# High School Chemistry/The Bohr Model

In the last lesson, you learned that atoms of different elements produce different atomic spectra when they are struck by an electric spark or an electric current. This phenomenon is really rather puzzling. Why do atoms emit light when they are exposed to an electric current? Why is the emitted light only at specific wavelengths? Why do different elements have different atomic spectra? Surely these atomic spectra must tell us something about the atoms that they came from, but what does it all mean? These were the types of questions that scientists were asking in the early 1900s when a Danish physicist named Niels Bohr (Figure 5.15) became interested in atomic spectra and the nature of the atom.

Figure 5.15: A photograph of Neils Bohr.

## Lesson Objectives

• Define an energy level in terms of the Bohr model.
• Find the energy of a given Bohr orbit using the equation $E_n = \tfrac{-Rhc}{n^2}$
• Discuss how the Bohr model can be used to explain atomic spectra.

## Bohr Used Atomic Spectra to Develop His Model

We have learned that light travels as a wave and gives up its energy as a particle. As a result, the wavelength (or color) of a light is related to the light's frequency which is, in turn, related to the light's energy. All of these relationships were summarized in the electromagnetic spectrum. Remember the smaller the wavelength of the light (blue end of the visible spectrum), the larger the frequency and the larger energy. Similarly, the larger the wavelength of the light (red end of the visible spectrum), the smaller the frequency and the smaller the energy.

When Niels Bohr began thinking about the atom and atomic spectra in general, he was aware of the wave-particle duality of light that had just been discovered. He knew that a specific wavelength (or color) of light was related to the light's energy. As a result, Bohr realized that the lines of color appearing in an element's atomic spectrum corresponded not only to those wavelengths of light but also, to the specific frequencies and, more importantly, to the specific energies of light. The important question, then, was why the atoms were only emitting light at very specific energies. Niels Bohr realized that this unusual result could be explained by proposing what he termed energy levels. What are energy levels?

## Electron Energy Is Quantized

You have learned that a rock held above the ground would release potential energy as it fell. Niels Bohr realized that in order for atoms to release energy, there must be a similar "falling" process going on inside the atom. Since Bohr, like Rutherford, knew that the protons in the atom were bound up in the tiny nucleus, the obvious sub-atomic objects that could be "falling" inside the atom were the electrons. Therefore, Bohr proposed a model in which electrons circled around the nucleus and, on occasion, fell closer to the nucleus, releasing energy in the process. According to Bohr, the energy that came out of the atoms as their electrons fell towards the nucleus appeared as light. This light, he argued, produced the atomic spectra that could be seen whenever electric current was passed through an element.

Of course, none of Bohr's arguments thus far explain why only certain energies appear in each element's atomic spectrum. If you hold a rock above the ground and drop it, it will release a specific amount of energy. If you raise that rock higher by a tiny amount, it will release slightly more energy. If you lower that rock by a tiny amount, it will release slightly less energy. In fact, it would seem you can get that rock to release any quantity of energy that you'd like, just by raising and lowering it to different levels above ground. But with electrons, the situation is clearly different. With electrons, it seems as if there are only a few levels from which electrons can fall. At least, that's what Bohr decided, and that's why he proposed the existence of the atomic energy level.

According to Bohr, the electrons in an atom were only allowed to exist at certain energy levels. The electrons could jump from a lower level to a higher level when they gained energy (which they could get from a passing electric current, electric spark, heat, or light), and they could drop from a higher level to a lower level when they lost energy (which they released in the form of light). Most importantly, though, the electrons could not exist in between the allowed energy levels. Many people have compared Bohr's energy levels to a set of stairs. Think about a child jumping up and down a set of stairs. The child can rest at any one of the stairs and stay there. If he puts energy in, he can jump up to a higher stair. If he allows himself to fall, he can drop to a lower stair. The child cannot, however, hover at a level in between two of the stairs, just as an electron cannot hover in between two of the atom's energy levels.

Just as children cannot hover between two steps on a staircase, Bohr suggested that electrons cannot hover between to energy levels in the atom.

