the orbits of the electrons, and almost the entire mass of the atom is supposed to be concentrated in the nucleus. ...

Let us now assume that a hydrogen atom simply consists of an electron revolving around a nucleus of equal and opposite charge, and of a mass which is very large in comparison with that of the electron. It is evident that this assumption may explain the peculiar position already referred to which hydrogen occupies among the elements, but it appears at the outset completely hopeless to attempt to explain anything at all of the special properties of hydrogen, still less its line spectrum, on the basis of considerations relating to such a simple system.

Let us imagine for the sake of brevity that the mass of the nucleus is infinitely large in proportion to that of the electron, and that the velocity of the electron is very small in comparison with that of light. If we now temporarily disregard the energy radiation, which, according to the ordinary electrodynamics, will accompany the accelerated motion of the electron, the latter in accordance with Kepler's first law will describe an ellipse with the nucleus at one of the foci.

These expressions are extremely simple and they show that the magnitude of the frequency of revolution as well as the length of the major axis depend only on W, the work which must be added to the system in order to remove the electron to an infinite distance from the nucleus; and are independent of the eccentricity of the orbit. By varying W we may obtain all possible values for the frequency of revolution and the major axis of the ellipse. This condition shows, however, that it is not possible to employ Kepler's formula directly in calculating the orbit of the electron in a hydrogen atom.

For this it will be necessary to assume that the orbit of the electron cannot take on all values, and in any event the line spectrum clearly indicates that the oscillations of the electron cannot vary continuously between limits. The impossibility of making any progress with a simple system like the one considered here might have been foretold from a consideration of the dimensions involved.

It can be seen that it is impossible to employ Rutherford's atomic model so long as we confine ourselves exclusively to the ordinary electrodynamics. But this is nothing more than might have been expected. As I have mentioned, we may consider it to be an established fact that it is impossible to obtain a satisfactory explanation of the experiments on temperature radiation with the aid of electrodynamics, no matter what atomic model be employed. The fact that the deficiencies of the atomic model we are considering stand out so plainly is therefore perhaps no serious drawback; even though the defects of other atomic models are much better concealed they must nevertheless be present and will be just as serious.

Quantum theory of Spectra

In assuming Planck's theory we have manifestly acknowledged the inadequacy of the ordinary electrodynamics and have definitely parted with the coherent group of ideas on which the latter theory is based. In fact in taking such a step we cannot expect that all cases of disagreement between the theoretical conceptions hitherto employed and experiment will be removed by the use of Planck's assumption regarding the quantum of the energy momentarily present in an oscillating system. We stand here almost entirely on virgin ground, and upon introducing new assumptions we need only take care not to get into contradiction with experiment. Time will have to show to what extent this can be avoided; but the safest way is, of course, to make as few assumptions as possible.

With this in mind let us first examine the experiments on temperature radiation. The subject of direct observation is the distribution of radiant energy over oscillations of the various wave lengths. Even though we may assume that this energy comes from systems of oscillating particles, we know little or nothing about these systems. No one has ever seen a Planck's resonator, nor indeed even measured its frequency of oscillation; we can observe only the period of oscillation of the radiation which is emitted. It is therefore very convenient that it is possible to show that to obtain the laws of temperature radiation it is not necessary to make any assumptions about the systems which emit the radiation except that the amount of energy emitted each time shall be equal to h ν, where h is Planck's constant and ν is the frequency of the radiation.

During the emission of the radiation the system may be regarded as passing from one state to another; in order to introduce a name for these states we shall call them "stationary" states, simply indicating thereby that they form some kind of waiting places between which occurs the emission of the energy corresponding to the various spectral lines. ...

Under ordinary circumstances a hydrogen atom will probably exist only in the state corresponding to n = 1. For this state W will have its greatest value and, consequently, the atom will have emitted the largest amount of energy possible; this will therefore represent the most stable state of the atom from which the system cannot be transferred except by adding energy to it from without.

I shall not tire you any further with more details; I hope to return to these questions here in the Physical Society, and to show how, on the basis of the underlying ideas, it is possible to develop a theory for the structure of atoms and molecules.

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180 Bohr’s Theory of the Hydrogen Atom

[latexpage]

Learning Objectives

• Describe the mysteries of atomic spectra.
• Explain Bohr’s theory of the hydrogen atom.
• Explain Bohr’s planetary model of the atom.
• Illustrate energy state using the energy-level diagram.
• Describe the triumphs and limits of Bohr’s theory.

The great Danish physicist Niels Bohr (1885–1962) made immediate use of Rutherford’s planetary model of the atom. ( (Figure) ). Bohr became convinced of its validity and spent part of 1912 at Rutherford’s laboratory. In 1913, after returning to Copenhagen, he began publishing his theory of the simplest atom, hydrogen, based on the planetary model of the atom. For decades, many questions had been asked about atomic characteristics. From their sizes to their spectra, much was known about atoms, but little had been explained in terms of the laws of physics. Bohr’s theory explained the atomic spectrum of hydrogen and established new and broadly applicable principles in quantum mechanics.

Mysteries of Atomic Spectra

As noted in Quantization of Energy , the energies of some small systems are quantized. Atomic and molecular emission and absorption spectra have been known for over a century to be discrete (or quantized). (See (Figure) .) Maxwell and others had realized that there must be a connection between the spectrum of an atom and its structure, something like the resonant frequencies of musical instruments. But, in spite of years of efforts by many great minds, no one had a workable theory. (It was a running joke that any theory of atomic and molecular spectra could be destroyed by throwing a book of data at it, so complex were the spectra.) Following Einstein’s proposal of photons with quantized energies directly proportional to their wavelengths, it became even more evident that electrons in atoms can exist only in discrete orbits.

In some cases, it had been possible to devise formulas that described the emission spectra. As you might expect, the simplest atom—hydrogen, with its single electron—has a relatively simple spectrum. The hydrogen spectrum had been observed in the infrared (IR), visible, and ultraviolet (UV), and several series of spectral lines had been observed. (See (Figure) .) These series are named after early researchers who studied them in particular depth.

The observed hydrogen-spectrum wavelengths can be calculated using the following formula:

where \(\lambda \) is the wavelength of the emitted EM radiation and \(R\) is the Rydberg constant , determined by the experiment to be

The constant \({n}_{\text{f}}\) is a positive integer associated with a specific series. For the Lyman series, \({n}_{\text{f}}=1\); for the Balmer series, \({n}_{\text{f}}=2\); for the Paschen series, \({n}_{\text{f}}=3\); and so on. The Lyman series is entirely in the UV, while part of the Balmer series is visible with the remainder UV. The Paschen series and all the rest are entirely IR. There are apparently an unlimited number of series, although they lie progressively farther into the infrared and become difficult to observe as \({n}_{\text{f}}\) increases. The constant \({n}_{\text{i}}\) is a positive integer, but it must be greater than \({n}_{\text{f}}\). Thus, for the Balmer series, \({n}_{\text{f}}=2\) and \({n}_{\text{i}}=3, 4, 5, 6, …\text{}\). Note that \({n}_{\text{i}}\) can approach infinity. While the formula in the wavelengths equation was just a recipe designed to fit data and was not based on physical principles, it did imply a deeper meaning. Balmer first devised the formula for his series alone, and it was later found to describe all the other series by using different values of \({n}_{\text{f}}\). Bohr was the first to comprehend the deeper meaning. Again, we see the interplay between experiment and theory in physics. Experimentally, the spectra were well established, an equation was found to fit the experimental data, but the theoretical foundation was missing.

What is the distance between the slits of a grating that produces a first-order maximum for the second Balmer line at an angle of \(\text{15º}\)?

Strategy and Concept

For an Integrated Concept problem, we must first identify the physical principles involved. In this example, we need to know (a) the wavelength of light as well as (b) conditions for an interference maximum for the pattern from a double slit. Part (a) deals with a topic of the present chapter, while part (b) considers the wave interference material of Wave Optics .

Solution for (a)

Hydrogen spectrum wavelength . The Balmer series requires that \({n}_{\text{f}}=2\). The first line in the series is taken to be for \({n}_{\text{i}}=3\), and so the second would have \({n}_{\text{i}}=4\).

The calculation is a straightforward application of the wavelength equation. Entering the determined values for \({n}_{\text{f}}\) and \({n}_{\text{i}}\) yields

Inverting to find \(\lambda \) gives

Discussion for (a)

This is indeed the experimentally observed wavelength, corresponding to the second (blue-green) line in the Balmer series. More impressive is the fact that the same simple recipe predicts all of the hydrogen spectrum lines, including new ones observed in subsequent experiments. What is nature telling us?

Solution for (b)

Double-slit interference ( Wave Optics ). To obtain constructive interference for a double slit, the path length difference from two slits must be an integral multiple of the wavelength. This condition was expressed by the equation

where \(d\) is the distance between slits and \(\theta \) is the angle from the original direction of the beam. The number \(m\) is the order of the interference; \(m=1\) in this example. Solving for \(d\) and entering known values yields

Discussion for (b)

This number is similar to those used in the interference examples of Introduction to Quantum Physics (and is close to the spacing between slits in commonly used diffraction glasses).

Bohr’s Solution for Hydrogen

Bohr was able to derive the formula for the hydrogen spectrum using basic physics, the planetary model of the atom, and some very important new proposals. His first proposal is that only certain orbits are allowed: we say that the orbits of electrons in atoms are quantized . Each orbit has a different energy, and electrons can move to a higher orbit by absorbing energy and drop to a lower orbit by emitting energy. If the orbits are quantized, the amount of energy absorbed or emitted is also quantized, producing discrete spectra. Photon absorption and emission are among the primary methods of transferring energy into and out of atoms. The energies of the photons are quantized, and their energy is explained as being equal to the change in energy of the electron when it moves from one orbit to another. In equation form, this is

Here, \(\Delta E\) is the change in energy between the initial and final orbits, and \(\text{hf}\) is the energy of the absorbed or emitted photon. It is quite logical (that is, expected from our everyday experience) that energy is involved in changing orbits. A blast of energy is required for the space shuttle, for example, to climb to a higher orbit. What is not expected is that atomic orbits should be quantized. This is not observed for satellites or planets, which can have any orbit given the proper energy. (See (Figure) .)

(Figure) shows an energy-level diagram , a convenient way to display energy states. In the present discussion, we take these to be the allowed energy levels of the electron. Energy is plotted vertically with the lowest or ground state at the bottom and with excited states above. Given the energies of the lines in an atomic spectrum, it is possible (although sometimes very difficult) to determine the energy levels of an atom. Energy-level diagrams are used for many systems, including molecules and nuclei. A theory of the atom or any other system must predict its energies based on the physics of the system.

Bohr was clever enough to find a way to calculate the electron orbital energies in hydrogen. This was an important first step that has been improved upon, but it is well worth repeating here, because it does correctly describe many characteristics of hydrogen. Assuming circular orbits, Bohr proposed that the angular momentum \(L\) of an electron in its orbit is quantized , that is, it has only specific, discrete values. The value for \(L\) is given by the formula

where \(L\) is the angular momentum, \({m}_{e}\) is the electron’s mass, \({r}_{n}\) is the radius of the \(n\) th orbit, and \(h\) is Planck’s constant. Note that angular momentum is \(L=\mathrm{I\omega }\). For a small object at a radius \(r,\phantom{\rule{0.25em}{0ex}}I={\text{mr}}^{2}\) and \(\omega =v/r\), so that \(L=\left({\text{mr}}^{2}\right)\left(v/r\right)=\text{mvr}\). Quantization says that this value of \(\text{mvr}\) can only be equal to \(h/2,\phantom{\rule{0.25em}{0ex}}2h/2,\phantom{\rule{0.25em}{0ex}}3h/2\), etc. At the time, Bohr himself did not know why angular momentum should be quantized, but using this assumption he was able to calculate the energies in the hydrogen spectrum, something no one else had done at the time.

From Bohr’s assumptions, we will now derive a number of important properties of the hydrogen atom from the classical physics we have covered in the text. We start by noting the centripetal force causing the electron to follow a circular path is supplied by the Coulomb force. To be more general, we note that this analysis is valid for any single-electron atom. So, if a nucleus has \(Z\) protons (\(Z=1\) for hydrogen, 2 for helium, etc.) and only one electron, that atom is called a hydrogen-like atom . The spectra of hydrogen-like ions are similar to hydrogen, but shifted to higher energy by the greater attractive force between the electron and nucleus. The magnitude of the centripetal force is \({m}_{e}{v}^{2}/{r}_{n}\), while the Coulomb force is \(k\left({\text{Zq}}_{e}\right)\left({q}_{e}\right)/{r}_{n}^{2}\). The tacit assumption here is that the nucleus is more massive than the stationary electron, and the electron orbits about it. This is consistent with the planetary model of the atom. Equating these,

Angular momentum quantization is stated in an earlier equation. We solve that equation for \(v\), substitute it into the above, and rearrange the expression to obtain the radius of the orbit. This yields:

where \({a}_{\text{B}}\) is defined to be the Bohr radius , since for the lowest orbit \(\left(n=1\right)\) and for hydrogen \(\left(Z=1\right)\), \({r}_{1}={a}_{\text{B}}\). It is left for this chapter’s Problems and Exercises to show that the Bohr radius is

These last two equations can be used to calculate the radii of the allowed (quantized) electron orbits in any hydrogen-like atom . It is impressive that the formula gives the correct size of hydrogen, which is measured experimentally to be very close to the Bohr radius. The earlier equation also tells us that the orbital radius is proportional to \({n}^{2}\), as illustrated in (Figure) .

To get the electron orbital energies, we start by noting that the electron energy is the sum of its kinetic and potential energy:

Kinetic energy is the familiar \(\text{KE}=\left(1/2\right){m}_{e}{v}^{2}\), assuming the electron is not moving at relativistic speeds. Potential energy for the electron is electrical, or \(\text{PE}={q}_{e}V\), where \(V\) is the potential due to the nucleus, which looks like a point charge. The nucleus has a positive charge \({\text{Zq}}_{e}\) ; thus, \(V={\text{kZq}}_{e}/{r}_{n}\), recalling an earlier equation for the potential due to a point charge. Since the electron’s charge is negative, we see that \(\text{PE}=-{\text{kZq}}_{e}/{r}_{n}\) . Entering the expressions for \(\text{KE}\) and \(\text{PE}\), we find

Now we substitute \({r}_{n}\) and \(v\) from earlier equations into the above expression for energy. Algebraic manipulation yields

for the orbital energies of hydrogen-like atoms . Here, \({E}_{0}\) is the ground-state energy \(\left(n=1\right)\) for hydrogen \(\left(Z=1\right)\) and is given by

Thus, for hydrogen,

(Figure) shows an energy-level diagram for hydrogen that also illustrates how the various spectral series for hydrogen are related to transitions between energy levels.

Electron total energies are negative, since the electron is bound to the nucleus, analogous to being in a hole without enough kinetic energy to escape. As \(n\) approaches infinity, the total energy becomes zero. This corresponds to a free electron with no kinetic energy, since \({r}_{n}\) gets very large for large \(n\), and the electric potential energy thus becomes zero. Thus, 13.6 eV is needed to ionize hydrogen (to go from –13.6 eV to 0, or unbound), an experimentally verified number. Given more energy, the electron becomes unbound with some kinetic energy. For example, giving 15.0 eV to an electron in the ground state of hydrogen strips it from the atom and leaves it with 1.4 eV of kinetic energy.

Finally, let us consider the energy of a photon emitted in a downward transition, given by the equation to be

Substituting \({E}_{n}=\left(–\text{13.6 eV}/{n}^{2}\right)\), we see that

Dividing both sides of this equation by \(\text{hc}\) gives an expression for \(1/\lambda \):

It can be shown that

is the Rydberg constant . Thus, we have used Bohr’s assumptions to derive the formula first proposed by Balmer years earlier as a recipe to fit experimental data.

We see that Bohr’s theory of the hydrogen atom answers the question as to why this previously known formula describes the hydrogen spectrum. It is because the energy levels are proportional to \(1/{n}^{2}\), where \(n\) is a non-negative integer. A downward transition releases energy, and so \({n}_{i}\) must be greater than \({n}_{\text{f}}\). The various series are those where the transitions end on a certain level. For the Lyman series, \({n}_{\text{f}}=1\) — that is, all the transitions end in the ground state (see also (Figure) ). For the Balmer series, \({n}_{\text{f}}=2\), or all the transitions end in the first excited state; and so on. What was once a recipe is now based in physics, and something new is emerging—angular momentum is quantized.

Triumphs and Limits of the Bohr Theory

Bohr did what no one had been able to do before. Not only did he explain the spectrum of hydrogen, he correctly calculated the size of the atom from basic physics. Some of his ideas are broadly applicable. Electron orbital energies are quantized in all atoms and molecules. Angular momentum is quantized. The electrons do not spiral into the nucleus, as expected classically (accelerated charges radiate, so that the electron orbits classically would decay quickly, and the electrons would sit on the nucleus—matter would collapse). These are major triumphs.

But there are limits to Bohr’s theory. It cannot be applied to multielectron atoms, even one as simple as a two-electron helium atom. Bohr’s model is what we call semiclassical . The orbits are quantized (nonclassical) but are assumed to be simple circular paths (classical). As quantum mechanics was developed, it became clear that there are no well-defined orbits; rather, there are clouds of probability. Bohr’s theory also did not explain that some spectral lines are doublets (split into two) when examined closely. We shall examine many of these aspects of quantum mechanics in more detail, but it should be kept in mind that Bohr did not fail. Rather, he made very important steps along the path to greater knowledge and laid the foundation for all of atomic physics that has since evolved.

How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental results.

Section Summary

where \(\lambda \) is the wavelength of the emitted EM radiation and \(R\) is the Rydberg constant, which has the value

• The constants \({n}_{i}\) and \({n}_{f}\) are positive integers, and \({n}_{i}\) must be greater than \({n}_{f}\).

where \(L\) is the angular momentum, \({r}_{n}\) is the radius of the \(n\text{th}\) orbit, and \(h\) is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is given by

\(Z\) is the atomic number of an element (the number of electrons is has when neutral) and \({a}_{\text{B}}\) is defined to be the Bohr radius, which is

where \({E}_{0}\) is the ground-state energy and is given by

• The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.

Conceptual Questions

How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.

Explain how Bohr’s rule for the quantization of electron orbital angular momentum differs from the actual rule.

What is a hydrogen-like atom, and how are the energies and radii of its electron orbits related to those in hydrogen?

Problems & Exercises

By calculating its wavelength, show that the first line in the Lyman series is UV radiation.

\(\frac{1}{\lambda }=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right)⇒\lambda =\frac{1}{R}\left[\frac{\left({n}_{\text{i}}\cdot {n}_{\text{f}}{\right)}^{2}}{{n}_{\text{i}}^{2}-{n}_{\text{f}}^{2}}\right];\phantom{\rule{0.25em}{0ex}}{n}_{\text{i}}=2,\phantom{\rule{0.25em}{0ex}}{n}_{\text{f}}=1,\phantom{\rule{0.25em}{0ex}}\) so that

\(\lambda =\left(\frac{m}{1.097×{\text{10}}^{7}}\right)\left[\frac{\left(2×1{\right)}^{2}}{{2}^{2}-{1}^{2}}\right]=1\text{.}\text{22}×{\text{10}}^{-7}\phantom{\rule{0.25em}{0ex}}\text{m}=\text{122 nm}\) , which is UV radiation.

Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation.

Look up the values of the quantities in \({a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}{}^{}\), and verify that the Bohr radius \({a}_{\text{B}}\) is \(\text{0.529}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}\).

\({a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kZq}}_{e}^{2}}=\frac{\left(\text{6.626}×{\text{10}}^{-\text{34}}\phantom{\rule{0.25em}{0ex}}\text{J·s}{\right)}^{2}}{{4\pi }^{2}\left(9.109×{\text{10}}^{-\text{31}}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)\left(8.988×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{N}\text{·}{\text{m}}^{2}/{C}^{2}\right)\left(1\right)\left(1.602×{\text{10}}^{-\text{19}}\phantom{\rule{0.25em}{0ex}}\text{C}{\right)}^{2}}=\text{0.529}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}\)

Verify that the ground state energy \({E}_{0}\) is 13.6 eV by using \({E}_{0}=\frac{{2\pi }^{2}{q}_{e}^{4}{m}_{e}{k}^{2}}{{h}^{2}}\text{.}\)

If a hydrogen atom has its electron in the \(n=4\) state, how much energy in eV is needed to ionize it?

A hydrogen atom in an excited state can be ionized with less energy than when it is in its ground state. What is \(n\) for a hydrogen atom if 0.850 eV of energy can ionize it?

Find the radius of a hydrogen atom in the \(n=2\) state according to Bohr’s theory.

\(\text{2.12}×{\text{10}}^{\text{–10}}\phantom{\rule{0.25em}{0ex}}\text{m}\)

Show that \(\left(13.6 eV\right)/\text{hc}=\text{1.097}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{m}=R\) (Rydberg’s constant), as discussed in the text.

What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?

It is in the ultraviolet.

Show that the entire Paschen series is in the infrared part of the spectrum. To do this, you only need to calculate the shortest wavelength in the series.

Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.

(a) Which line in the Balmer series is the first one in the UV part of the spectrum?

(b) How many Balmer series lines are in the visible part of the spectrum?

(c) How many are in the UV?

A wavelength of \(4\text{.}\text{653 μm}\) is observed in a hydrogen spectrum for a transition that ends in the \({n}_{\text{f}}=5\) level. What was \({n}_{\text{i}}\) for the initial level of the electron?

A singly ionized helium ion has only one electron and is denoted \({\text{He}}^{+}\). What is the ion’s radius in the ground state compared to the Bohr radius of hydrogen atom?