Bohr developed his energy level model further using principles from physics. In physics, the energy of a positive charge and a negative charge depends on the distance between them. Therefore, Bohr decided that his energy levels must correspond to orbits, or circular paths centered around the nucleus of the atom. Since an electron that was trapped in one of these orbits remained a constant distance from the nucleus, Bohr stated that within an orbit an electron had a constant energy. Of course, the fact that only certain energy levels were allowed, also meant that only certain orbits were allowed. Figure 5.16 shows a schematic illustration of Bohr's model. In the diagram, each circle from n = 1 to n = 4 is an allowed orbit.

Figure 5.16: A schematic illustration of the Bohr model of the atom.

Notice how the negative electron, in any given orbit, is always the same distance from the positive nucleus at any place on the orbit. As a result electrons within an orbit always have the same energy. When an electron is hit by electricity, it can gain energy and can be bumped up to a higher energy orbit further away from the nucleus. On the other hand, when an electron loses energy, it falls back down to a lower energy orbit closer to the nucleus. The electron can never exist at distances in between allowed orbits. Notice how this limits the number of different transitions that the electron can make. Because the electron can only exist in certain orbits, and thus can have only certain energies, we say that the energy of the electron is quantized.

## Bohr Used a Formula to Determine Allowed Energy Levels

Although Bohr's descriptive model of atomic orbits gives a nice explanation as to why atoms should have discontinuous atomic spectra, it's hard to test experimentally. Luckily, using advanced physics, the Bohr model can be used to derive a mathematical expression for the energies of the allowed orbits in the hydrogen atom. As you'll see in the next section, the energies of these orbits actually determine which wavelengths of light appear in hydrogen's atomic spectrum, meaning that Bohr's model can, in fact, be tested experimentally. The equation predicting the energies of the allowed hydrogen orbits is:

$E_n = \frac{-Rhc}{n^2}$

Here En is the energy of the nth orbit (in other words the energy of the 1st, 2nd, 3rd, 4th, etc. orbit), R is the Rydberg constant for hydrogen (which always has the value R = 1.097×107 m−1), h is Planck's constant (which always has the value h = 6.63×10−34 J · s), c is the speed of light (which always has the value c = 3.00×108 m/s, and n is the number of the orbit you are interested in. n can have any integer value (no decimals!) from 1 to infinity, n = 1, 2, 3, 4 … ∞. As was shown in Figure 5.16, n = 1 is the orbit closest to the atomic nucleus, n = 2 is the next orbit out, n = 3 is the one after that, and so on. In other words, n increases with increasing distance from the nucleus. Let's take a look at several examples using this equation.

 Example 1 What is the energy of the 3rd allowed orbit in the hydrogen atom? Solution: Planck's constant, h = 6.63×10−34 J · s Rydberg constant for hydrogen R = 1.097×107 m−1 (Notice how you do not need to be given h, R, or c, since they are all constants) speed of light, c = 3.00×108 m/s n = 3 $E_n = \frac{-R \times h \times c}{n^2}$ $E_n = -\frac{(1.097 \times 10^7\,\text{m}^-1) \times (6.63 \times 10^{-34}\,\text{J}\cdot\text{s}) \times (3.00 \times 10^8\,\text{m/s})}{(3)^2}$ $E_n = -\frac{(7.27 \times 10^{-27}\,\text{J}\cdot\text{s}) \times (3.00 \times 10^8\,\text{m/s})}{9}$ $E_n = -\frac{(2.18 \times 10^{-18}\,\text{J})}{9}$ $E_n = -2.42 \times 10^{-19}\,\text{J}$ (Be careful not to drop the negative sign! These energies should always be negative.)
 Example 2 What is the energy of the 2nd allowed orbit in the hydrogen atom? Solution: Planck's constant, h = 6.63×10−34 J · s Rydberg constant for hydrogen R = 1.097×107 m−1 (Notice, again, how you do not need to be given h, R, or c, since they are all constants) speed of light, c = 3.00×108 m/s n = 2 $E_n = \frac{-R \times h \times c}{n^2}$ $E_n = -\frac{(1.097 \times 10^7\,\text{m}^-1) \times (6.63 \times 10^{-34}\,\text{J}\cdot\text{s}) \times (3.00 \times 10^8\,\text{m/s})}{(2)^2}$ $E_n = -\frac{(7.27 \times 10^{-27}\,\text{J}\cdot\text{s}) \times (3.00 \times 10^8\,\text{m/s})}{4}$ $E_n = -\frac{(2.18 \times 10^{-18}\,\text{J})}{4}$ $E_n = -5.45 \times 10^{-19}\,\text{J}$ (Again, be careful not to drop the negative sign!)