A beryllium ion with a single electron (denoted \({\text{Be}}^{3+}\)) is in an excited state with radius the same as that of the ground state of hydrogen.

(a) What is \(n\) for the \({\text{Be}}^{3+}\) ion?

(b) How much energy in eV is needed to ionize the ion from this excited state?

(b) 54.4 eV

Atoms can be ionized by thermal collisions, such as at the high temperatures found in the solar corona. One such ion is \({C}^{+5}\), a carbon atom with only a single electron.

(a) By what factor are the energies of its hydrogen-like levels greater than those of hydrogen?

(b) What is the wavelength of the first line in this ion’s Paschen series?

(c) What type of EM radiation is this?

Verify Equations \({r}_{n}=\frac{{n}^{2}}{Z}{a}_{\text{B}}\) and \({a}_{B}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\text{0.529}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}\) using the approach stated in the text. That is, equate the Coulomb and centripetal forces and then insert an expression for velocity from the condition for angular momentum quantization.

\(\frac{{\text{kZq}}_{e}^{2}}{{r}_{n}^{2}}=\frac{{m}_{e}{V}^{2}}{{r}_{n}}\text{,}\) so that \({r}_{n}=\frac{{\text{kZq}}_{e}^{2}}{{m}_{e}{V}^{2}}=\frac{{\text{kZq}}_{e}^{2}}{{m}_{e}}\frac{1}{{V}^{2}}\text{.}\) From the equation \({m}_{e}{\text{vr}}_{n}=n\frac{h}{2\pi }\text{,}\) we can substitute for the velocity, giving: \({r}_{n}=\frac{{\text{kZq}}_{e}^{2}}{{m}_{e}}\cdot \frac{{4\pi }^{2}{m}_{e}^{2}{r}_{n}^{2}}{{n}^{2}{h}^{2}}\) so that \({r}_{n}=\frac{{n}^{2}}{Z}\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\frac{{n}^{2}}{Z}{a}_{\text{B}},\) where \({a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}\).

The wavelength of the four Balmer series lines for hydrogen are found to be 410.3, 434.2, 486.3, and 656.5 nm. What average percentage difference is found between these wavelength numbers and those predicted by \(\frac{1}{\lambda }=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right)\)? It is amazing how well a simple formula (disconnected originally from theory) could duplicate this phenomenon.

Intro to Physics for Non-Majors Copyright © 2012 by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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In 1913, the Danish physicist Niels Bohr (1885 - 1962) managed to explain the spectrum of atomic hydrogen by an extension of Rutherford's description of the atom. In that model, the negatively charged electrons revolve about the positively charged atomic nucleus because of the attractive electrostatic force according to Coulomb's law. But the electron can be taken not only as a particle, but also as a de Broglie wave (wave of matter) which interferes with itself. The orbit is only stable, if it meets the condition for a standing wave: The circumference must be an integer multiple of the wavelength. The consequence is that only special values of radius and energy are allowed. The mathematical appendix explains how to calculate these values. According to classical electrodynamics, a charge, which is subject to centripetal acceleration on a circular orbit, should continuously radiate electromagnetic waves. Thus, because of the loss of energy, the electron should spiral into the nucleus very soon. By contast, an electron in Bohr's model emits no energy, as long as its energy has one of the above-mentioned values. However, an electron which is not in the lowest energy level (n = 1), can make a spontaneous change to a lower state and thereby emit the energy difference in the form of a photon (particle of light). By calculating the wavelengths of the corresponding electromagnetic waves, one will get the same results as by measuring the lines of the hydrogen spectrum. You must not take the idea of electrons, orbiting around the atomic nucleus, for reality. Bohr's model of the hydrogen atom was only an intermediate step on the way to a precise theory of the atomic structure, which was made possible by quantum mechanics and quantum electrodynamics. This applet illustrates a hydrogen atom according to particle or wave model. You can choose a principal quantum number n. The right part of the graphics represents the energy levels of the atom. Right down at the bottom you can read off the orbital radius r and the total energy E. If you try to vary the orbit's radius with pressed mouse button, this will generally lead to a non-stationary state. You can realize that by using the option "Wave model": The green wavy line which symbolizes the de Broglie wave will not be closed in most cases. Only if the circle's circumference is an integer multiple of the wavelength (blue), you will get a stationary state. URL: http://home.a-city.de/walter.fendt/phe/bohrh.htm © Walter Fendt, May 30, 1999 Last modification: December 2, 2001 Nuclear Physics Bohr Model Of The Hydrogen Atom Bohr Model of the Hydrogen Atom Bohr model of the hydrogen atom was the first atomic model to successfully explain the radiation spectra of atomic hydrogen. Niels Bohr introduced the atomic Hydrogen model in the year 1913. Bohr’s Model of the hydrogen atom attempts to plug in certain gaps as suggested by Rutherford’s model. It holds a special place in history as it gave rise to quantum mechanics by introducing the quantum theory. Planetary Model of the Atom Quantum mechanics emerged in the mid-1920s. Neil Bohr, one of the founders of quantum mechanics , was interested in the much-debated topic of the time – the structure of the atom. Numerous atomic models, including the theory postulated by J.J Thompson and the discovery of the nucleus by Ernest Rutherford, had emerged. But Bohr supported the planetary model, which asserted that electrons revolved around a positively charged nucleus just like the planets around the sun. Planetary Model of The Atom Nevertheless, scientists still had many unanswered questions such as Why didn’t the electrons drop into the nucleus as foretold by classical physics? Where are the electrons and what do they do there? How is the discrete emission lines produced by excited elements correlated to the internal structure of the atom? Bohr addressed all these questions using a seemingly simple assumption: What if electron orbits and energies, could exhibit only specific values?  You can check  Atomic Theory to learn about the various atomic theory put forward by scientists in the early 20 th century. You may also want to check out these topics given below! Energy level Bohr’s Theory of Hydrogen Atoms Bohr’s Equation Bohr Model of the hydrogen atom first proposed the planetary model, but later an assumption concerning the electrons was made. The assumption was the quantization of the structure of atoms. Bohr’s proposed that electrons orbited the nucleus in specific orbits or shells with a fixed radius. Only those shells with a radius provided by the equation below were allowed, and it was impossible for electrons to exist between these shells. Mathematically, the allowed value of the atomic radius is given by the equation: n is a positive integer r(1) is the smallest allowed radius for the hydrogen atom also known as the Bohr’s radius The Bohr’s radius has a value of: \(\begin{array}{l}r(1)=0.529\times 10^{-10}\,m\end{array} \) . Bohr calculated the energy of an electron in the nth level of hydrogen by considering the electrons in circular, quantized orbits as: 13.6 eV is the lowest possible energy of a hydrogen electron E(1). The energy obtained is always a negative number and the ground state n = 1, has the most negative value. The reason being that the energy of an electron in orbit is relative to the energy of an electron that is entirely separated from its nucleus, \(\begin{array}{l}n=\infty\end{array} \) and it is recognised to have an energy of 0 eV. Since the electron in a fixed orbit around the nucleus is more stable than an electron that is extremely far from its nucleus, the energy of the electron in orbit is always negative. Absorption and Emission According to Bohr’s model, an electron would absorb energy in the form of photons to get excited to a higher energy level . After escaping to the higher energy level, also known as the excited state, the excited electron is less stable, and therefore, would rapidly emit a photon to come back to a lower, more stable energy level. The energy of the emitted photon is equal to the difference in energy between the two energy levels for a specific transition. The energy can be calculated using the equation Atomic Excitation and De-excitation Limitations of the Bohr Model of the Hydrogen Atom: Bohr’s model doesn’t work well for complex atoms. It couldn’t explain why some spectral lines are more intense than others. It could not explain why some spectral lines split into multiple lines in the presence of a magnetic field. Heisenberg’s uncertainty principle contradicts Bohr’s idea of electrons existing in specific orbits with a known radius and velocity. Although the modern quantum mechanical model and the Bohr Model of the Hydrogen Atom may seem vastly different, the fundamental idea is the same in both. Classical physics isn’t sufficient to describe all the phenomena that occur on an atomic level. But, Bohr was the first to realise the quantization of electronic shells by fusing the idea of quantization into the electronic structure of the hydrogen atom and was successfully able to explain the emission spectra of hydrogen as well as other one-electron systems. Bohr proposed that electrons travel in specific orbits, shells around the nucleus. According to Bohr’s calculation, the energy for an electron in the shell is given by the expression: The hydrogen spectrum is explained in terms of electrons absorbing and emitting photons to change energy levels, where the photon energy is: Bohr’s Model of the Hydrogen Atom isn’t applicable for systems with more than one electron. If you wish to learn more Physics-concepts with the help of interactive video lessons, download BYJU’S – The Learning App. Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin! Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz Visit BYJU’S for all Physics related queries and study materials Your result is as below Request OTP on Voice Call PHYSICS Related Links Leave a Comment Cancel reply Your Mobile number and Email id will not be published. Required fields are marked * Post My Comment Register with BYJU'S & Download Free PDFs Register with byju's & watch live videos. How Two Rebel Physicists Changed Quantum Theory David Bohm and Hugh Everett were once ostracized for challenging the dominant thinking in physics. Now, science accepts their ideas, which are said to enrich our understanding of the universe. The field of quantum mechanics dates to 1900, the year German scientist Max Planck (1858–1947) discovered that energy could come in discrete packages called quanta. It advanced in 1913, when Danish physicist Niels Bohr (1885–1962) used quantum principles to explain what had until then been inexplicable, the exact wavelengths of light emitted or absorbed by a gas of hydrogen atoms. And since the 1920s, when Werner Heisenberg (1901–1976) and Erwin Schrödinger (1887–1961) built new quantum theories, quantum mechanics has consistently proven its value as the fundamental theory of the nanoscale and as a source of technology, from computer chips and lasers to LED bulbs and solar panels. One question, however, still puzzles: how does the quantum world relate to the more familiar human-scale one? For a century, the Copenhagen interpretation , chiefly developed by Bohr and Heisenberg in that city, has been the standard answer taught in physics courses. It posits that the quantum scale is indeterminate; that is, operates according to the laws of probability. This world is utterly different from the deterministic and predictable “classical” human scale, yet the Copenhagen interpretation doesn’t clearly explain how reality changes between the two worlds. Heisenberg and Bohr developed the Copenhagen interpretation amidst the blossoming of new quantum theories in the first half of the twentieth century. In 1927, Heisenberg announced his important uncertainty principle : at the quantum level, certain pairs of quantities, such as momentum and position, cannot be simultaneously measured to any desired degree of precision. The more exactly you measure one, the less well you know the other. Thus, we can never fully know the quantum world, a key feature of the Copenhagen interpretation. Indeterminism also appears in the Schrödinger wave equation at the heart of the Copenhagen view. Einstein had shown that light waves can act like swarms of particles, later called photons; in 1924, Louis de Broglie assumed the inverse, that tiny particles are also wave-like. In 1926, Schrödinger published his equation for these “matter waves.” Its solution, the “wave function” denoted by the Greek letter Ψ (psi), contains all possible information about a quantum entity such as an electron in an atom. But the information is indeterminate: Ψ is only a list of probable values for all the different physical properties, such as position or momentum, that the electron could have in its particular surroundings. The electron is said be in a superposition , simultaneously present in all its potential states of actual being. This superposition exists until an observer measures the properties of the electron, which makes its wave function “collapse”; the cloud of possible outcomes yields just one result, a definite value emerging into the classical world. It is as if, asked to pick a card out of a deck, the instant you select the three of hearts, the other fifty-one cards fade away. In this case, we know that the rejected cards still physically exist with definite properties, but in the Copenhagen view, subatomic particles aren’t real until they’re observed. Another problem is that the notion of a sudden wave function collapse seems an arbitrary addition to the Copenhagen interpretation; it contradicts the smooth evolution in time built into the Schrödinger equation. These troubling features, called “the measurement problem,” were hotly debated in the 1920s. But overwhelming any objections was the fact that the Copenhagen interpretation works! Its results agree precisely with experiments, the final test of any theory, and inspire real devices. Even so, David Joseph Bohm (1917–1992) and Hugh Everett III (1930–1982) sought equally valid theories without any incongruities. In the 1950s, these two American physicists dared to challenge the conventional Copenhagen interpretation with their “pilot wave” and “many-worlds” theories, respectively. Though from different backgrounds, Bohm and Everett shared characteristics that helped them seek answers: mathematical aptitude, necessary to manipulate quantum theory; and unconventional career paths, which separated them from the orthodoxy of academic physics. Bohm was a second-generation American, born into an immigrant family from Europe that operated a furniture store in Wilkes-Barre, Pennsylvania. In high school, where his physics instructor described him as “outstanding” and “brilliant,” Bohm developed his own alternative ideas about Bohr’s hydrogen atom. After undergraduate work at Penn State, he began earning a PhD in nuclear physics in 1941 under J. Robert Oppenheimer (1904–1967) at the University of California, Berkeley. The United States was engaged in World War II at the time and was about to build an atomic bomb. Bohm’s doctoral research was classified, and he was awarded his degree in 1943 without writing a dissertation. Though Oppenheimer wanted Bohm to work with him at Los Alamos, Bohm couldn’t get security clearance as he had briefly been, in the early 1940s, a member of the Communist Party. In 1947, supported by theorist John Wheeler, Bohm became an assistant professor at Princeton. There he taught quantum mechanics and wrote Quantum Theory (1951), in which he presented the Copenhagen interpretation, only to disavow it the next year, when he published his alternative theory in a pair of papers in the Physical Review (in 1957, he expounded his ideas further in his book Causality and Chance in Modern Physics ). But in 1951, his life had taken a serious turn. In that Cold War era of McCarthyism, Bohm was brought before the House Committee on Un-American Activities (HUAC). He pleaded the Fifth Amendment against self-incrimination, and he was first indicted and jailed for contempt of Congress and then acquitted when the Supreme Court decriminalized this action. Still, the damage was done. Princeton didn’t renew Bohm’s contract and banned him from campus in June 1951. Unable to obtain a new academic position in the US, he began a life-long exile, taking temporary teaching positions in Brazil and elsewhere. Finally, in 1961, he accepted the offer of a chaired professorship in physics at Birkbeck College, London. He remained in that position until he retired in 1983, continuing to develop his new approach, the “pilot wave” theory. When de Broglie postulated that tiny particles are also wave-like, he proposed the role of the waves as guiding or piloting the motions of real physical particles. Bohm fleshed out this insight by relating the pilot wave to Schrödinger’s wave function Ψ. In Bohm’s view, Ψ doesn’t collapse, but shepherds real subatomic particles into specific trajectories. This scenario yields the same results as the Schrödinger equation and resolves a great wave-particle quantum paradox. In the famous double-slit experiment , a stream of electrons or photons sent through two slits produces a pattern that could arise only from interfering waves, not particles. Bohm’s solution is that each particle traversing one of the slits rides a wave that pilots it into a complex path and generates an interference pattern from the swarm of particles. For his part, Everett solved the measurement problem differently, as described by biographer Peter Byrne in an article , and later, a book . Born in Washington, DC, Everett showed an early interest in logical contradictions. At age twelve, he wrote to Einstein about the paradox “irresistible force meets immovable body,” and, as Everett reports, Einstein replied that there is no such paradox, but he noted Everett’s drive in attacking the problem. Everett graduated with honors from Catholic University as an engineer with strong backgrounds in math, operations research, and physics. In 1953, Everett went to Princeton for graduate work. There he met Bohr, whose visit at the nearby Institute for Advanced Study sparked discussions about quantum mechanics. Later Everett said that the idea for his new theory came during a sherry-fueled session with one of Bohr’s assistants, among others. Everett was soon working out the consequences of his idea in a dissertation under John Wheeler, who had mentored Bohm and also Nobel Laureate Richard Feynman (1918–1988), and who called Everett “highly original.” In the Copenhagen view, quantum reality as determined by the Schrödinger equation is separate from classical reality. Everett boldly asserted instead that the Schrödinger equation applies to everything, small or big, object or observer. The resulting universal wave function describes a reality without a boundary between microscopic and macroscopic or any need for the wave function to collapse. In his scheme, the measurement problem doesn’t exist. This, however, comes at the cost of accepting a highly complex universe. If large objects and their observers obey the Schrödinger equation, then the universal wave function includes all observers and objects and their links in superposition. As Byrne explains: if the object could exist at either point A or B, in one branch of the universal wave function the observer sees the measurement result as “A,” and in another branch, a nearly identical person sees the result as “B.” (Everett called different elements of the superposition “branches.”) Further, without the jarring disruption of wave function collapse, the Schrödinger equation tells us that the branches go smoothly forward in time and do not interact, so each observer separately sees a normally unfolding macroscopic world. In layman’s terms, this means that the universe, instead of being a unity that encompasses all reality, is filled with separate multiverses or bubbles of reality, each believed to be the entire universe by its inhabitants. The observer who saw result “A” now lives in that reality, and the person who saw “B” occupies a separately evolving reality according to their different outcome. Each of these unimaginable numbers of bubbles moves ahead into its own future, forming a totality filled with what have come to be called “parallel worlds.” Bohr and his group scorned this grandiose idea as an answer to the measurement problem, one of his circle calling it “theology,” and another deriding Everett as “ stupid .” Wheeler had Everett rewrite his dissertation so it didn’t directly criticize the Copenhagen interpretation or its proponents. His thesis was published in 1957 and, according to Byrne, “slipped into instant obscurity.” All this should have been no surprise. As Olival Freire Jr. points out , Bohm’s earlier work—which Everett cited—had also been badly received, even with hostility, in a community dominated by Bohr and champions of the Copenhagen interpretation. That was to change for both theories. In 1964, a bombshell result from theorist John Bell showed how to experimentally confirm the exceedingly strange quantum effect of entanglement, which Einstein called “ spooky action at a distance ”: the fact that two quantum entities can affect each other over arbitrary distances. Bell, it turns out, was strongly influenced by Bohm’s work, notes Freire. This shows that rethinking the foundations of quantum mechanics, downplayed by some physicists as only a philosophical exercise, can pay off in deep theoretical insights as well as in technology; entanglement today is used in quantum computation, communication, and cryptography. Everett’s ideas too came to be more appreciated after The Many-Worlds Interpretation of Quantum Mechanics (Bryce DeWitt and Neill Graham, editors) was published in 1973. It included Everett’s original dissertation and related papers. This and DeWitt’s evocative phrase “many-worlds interpretation” brought new interest in Everett’s work and linked it to multiverse theory , which has been developed to solve certain problems in cosmology and as an outcome of string theory. Everett won further recognition—this time in popular culture—in 1976, when his work appeared in Analog , a leading science fiction magazine. (In fact, multiverses and parallel worlds have become staples of popular culture, as the film Everything Everywhere All at Once (2022) and the streaming series Dark Matter (2024), based on the novel by Blake Crouch, make clear.) By 2023, Bohm’s and Everett’s seminal papers had each amassed tens of thousands of citations in the scientific literature. Surveys have also asked hundreds of physicists which interpretation of quantum mechanics they consider best. Many chose the Copenhagen view, but an equal number favor either the pilot wave or many-worlds interpretation. It’s striking that what in the 1950s were outlaw ideas, met with disbelief and antagonism, today have a significant degree of acceptance and have greatly expanded our view of the quantum world and the universe. That Bohm and Everett could produce novel theories reflects their special circumstances and their times as well as their abilities. McCarthyism interrupted Bohm’s career but also freed him from conventional views of quantum mechanics. Historian of science Christian Forstner cites a 1981 interview in which Bohm acknowledges the upside to his departure from Princeton. It “liberated me,” Bohm admitted. “I was able to think more easily and more freely…without having to talk the language of other people.” Forstner notes that in exile, the physicist had the freedom to choose like-minded colleagues so that “the US-community and its thought-style lost importance for Bohm.” Indeed, Bohm’s exile was highly productive. Weekly Newsletter Get your fix of JSTOR Daily’s best stories in your inbox each Thursday. Privacy Policy   Contact Us You may unsubscribe at any time by clicking on the provided link on any marketing message. Everett’s expertise in operations research brought him the offer of a position with the Pentagon’s Weapons Systems Evaluation Group (WSEG) to analyze nuclear warfare after finishing his PhD. Wheeler wanted him to continue at Princeton but also knew, Byrne writes, that the lack of recognition for Everett’s ideas had left him “disappointed, perhaps bitter.” Nor did Everett enjoy truncating his thesis to mollify Bohr, and he must have realized that advocating an unpopular theory would cloud his academic career. In the end, he chose WSEG and never again worked in theoretical physics, but perhaps having this alternate possibility stiffened his resolve in presenting and defending his audacious idea. His talents shone at WSEG, but, according to Byrne, he was an alcoholic and died of a heart attack at age fifty-one. The co-existing Copenhagen, Bohm, and Everett interpretations give the same results for many different tests of quantum behavior; and so we await the subtle experiment that distinguishes among them, showing which one is physically true and might give philosophers new insight into the nature of reality. Bohm’s and Everett’s sagas provide another valuable lesson. Science prides itself on being self-correcting; wrong theories are eventually made right, as in the old notion of a geocentric universe giving way to the modern view. The Copenhagen interpretation became unquestioned orthodoxy, but Bohm and Everett challenged it even at personal cost. That reflects the highest aspirations of science and deserves to be recognized in 2025, the upcoming International Year of Quantum Science and Technology. Support JSTOR Daily! Join our membership program on Patreon today. JSTOR is a digital library for scholars, researchers, and students. JSTOR Daily readers can access the original research behind our articles for free on JSTOR. Get Our Newsletter More stories. Growing Quickly Helped the Earliest Dinosaurs The Huts of the Appalachian Trail Saturn’s Ocean Moon Enceladus Is Able to Support Life Aurorae and the Green of the Night Sky Recent posts. The Long Shadow of the Jolly Bachelors The Psychological Problems of Modern Warfare Reading for Juneteenth Elephants, Decadence, and LGBTQ Records Support JSTOR Daily Sign up for our weekly newsletter. You are using an outdated browser. Please upgrade your browser to improve your experience and security. Enhanced Page Navigation Niels Bohr - Biographical Biographical N iels Henrik David Bohr was born in Copenhagen on October 7, 1885, as the son of Christian Bohr, Professor of Physiology at Copenhagen University, and his wife Ellen, née Adler. Niels, together with his younger brother Harald (the future Professor in Mathematics), grew up in an atmosphere most favourable to the development of his genius – his father was an eminent physiologist and was largely responsible for awakening his interest in physics while still at school, his mother came from a family distinguished in the field of education. After matriculation at the Gammelholm Grammar School in 1903, he entered Copenhagen University where he came under the guidance of Professor C. Christiansen, a profoundly original and highly endowed physicist, and took his Master’s degree in Physics in 1909 and his Doctor’s degree in 1911. While still a student, the announcement by the Academy of Sciences in Copenhagen of a prize to be awarded for the solution of a certain scientific problem, caused him to take up an experimental and theoretical investigation of the surface tension by means of oscillating fluid jets. This work, which he carried out in his father’s laboratory and for which he received the prize offered (a gold medal), was published in the Transactions of the Royal Society , 1908. Bohr’s subsequent studies, however, became more and more theoretical in character, his doctor’s disputation being a purely theoretical piece of work on the explanation of the properties of the metals with the aid of the electron theory, which remains to this day a classic on the subject. It was in this work that Bohr was first confronted with the implications of Planck ‘s quantum theory of radiation. In the autumn of 1911 he made a stay at Cambridge, where he profited by following the experimental work going on in the Cavendish Laboratory under Sir J.J. Thomson’s guidance, at the same time as he pursued own theoretical studies. In the spring of 1912 he was at work in Professor Rutherford’s laboratory in Manchester, where just in those years such an intensive scientific life and activity prevailed as a consequence of that investigator’s fundamental inquiries into the radioactive phenomena. Having there carried out a theoretical piece of work on the absorption of alpha rays which was published in the Philosophical Magazine , 1913, he passed on to a study of the structure of atoms on the basis of Rutherford’s discovery of the atomic nucleus. By introducing conceptions borrowed from the Quantum Theory as established by Planck, which had gradually come to occupy a prominent position in the science of theoretical physics, he succeeded in working out and presenting a picture of atomic structure that, with later improvements (mainly as a result of Heisenberg ‘s ideas in 1925), still fitly serves as an elucidation of the physical and chemical properties of the elements. In 1913-1914 Bohr held a Lectureship in Physics at Copenhagen University and in 1914-1916 a similar appointment at the Victoria University in Manchester. In 1916 he was appointed Professor of Theoretical Physics at Copenhagen University, and since 1920 (until his death in 1962) he was at the head of the Institute for Theoretical Physics, established for him at that university. Recognition of his work on the structure of atoms came with the award of the Nobel Prize for 1922. Bohr’s activities in his Institute were since 1930 more and more directed to research on the constitution of the atomic nuclei, and of their transmutations and disintegrations. In 1936 he pointed out that in nuclear processes the smallness of the region in which interactions take place, as well as the strength of these interactions, justify the transition processes to be described more in a classical way than in the case of atoms (Cf.  »Neutron capture and nuclear constitution«, Nature , 137 (1936) 344). A liquid drop would, according to this view, give a very good picture of the nucleus. This so-called liquid droplet theory permitted the understanding of the mechanism of nuclear fission, when the splitting of uranium was discovered by Hahn and Strassmann, in 1939, and formed the basis of important theoretical studies in this field (among others, by Frisch and Meitner). Bohr also contributed to the clarification of the problems encountered in quantum physics, in particular by developing the concept of complementarity . Hereby he could show how deeply the changes in the field of physics have affected fundamental features of our scientific outlook and how the consequences of this change of attitude reach far beyond the scope of atomic physics and touch upon all domains of human knowledge. These views are discussed in a number of essays, written during the years 1933-1962. They are available in English, collected in two volumes with the title Atomic Physics and Human Knowledge and Essays 1958-1962 on Atomic Physics and Human Knowledge , edited by John Wiley and Sons, New York and London, in 1958 and 1963, respectively. Among Professor Bohr’s numerous writings (some 115 publications), three appearing as books in the English language may be mentioned here as embodying his principal thoughts: The Theory of Spectra and Atomic Constitution , University Press, Cambridge, 1922/2nd. ed., 1924; Atomic Theory and the Description of Nature , University Press, Cambridge, 1934/reprint 1961; The Unity of Knowledge , Doubleday & Co., New York, 1955. During the Nazi occupation of Denmark in World War II, Bohr escaped to Sweden and spent the last two years of the war in England and America, where he became associated with the Atomic Energy Project. In his later years, he devoted his work to the peaceful application of atomic physics and to political problems arising from the development of atomic weapons. In particular, he advocated a development towards full openness between nations. His views are especially set forth in his Open Letter to the United Nations , June 9, 1950. Until the end, Bohr’s mind remained alert as ever; during the last few years of his life he had shown keen interest in the new developments of molecular biology. The latest formulation of his thoughts on the problem of Life appeared in his final (unfinished) article, published after his death: “Licht und Leben-noch einmal”, Naturwiss ., 50 (1963) 72: (in English: “Light and Life revisited”, ICSU Rev ., 5 ( 1963) 194). Niels Bohr was President of the Royal Danish Academy of Sciences, of the Danish Cancer Committee, and Chairman of the Danish Atomic Energy Commission. He was a Foreign Member of the Royal Society (London), the Royal Institution, and Academies in Amsterdam, Berlin, Bologna, Boston, Göttingen, Helsingfors, Budapest, München, Oslo, Paris, Rome, Stockholm , Uppsala, Vienna, Washington, Harlem, Moscow, Trondhjem, Halle, Dublin, Liege, and Cracow. He was Doctor, honoris causa , of the following universities, colleges, and institutes: (1923-1939) – Cambridge, Liverpool, Manchester, Oxford, Copenhagen, Edinburgh, Kiel, Providence, California, Oslo, Birmingham, London; (1945-1962) – Sorbonne (Paris), Princeton, Mc. Gill (Montreal), Glasgow, Aberdeen, Athens, Lund, New York, Basel, Aarhus, Macalester (St. Paul), Minnesota, Roosevelt (Chicago, Ill.), Zagreb, Technion (Haifa), Bombay, Calcutta, Warsaw, Brussels, Harvard, Cambridge (Mass.), and Rockefeller (New York). Professor Bohr was married, in 1912, to Margrethe Nørlund, who was for him an ideal companion. They had six sons, of whom they lost two; the other four have made distinguished careers in various professions – Hans Henrik (M.D.), Erik (chemical engineer), Aage (Ph.D., theoretical physicist, following his father as Director of the Institute for Theoretical Physics), Ernest (lawyer). Niels Bohr died in Copenhagen on November 18, 1962. This autobiography/biography was written at the time of the award and first published in the book series Les Prix Nobel . It was later edited and republished in Nobel Lectures . To cite this document, always state the source as shown above. Nobel Prizes and laureates Nobel prizes 2023. Explore prizes and laureates 13.1 Discovery of the Atom 13.2 Discovery of the Parts of the Atom: Electrons and Nuclei 13.3 Bohr’s Theory of the Hydrogen Atom 13.4 X-Rays: Atomic Origins and Applications 13.5 Applications of Atomic Excitations and De-Excitations 13.6 The Wave Nature of Matter Causes Quantization 13.7 Patterns in Spectra Reveal More Quantization 13.8 Quantum Numbers and Rules 13.9 The Pauli Exclusion Principle Section Summary Conceptual Questions Problems & Exercises Test Prep for AP ® Courses Learning Objectives By the end of this section, you will be able to do the following: Describe the mysteries of atomic spectra Explain Bohr’s theory of the hydrogen atom Explain Bohr’s planetary model of the atom Illustrate the energy state using the energy-level diagram Describe the triumphs and limits of Bohr’s theory The information presented in this section supports the following AP® learning objectives and science practices: 1.A.4.1 The student is able to construct representations of the energy-level structure of an electron in an atom and to relate this to the properties and scales of the systems being investigated. (S.P. 1.1, 7.1) 5.B.8.1 The student is able to describe emission or absorption spectra associated with electronic or nuclear transitions as transitions between allowed energy states of the atom in terms of the principle of energy conservation, including characterization of the frequency of radiation emitted or absorbed. (S.P. 1.2, 7.2) The great Danish physicist Niels Bohr (1885–1962) made immediate use of Rutherford’s planetary model of the atom ( Figure 13.14 ). Bohr became convinced of its validity and spent part of 1912 at Rutherford’s laboratory. In 1913, after returning to Copenhagen, he began publishing his theory of the simplest atom, hydrogen, based on the planetary model of the atom. For decades, many questions had been asked about atomic characteristics. From their sizes to their spectra, much was known about atoms, but little had been explained in terms of the laws of physics. Bohr’s theory explained the atomic spectrum of hydrogen and established new and broadly applicable principles in quantum mechanics. Mysteries of Atomic Spectra As noted in Quantization of Energy , the energies of some small systems are quantized. Atomic and molecular emission and absorption spectra have been known for over a century to be discrete or quantized (see Figure 13.15 ). Maxwell and others had realized that there must be a connection between the spectrum of an atom and its structure, something like the resonant frequencies of musical instruments. But, in spite of years of efforts by many great minds, no one had a workable theory. It was a running joke that any theory of atomic and molecular spectra could be destroyed by throwing a book of data at it, so complex were the spectra. Following Einstein’s proposal of photons with quantized energies directly proportional to their wavelengths, it became even more evident that electrons in atoms can exist only in discrete orbits. In some cases, it had been possible to devise formulas that described the emission spectra. As you might expect, the simplest atom—hydrogen, with its single electron—has a relatively simple spectrum. The hydrogen spectrum had been observed in the infrared (IR), visible, and ultraviolet (UV), and several series of spectral lines had been observed (see Figure 13.16 ). These series are named after early researchers who studied them in particular depth. The observed hydrogen-spectrum wavelengths can be calculated using the following formula where λ λ size 12{λ} {} is the wavelength of the emitted EM radiation and R R size 12{R} {} is the Rydberg constant , determined by the experiment to be The constant n f n f is a positive integer associated with a specific series. For the Lyman series, n f = 1 ; n f = 1 ; for the Balmer series, n f = 2 ; n f = 2 ; for the Paschen series, n f = 3 ; n f = 3 ; and so on. The Lyman series is entirely in the UV, while part of the Balmer series is visible with the remainder UV. The Paschen series and all the rest are entirely IR. There are apparently an unlimited number of series, although they lie progressively farther into the infrared and become difficult to observe as n f n f increases. The constant n i n i is a positive integer, but it must be greater than n f . n f . Thus, for the Balmer series, n f = 2 n f = 2 and n i = 3, 4, 5, 6, . . . . n i = 3, 4, 5, 6, . . . . Note that n i n i size 12{n rSub { size 8{i} } } {} can approach infinity. While the formula in the wavelengths equation was just a recipe designed to fit data and was not based on physical principles, it did imply a deeper meaning. Balmer first devised the formula for his series alone, and it was later found to describe all the other series by using different values of n f. n f. size 12{n rSub { size 8{f} } } {} Bohr was the first to comprehend the deeper meaning. Again, we see the interplay between experiment and theory in physics. Experimentally, the spectra were well established, an equation was found to fit the experimental data, but the theoretical foundation was missing. Example 13.1 Calculating Wave Interference of a Hydrogen Line What is the distance between the slits of a grating that produces a first-order maximum for the second Balmer line at an angle of 15º ? 15º ? size 12{"15"°} {} Strategy and Concept For an Integrated Concept problem, we must first identify the physical principles involved. In this example, we need to know (a) the wavelength of light as well as (b) conditions for an interference maximum for the pattern from a double slit. Part (a) deals with a topic of the present chapter, while part (b) considers the wave interference material of Wave Optics . Solution for (a) Hydrogen spectrum wavelength . The Balmer series requires that n f = 2 . n f = 2 . size 12{n rSub { size 8{f} } =2} {} The first line in the series is taken to be for n i = 3 , n i = 3 , size 12{n rSub { size 8{i} } =3} {} and so the second would have n i = 4 . n i = 4 . size 12{n rSub { size 8{i} } =4} {} The calculation is a straightforward application of the wavelength equation. Entering the determined values for n f n f size 12{n rSub { size 8{f} } } {} and n i n i size 12{n rSub { size 8{i} } } {} yields Inverting to find λ λ size 12{λ} {} gives Discussion for (a) This is indeed the experimentally observed wavelength, corresponding to the second blue-green line in the Balmer series. More impressive is the fact that the same simple recipe predicts all of the hydrogen spectrum lines, including new ones observed in subsequent experiments. What is nature telling us? Solution for (b) Double-slit interference ( Wave Optics ). To obtain constructive interference for a double slit, the path length difference from two slits must be an integral multiple of the wavelength. This condition was expressed by the equation where d d size 12{d} {} is the distance between slits and θ θ size 12{θ} {} is the angle from the original direction of the beam. The number m m size 12{m} {} is the order of the interference; m = 1 m = 1 size 12{m=1} {} in this example. Solving for d d size 12{d} {} and entering known values yields Discussion for (b) This number is similar to those used in the interference examples of Introduction to Quantum Physics and is close to the spacing between slits in commonly used diffraction glasses. Bohr’s Solution for Hydrogen Bohr was able to derive the formula for the hydrogen spectrum using basic physics, the planetary model of the atom, and some very important new proposals. His first proposal is that only certain orbits are allowed: We say that the orbits of electrons in atoms are quantized . Each orbit has a different energy, and electrons can move to a higher orbit by absorbing energy and drop to a lower orbit by emitting energy. If the orbits are quantized, the amount of energy absorbed or emitted is also quantized, producing discrete spectra. Photon absorption and emission are among the primary methods of transferring energy into and out of atoms. The energies of the photons are quantized, and their energy is explained as being equal to the change in energy of the electron when it moves from one orbit to another. In equation form, this is Here, Δ E Δ E size 12{ΔE} {} is the change in energy between the initial and final orbits, and hf hf size 12{ ital "hf"} {} is the energy of the absorbed or emitted photon. It is quite logical, that is, expected from our everyday experience, that energy is involved in changing orbits. A blast of energy is required for the space shuttle, for example, to climb to a higher orbit. What is not expected is that atomic orbits should be quantized. This is not observed for satellites or planets, which can have any orbit given the proper energy (see Figure 13.17 ). Figure 13.18 shows an energy-level diagram , a convenient way to display energy states. In the present discussion, we take these to be the allowed energy levels of the electron. Energy is plotted vertically with the lowest or ground state at the bottom and with excited states above. Given the energies of the lines in an atomic spectrum, it is possible, although sometimes very difficult, to determine the energy levels of an atom. Energy-level diagrams are used for many systems, including molecules and nuclei. A theory of the atom or any other system must predict its energies based on the physics of the system. Bohr was clever enough to find a way to calculate the electron orbital energies in hydrogen. This was an important first step that has been improved upon, but it is well worth repeating here, because it does correctly describe many characteristics of hydrogen. Assuming circular orbits, Bohr proposed that the angular momentum L L size 12{L} {} of an electron in its orbit is quantized , that is, it has only specific, discrete values. The value for L L size 12{L} {} is given by the formula where L L is the angular momentum, m e m e is the electron’s mass, r n r n is the radius of the n n th orbit, and h h is Planck’s constant. Note that angular momentum is L = Iω . L = Iω . For a small object at a radius r , I = mr 2 r , I = mr 2 and ω = v / r , ω = v / r , so that L = mr 2 v / r = mvr . L = mr 2 v / r = mvr . Quantization says that this value of mvr mvr can only be equal to h / 2, 2 h / 2, 3 h / 2 , h / 2, 2 h / 2, 3 h / 2 , size 12{h/2,` 2h/2, `3h/2} {} etc. At the time, Bohr himself did not know why angular momentum should be quantized, but using this assumption he was able to calculate the energies in the hydrogen spectrum, something no one else had done at the time. From Bohr’s assumptions, we will now derive a number of important properties of the hydrogen atom from the classical physics we have covered in the text. We start by noting the centripetal force causing the electron to follow a circular path is supplied by the Coulomb force. To be more general, we note that this analysis is valid for any single-electron atom. So, if a nucleus has Z Z size 12{Z} {} protons ( Z = 1 ( Z = 1 size 12{Z=1} {} for hydrogen, 2 for helium, etc.) and only one electron, that atom is called a hydrogen-like atom . The spectra of hydrogen-like ions are similar to hydrogen, but shifted to higher energy by the greater attractive force between the electron and nucleus. The magnitude of the centripetal force is m e v 2 / r n , m e v 2 / r n , size 12{m rSub { size 8{e} } v rSup { size 8{2} } /r rSub { size 8{n} } } {} while the Coulomb force is k Zq e q e / r n 2 . k Zq e q e / r n 2 . size 12{k left ( ital "Zq" rSub { size 8{e} } right ) left (q rSub { size 8{e} } right )/r rSub { size 8{n} } rSup { size 8{2} } } {} The tacit assumption here is that the nucleus is more massive than the stationary electron, and the electron orbits about it. This is consistent with the planetary model of the atom. Equating these Angular momentum quantization is stated in an earlier equation. We solve that equation for v , v , size 12{v} {} substitute it into the above, and rearrange the expression to obtain the radius of the orbit. This yields where a B a B size 12{a rSub { size 8{B} } } {} is defined to be the Bohr radius , since for the lowest orbit n = 1 n = 1 size 12{ left (n=1 right )} {} and for hydrogen Z = 1 , Z = 1 , size 12{ left (Z=1 right )} {} r 1 = a B. r 1 = a B. size 12{r rSub { size 8{1} } =a rSub { size 8{B} } } {} It is left for this chapter’s Problems and Exercises to show that the Bohr radius is These last two equations can be used to calculate the radii of the allowed (quantized) electron orbits in any hydrogen-like atom . It is impressive that the formula gives the correct size of hydrogen, which is measured experimentally to be very close to the Bohr radius. The earlier equation also tells us that the orbital radius is proportional to n 2 , n 2 , size 12{n rSup { size 8{2} } } {} as illustrated in Figure 13.19 . To get the electron orbital energies, we start by noting that the electron energy is the sum of its kinetic and potential energy Kinetic energy is the familiar KE = 1 / 2 m e v 2 , KE = 1 / 2 m e v 2 , size 12{ ital "KE"= left (1/2 right )m rSub { size 8{e} } v rSup { size 8{2} } } {} assuming the electron is not moving at relativistic speeds. Potential energy for the electron is electrical, or PE = q e V , PE = q e V , size 12{ ital "PE"=q rSub { size 8{e} } V} {} where V V size 12{V} {} is the potential due to the nucleus, which looks like a point charge. The nucleus has a positive charge Zq e Zq e size 12{ ital "Zq" rSub { size 8{e} } } {} ; thus, V = kZq e / r n , V = kZq e / r n , recalling an earlier equation for the potential due to a point charge. Since the electron’s charge is negative, we see that PE = − kZq e / r n , PE = − kZq e / r n , size 12{ ital "PE"= - ital "kZq" rSub { size 8{e/r rSub { size 6{n} } } } } {} Entering the expressions for KE KE size 12{ ital "KE"} {} and PE, PE, size 12{ ital "PE"} {} we find Now we substitute r n r n size 12{r rSub { size 8{n} } } {} and v v size 12{v} {} from earlier equations into the above expression for energy. Algebraic manipulation yields for the orbital energies of hydrogen-like atoms . Here, E 0 E 0 size 12{E rSub { size 8{0} } } {} is the ground-state energy n = 1 n = 1 size 12{ left (n=1 right )} {} for hydrogen Z = 1 Z = 1 size 12{ left (Z=1 right )} {} and is given by Thus, for hydrogen, Figure 13.20 shows an energy-level diagram for hydrogen that also illustrates how the various spectral series for hydrogen are related to transitions between energy levels. Electron total energies are negative, since the electron is bound to the nucleus, analogous to being in a hole without enough kinetic energy to escape. As n n size 12{n} {} approaches infinity, the total energy becomes zero. This corresponds to a free electron with no kinetic energy, since r n r n size 12{r rSub { size 8{n} } } {} gets very large for large n , n , size 12{n} {} and the electric potential energy thus becomes zero. Thus, 13.6 eV is needed to ionize hydrogen to go from –13.6 eV to 0, or unbound, an experimentally verified number. Given more energy, the electron becomes unbound with some kinetic energy. For example, giving 15.0 eV to an electron in the ground state of hydrogen strips it from the atom and leaves it with 1.4 eV of kinetic energy. Finally, let us consider the energy of a photon emitted in a downward transition, given by the equation to be Substituting E n = ( – 13.6 eV / n 2 ) , E n = ( – 13.6 eV / n 2 ) , size 12{E rSub { size 8{n} } = - "13" "." 6``"eV"/n rSup { size 8{2} } } {} we see that Dividing both sides of this equation by hc hc size 12{ ital "hc"} {} gives an expression for 1 / λ 1 / λ size 12{1/λ} {} It can be shown that is the Rydberg constant . Thus, we have used Bohr’s assumptions to derive the formula first proposed by Balmer years earlier as a recipe to fit experimental data. We see that Bohr’s theory of the hydrogen atom answers the question as to why this previously known formula describes the hydrogen spectrum. It is because the energy levels are proportional to 1 / n 2 , 1 / n 2 , size 12{1/n rSup { size 8{2} } } {} where n n size 12{n} {} is a non-negative integer. A downward transition releases energy, and so n i n i size 12{n rSub { size 8{i} } } {} must be greater than n f . n f . size 12{n rSub { size 8{f} } } {} The various series are those where the transitions end on a certain level. For the Lyman series, n f = 1 n f = 1 size 12{n rSub { size 8{f} } =1} {} —that is, all the transitions end in the ground state (see also Figure 13.20 ). For the Balmer series, n f = 2 , n f = 2 , size 12{n rSub { size 8{f} } =2} {} or all the transitions end in the first excited state and so on. What was once a recipe is now based in physics, and something new is emerging—angular momentum is quantized. Triumphs and Limits of the Bohr Theory Bohr did what no one had been able to do before. Not only did he explain the spectrum of hydrogen, he correctly calculated the size of the atom from basic physics. Some of his ideas are broadly applicable. Electron orbital energies are quantized in all atoms and molecules. Angular momentum is quantized. The electrons do not spiral into the nucleus, as expected classically—accelerated charges radiate, so that the electron orbits classically would decay quickly, and the electrons would sit on the nucleus—matter would collapse. These are major triumphs. But there are limits to Bohr’s theory. It cannot be applied to multielectron atoms, even one as simple as a two-electron helium atom. Bohr’s model is what we call semiclassical . The orbits are quantized—nonclassical—but are assumed to be simple circular paths—classical. As quantum mechanics was developed, it became clear that there are no well-defined orbits; rather, there are clouds of probability. Bohr’s theory also did not explain that some spectral lines are doublets—split into two—when examined closely. We shall examine many of these aspects of quantum mechanics in more detail, but it should be kept in mind that Bohr did not fail. Rather, he made very important steps along the path to greater knowledge and laid the foundation for all of atomic physics that has since evolved. PhET Explorations: Models of the Hydrogen Atom How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental results. Copy and paste the link code above. Related Items Niels Bohr Atomic Model Theory Experiment Niels Bohr Education & Life Niels Bohr is a well-known Danish physicist that spent the majority of his life studying the atomic model. The atomic model is a theory that holds that the atoms in an element are different from one another and contain protons, electrons, and neutrons. What Was Niels Bohr Experiment? What Did Niels Bohr Discover? The Niels Bohr Atomic Model theory is a model that was introduced by Niels Bohr in 1913 to describe the atom. It was a postulation of Bohr that the electrons rotated in a circular orbit around the nucleus of the atom. Niels Bohr’s atomic model was created based on previous research by Rutherford, Rutherford’s gold foil experiment, and Ernest Rutherford’s model of the atom. In his model, Bohr postulated that electrons were placed in orbits that are referred to as orbitals. Atoms consist of a central nucleus, surrounded by electrons in orbital shells. The electrons sit in energy levels around the nucleus, with the lowest possible energy level being electron number one and the highest being electron number eight. The Bohr atomic model theory states that atoms are composed of a nucleus, which consists of one or more protons and neutrons that are held together by nuclear forces. It is also known as a hydrogen atom model or the Rutherford-Bohr Atomic Model Theory. Niels Bohr was a Danish physicist who had a theory about atoms that he called the “atomic model”. Bohr’s atomic model had a nucleus with a certain number of positively charged particles that were held together by negatively charged particles. The electrons would orbit around the nucleus of the atom. Atomic Model Theory is the idea that the electrons orbiting the atom don’t orbit around a stationary nucleus like they were on the earth in a solar system. Instead, the electrons orbit around the nucleus of the atom, which is constantly moving. This is what Bohr called his quantum leap. Bohr’s theory helped to explain the interference experiment and helped to create quantum theories, like the wave-particle duality Niels Bohr came up with a model of the atom that was entirely radical for its time. It contradicted much of what was previously believed about atoms and electrons. He believed that an electron orbits a nucleus, which is made up of a group of subatomic particles. Bohr received the Nobel Prize in 1922 for his theory. Niels Bohr As A Physicist Niels Bohr is considered to be one of the greatest physicists in history. He worked for many years on physics, teaching, and management. This work led him to become a professor at the University of Copenhagen for thirty years. In 1912, he was offered a professorship at the Institute of Theoretical Physics in Stockholm. However, there was a problem with his salary because he was not on an equal footing with his counterpart at Uppsala University. In 1920, Bohr returned to the Institute of Theoretical Physics in Copenhagen. To this day, Bohr remains one of the most celebrated people in Danish history. Niels Bohr as a Father and a Husband In 1908, Niels Bohr married Margrethe Nørlund. They had two sons, Aage Nørlund (1909) and Harald Bohr (1911). In 1920, they moved to King’s Gate No.1. They remained there for the rest of their lives. Bohr was a caring husband and father, who did not like to leave home too often because he missed his family. Bohr also liked to play classical music, and he was a good enough pianist to give concerts in Copenhagen. Niels Bohr’s Death In 1942, Niels Bohr became increasingly ill and was diagnosed with an incurable muscle disease, which caused him great pain and robbed him of his ability to walk. In September 1948, Bohr became very ill. He developed a blood clot in his leg and he could no longer move around on his own. On October 17, he suffered a severe stroke. He passed away on 18 November. After his death, the Danish king said about Bohr: “I know of no one who has contributed more to the knowledge and to the progress of mankind than Niels Bohr”. Niels Bohr’s Legacy One of the most important things that Niels Bohr did was to create a new model of the atom. He realized that electrons could exist in ‘allowed’ orbits, but they could also ‘jump’, or transition, to higher energy orbits. One way that people continued to think about Bohr’s ideas was through the use of his concept of quantum jumps. Bohr also believed that the electron didn’t exist in any particular orbit, but instead was found in all orbits all at the same time, and that only when we looked at an atom would it ‘decide’ which orbit to be in. He was awarded the Nobel Prize for physics in 1922 for this work. The Bohr Model of The Atom Bohr’s model of the atom was one of the most important contributions of his career because it helped us to understand why atoms didn’t collapse. However, Bohr’s model didn’t explain all the properties of an atom. For example, in the ‘old model of the atom, electrons were stationary (always in the same orbit), and they were at a fixed distance from their nucleus. In other words, they orbited at a fixed distance from their nucleus. Now, with Bohr’s model, this wasn’t true anymore – electrons could jump around to different orbits. It’s easy to understand that if electrons can jump around, then they can’t have a fixed distance from the nucleus. They would also have to be influenced by the nucleus. So, when you measure any of the properties of an atom (e.g. the position of an electron), you can never measure it as if it were in ‘absolute space’, but only as how things are relative to each other (relative motion). What Is Niels Bohr Known For? The physics community remembers Niels Bohr for his work with the Bohr model of the atom. He was able to explain and interpret vast amounts of experimental data in terms of his atomic model. The Bohr atomic model consists of one positively charged nucleus surrounded by electrons, which are negatively charged. The positive charge in the nucleus is balanced by negative charge in the electron. Bohr argued that electrons move around the atom by radially oscillating, which wiggles their position in space. Bohr also thought that atoms could be described as a series of stationary orbitals. An orbital can be considered a “shell” around an electron and “is filled” with electrons. Energy can be transferred between an orbital and the electron by oscillations. Bohr provided the mathematical description of his model by applying quantum mechanics. For example, the electron orbits are given by Schrödinger wave equations. The radius of the orbits is related to energy levels in a very simple way. These are the most basic atomic model equations ever published. All other models have been derived from these basic ones. Bohr himself made sure that the model could be applied to spectroscopy and other measurements. What Is Niels Bohr Famous For? Niels Bohr was a physicist who made fundamental contributions to the theory of the atom, quantum mechanics, and chemical bonding. He is also known as the father of modern quantum physics. Bohr was one of the first to apply mathematics to physics. He was able to think in terms of waves and positions instead of just particles and points. Niels Bohr’s Influence On Chemistry Bohr’s influence also extended beyond physics. In fact, he made some interesting contributions to chemistry. For example, he correctly predicted that helium atoms would absorb high-frequency light in a series of elements (helium, neon, argon, and krypton). He also predicted that they would emit light in a series of elements (for example sodium). But perhaps his most important contribution to chemistry was helping to explain why certain chemical reactions occur. Bohr’s ideas about quantum jumps also helped us to understand how hydrogen, which has a very large atomic mass, could be broken up into its component atoms. He explained that a hydrogen atom consists of only one electron which moves around the nucleus. The electron orbits the nucleus and then jumps to a new energy level. Another of Bohr’s greatest contributions was his work in spectroscopy. He correctly predicted that the frequency of light would increase when light passed through a series of metals (such as helium and sodium). He also predicted that these elements would emit photons at visible frequencies when heated. The Bohr Model And Quantum Mechanics While the basic idea behind Bohr’s model (the atom is made up of electrons that move around a nucleus) is still in use today, it was eventually superseded by quantum mechanics . However, Bohr’s ideas were very important for understanding how atoms worked. He showed how the strangeness of quantum physics explained why atoms didn’t collapse. He also showed how the strangeness of quantum physics could be used to explain how atoms absorb and emit light. While Bohr’s model did not explain some of the properties of atoms (mass, charge, or size), it had a major influence on the way that we think about and study atoms today. Niels Bohr And Experimental Data Bohr was a physicist who was very important to experimentalists. His contributions helped to explain how electrons could jump from one orbit to another in an atom. It also helped explain why different atoms have different masses and predicted light emission colors for various kinds of spectroscopy. In addition, Bohr was one of the first to suggest that the cathode rays (later to be called electrons) do not actually have a definite trajectory but instead travel in a broad wave with peaks and troughs. The wave theory described the behavior of electrons much better than the Newtonian particle model, which had been used up until then. What Did Niels Bohr Think About The Atom And Quantum Mechanics? According to Bohr, an atom is composed of a charged nucleus and a cloud of electrons. The nucleus is fixed in space, while the electrons can move around inside the atom. This movement happens very quickly but is maintained by electromagnetic forces. It is also maintained by the energy which keeps the electrons in their orbits. Ionization occurs when an electron jumps from one orbit to another – or when light from a specific wavelength enters an atom. Bohr was very conscious of the fact that he was a ‘complementary’ physicist. This means that he accepted quantum theory, but also believed in the classical view (which has all particles having definite locations). In his day, this challenged the idea of quantum mechanics, since it meant that Bohr himself did not believe in quantum theory. This is because Bohr did not equate the accuracy of his predictions with the validity of theoretical physics. However, since he never really discussed these views with his colleagues, and because the laws of quantum mechanics were absolutely consistent with all of his predictions, Bohr did not suffer any significant criticism. What Was Niels Bohr’s Contribution To Quantum Mechanics? In 1913 Bohr began working on what we now call the “old” model of an atom. Before this time, it was thought that electrons orbited the nucleus in evenly spaced orbits. It was also thought that electrons jumped to a new orbit when they gained or lost energy. Bohr changed this view completely by introducing the idea of stationary, allowed orbits. This meant that electrons had a certain angular momentum inside the atom, which was ever-changing. An electron could jump to another orbit by losing or gaining energy but did not jump because of an external push or pull. In other words, electrons jump because they are excited by the electromagnetic radiation of an atom. The idea of stationary allowed orbits was revolutionary. It meant that atoms could emit and absorb energy in a continuous way, rather than in individual packets (which is what happened when people used the Bohr-Ellsberg-Slater theory). In 1914, Bohr suggested that electrons could exist only in certain orbits inside the atom. This meant that there was a mathematical connection between atomic orbitals and wavelengths or frequencies of light. Later, in 1916, Bohr suggested that the atom is mainly made of neutrons. He also introduced the idea of electron jumping. This was significant because it was one of the first models to combine quantum theory and classical physics. In 1918 Bohr published an explanation for atomic structure based on a “postulate” about what happened when electrons jumped from one orbit to another. According to Bohr, electrons could exist only in certain orbits (i.e., certain energy levels). Electrons could also jump from one orbit into another. This was an important development in quantum mechanics because it helped to explain why the atom would not collapse. What Did Niels Bohr Contribute To Society? Bohr was one of the founders of quantum mechanics. This theory is still in use today. In addition, Bohr was one of the first people to think about atoms – what they might be like and how we can observe them. He developed models which are still used today. Besides this, Bohr was a very successful teacher and mentor. Many young scientists (including future Nobel Prize winners) studied with him in Copenhagen and benefited from his advice and guidance. Niels Bohr was one of the first people to suggest that the laws of classical physics could be thought of as being the same as the laws of quantum physics. This was a revolutionary idea, and it showed that everything in our world is quantifiable. In other words, nothing in our world can escape quantification – or measurement. This view of reality – or what we would call ‘the scientific method’ – has had a huge influence on modern thinking about how our society works. Similar Posts Non-Ferrous Metals List | Properties of Non Ferrous Metals Non-Ferrous Metals List | Properties of Non Ferrous Metals Non-Ferrous Metals What is Non Ferrous Metals? Non-ferrous metals  can be defined as metals that  do not have iron content in them. Non-ferrous metals  examples include aluminum, Bronze, copper, Brass, lead, zinc and tin, silver and gold. 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It is the standard test by which the… Ernest Rutherford Atomic Theory Model & Experiment Ernest Rutherford Atomic Theory Model & Experiment Ernest Rutherford Contribution Ernest Rutherford was a New Zealand-born British chemist and physicist known for his pioneering work in the study of radioactivity. In 1903, he became the first person to demonstrate that radioactive materials emit energy in a continuous stream of particles and are not merely energy… 492 IMAGES Bohr Model of the Atom PPT Bohr Model of the Hydrogen Atom Bohr Model of the Hydrogen Atom Describe Bohr’s model of the hydrogen atom. Bohr Atomic Model Of Hydrogen VIDEO HYDROGEN ATOM Biography of Niels Bohr Exploring the Bohr-Einstein Debate Best HYDROGEN experiment ever! (maybe...) --DON'T TRY THIS AT HOME Niels Bohr (1937) [Live-action footage] Bohr's Atomic Model #bohr #bohrmodelofatom #atomicmodel #chemistry #shortvideo #shorts COMMENTS Bohr's model of hydrogen (article) Key points. Bohr's model of hydrogen is based on the nonclassical assumption that electrons travel in specific shells, or orbits, around the nucleus. E ( n) = − 1 n 2 ⋅ 13.6 eV. h ν = Δ E = ( 1 n l o w 2 − 1 n h i g h 2) ⋅ 13.6 eV. Bohr's model does not work for systems with more than one electron. Bohr model Bohr model, description of the structure of atoms, especially that of hydrogen, proposed (1913) by the Danish physicist Niels Bohr.The Bohr model of the atom, a radical departure from earlier, classical descriptions, was the first that incorporated quantum theory and was the predecessor of wholly quantum-mechanical models. The Bohr model and all of its successors describe the properties of ... Bohr model The Bohr model of the hydrogen atom (Z = 1) or a hydrogen-like ion (Z > 1), where the negatively charged electron confined to an atomic shell encircles a small, positively charged atomic nucleus and where an electron jumps between orbits, is accompanied by an emitted or absorbed amount of electromagnetic energy (hν). The orbits in which the electron may travel are shown as grey circles; their ... 30.3: Bohr's Theory of the Hydrogen Atom Bohr's theory explained the atomic spectrum of hydrogen and established new and broadly applicable principles in quantum mechanics. Figure 30.3.1. Niels Bohr, Danish physicist, used the planetary model of the atom to explain the atomic spectrum and size of the hydrogen atom. Niels Bohr In 1913, Niels Bohr proposed a theory for the hydrogen atom, based on quantum theory that some physical quantities only take discrete values. Electrons move around a nucleus, but only in prescribed orbits, and If electrons jump to a lower-energy orbit, the difference is sent out as radiation. Bohr's model explained why atoms only emit light ... 12.7: Bohr's Theory of the Hydrogen Atom The great Danish physicist Niels Bohr (1885-1962) made immediate use of Rutherford's planetary model of the atom. (Figure 12.7.1 ). Bohr became convinced of its validity and spent part of 1912 at Rutherford's laboratory. In 1913, after returning to Copenhagen, he began publishing his theory of the simplest atom, hydrogen, based on the ... 30.3 Bohr's Theory of the Hydrogen Atom The great Danish physicist Niels Bohr (1885-1962) made immediate use of Rutherford's planetary model of the atom. ( Figure 1 ). Bohr became convinced of its validity and spent part of 1912 at Rutherford's laboratory. In 1913, after returning to Copenhagen, he began publishing his theory of the simplest atom, hydrogen, based on the ... 30.3 Bohr's Theory of the Hydrogen Atom Figure 30.13 Niels Bohr, Danish physicist, used the planetary model of the atom to explain the atomic spectrum and size of the hydrogen atom. His many contributions to the development of atomic physics and quantum mechanics, his personal influence on many students and colleagues, and his personal integrity, especially in the face of Nazi oppression, earned him a prominent place in history. Atomic flashback: A century of the Bohr model Bohr, one of the pioneers of quantum theory, had taken the atomic model presented a few years earlier by physicist Ernest Rutherford and given it a quantum twist. Rutherford had made the startling discovery that most of the atom is empty space. The vast majority of its mass is located in a positively charged central nucleus, which is 10,000 ... Bohr's atomic model At the beginning of 1913, his colleague H. M. Hansen brought to his attention physicist J. J. Balmer's formula in experimental spectroscopy, an empirically derived formula that described, but did not explain, the spectrum of the hydrogen atom. It turned out that Niels Bohr's theory accurately predicted this formula. Niels Bohr Niels Henrik David Bohr (Danish: [ˈne̝ls ˈpoɐ̯ˀ]; 7 October 1885 - 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. Bohr was also a philosopher and a promoter of scientific research.. Bohr developed the Bohr model of the atom, in which he proposed ... Bohr's Theory of the Hydrogen Atom The great Danish physicist Niels Bohr (1885-1962) made immediate use of Rutherford's planetary model of the atom. (Figure 1). Bohr became convinced of its validity and spent part of 1912 at Rutherford's laboratory. In 1913, after returning to Copenhagen, he began publishing his theory of the simplest atom, hydrogen, based on the planetary ... Bohr Model of the Atom The simplest example of the Bohr Model is for the hydrogen atom (Z = 1) or for a hydrogen-like ion (Z > 1), in which a negatively charged electron orbits a small positively charged nucleus. Electromagnetic energy will be absorbed or emitted if an electron moves from one orbit to another. Only certain electron orbits are permitted. 1.8: The Bohr Theory of the Hydrogen Atom 1.8: The Bohr Theory of the Hydrogen Atom is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts. The model we will describe here, due to Niels Bohr in 1913, is an early attempt to predict the allowed energies for single-electron atoms. Bohr's Hydrogen Atom Niels Bohr introduced the atomic Hydrogen model in 1913. He described it as a positively charged nucleus, comprised of protons and neutrons, surrounded by a negatively charged electron cloud. In the model, electrons orbit the nucleus in atomic shells. The atom is held together by electrostatic forces between the positive nucleus and negative ... Niels Bohr Niels Bohr (1885-1962) On the Spectrum of Hydrogen address to the Physical Society of Copenhagen, December 20, 1913 [Fysisk Tidsskrift 12, 97 (1914) translated by A. D. Udden ("The Theory of Spectra and Atomic Constitution--Three Essays", 1922) from Forest Ray Moulton and Justus J. Schifferes, Eds., Autobiography of Science (New York: Doubleday ... Bohr's Theory of the Hydrogen Atom The great Danish physicist Niels Bohr (1885-1962) made immediate use of Rutherford's planetary model of the atom. ( (Figure) ). Bohr became convinced of its validity and spent part of 1912 at Rutherford's laboratory. In 1913, after returning to Copenhagen, he began publishing his theory of the simplest atom, hydrogen, based on the ... Bohr's theory of the hydrogen atom In 1913, the Danish physicist Niels Bohr (1885 - 1962) managed to explain the spectrum of atomic hydrogen by an extension of Rutherford's description of the atom. In that model, the negatively charged electrons revolve about the positively charged atomic nucleus because of the attractive electrostatic force according to Coulomb's law. Bohr Model of the Hydrogen Atom Niels Bohr introduced the atomic Hydrogen model in the year 1913. Bohr's Model of the hydrogen atom attempts to plug in certain gaps as suggested by Rutherford's model. It holds a special place in history as it gave rise to quantum mechanics by introducing the quantum theory. How Two Rebel Physicists Changed Quantum Theory In high school, where his physics instructor described him as "outstanding" and "brilliant," Bohm developed his own alternative ideas about Bohr's hydrogen atom. After undergraduate work at Penn State, he began earning a PhD in nuclear physics in 1941 under J. Robert Oppenheimer (1904-1967) at the University of California, Berkeley. Niels Bohr N iels Henrik David Bohr was born in Copenhagen on October 7, 1885, as the son of Christian Bohr, Professor of Physiology at Copenhagen University, and his wife Ellen, née Adler. Niels, together with his younger brother Harald (the future Professor in Mathematics), grew up in an atmosphere most favourable to the development of his genius - his father was an eminent physiologist and was ... 13.3 Bohr's Theory of the Hydrogen Atom Figure 13.14 Niels Bohr, Danish physicist, used the planetary model of the atom to explain the atomic spectrum and size of the hydrogen atom. His many contributions to the development of atomic physics and quantum mechanics, his personal influence on many students and colleagues, and his personal integrity, earned him a prominent place in history. Niels Bohr Atomic Model Theory Experiment The Niels Bohr Atomic Model theory is a model that was introduced by Niels Bohr in 1913 to describe the atom. It was a postulation of Bohr that the electrons rotated in a circular orbit around the nucleus of the atom. Niels Bohr's atomic model was created based on previous research by Rutherford, Rutherford's gold foil experiment, and ... essay homework paper phd project resume review speech thesis