At the moment, these probably seem like fairly boring calculations, but in the next section, we'll see how energy level calculations like the two above can actually be used to predict the atomic spectrum of hydrogen. When scientists first did this in the early 1900s, they were amazed at how well this simple equation predicted the colors of light emitted by hydrogen atoms.

## Atomic Spectra Produced by Electrons Changing Energy Levels

Suppose you have $10 in your wallet when you go to the store, and you only have$4 when you come home. How much money have you spent at the store? The answer is simple – you've obviously spent \$6. With electrons, the situation is similar. If they start with 10 units of energy, and fall down to 4 units of energy, they lose 6 units of energy in the process. To state this mathematically, we write:

 $\Delta E_{i \to f}\,\!$ $=\,\!$ $E_f\,\!$ $-\,\!$ $E_i\,\!$ $\text{Change in energy from initial state,}\ i\text{, to final state,}\ f\,\!$ $\text{Energy in final state,}\ f\,\!$ $\text{Energy in the state,}\ i\,\!$

When electrons "lose" energy, though, they lose it by giving it off in the form of light. We have calculated the energy of an electron in the 3rd hydrogen orbit, −2.42×10−19 J, and the energy of an electron in the 2nd hydrogen orbit, −5.45×10−19 J. An electron falling from the 3rd hydrogen orbit to the 2nd hydrogen orbit begins with −2.42×10−19 J, and ends with −5.45×10−19 J, thus it loses −3.03×10−19 J in the process. This can be worked out mathematically. (Don't worry too much about the fact that this lost energy has a negative sign. Any time that an atom gains energy, the sign will be positive, and any time that an atom loses energy, the sign will be negative).

initial state, i = 3

final state, f = 2

 $\Delta E_{i \to f} =\,\!$ $\Delta E_{3 \to 2}\,\!$ $E_f =\,\!$ $E_2 = -5.45 \times 10^{-19}\,\text{J}\,\!$ $E_i =\,\!$ $E_3 = -2.42 \times 10^{-19}\,\text{J}\,\!$ $\Delta E_{3 \to 2} =\,\!$ $E_2 - E_3\,\!$ $\Delta E_{i \to f} =\,\!$ $(-5.45 \times 10^{-19}\,\text{J}) - (-2.42 \times 10^{-19}\,\text{J})\,\!$
Remember, subtracting a negative number is the same as adding a positive number!
 $\Delta E_{i \to f} =\,\!$ $(-5.45 \times 10^{-19}\,\text{J}) + (2.42 \times 10^{-19}\,\text{J})\,\!$ $\Delta E_{i \to f} =\,\!$ $-3.03 \times 10^{-19}\,\text{J}\,\!$

All of the energy lost when an electron falls from a higher energy orbit to a lower energy orbit is turned into light, thus when an electron falls from the 3rd hydrogen orbit to the 2nd hydrogen orbit, it emits a beam of light with an energy of 3.03×10−19 J. You should now be able convert this energy into a wavelength (first convert the energy into a frequency and then, convert that frequency into a wavelength ). The wavelength turns out to be 656 nm and, amazingly, if you look at hydrogen's atomic spectrum, shown below, you'll notice that hydrogen clearly has a line of red light at exactly 656 nm! What's more, the green 486 nm line in hydrogen's atomic spectrum corresponds to an electron falling from the 4th hydrogen orbit to the 2nd hydrogen orbit, the blue 434 nm line corresponds to an electron falling from the 5th hydrogen orbit to the 2nd hydrogen orbit, and the purple 410 nm line corresponds to an electron falling from the 6th hydrogen orbit to the 2nd hydrogen orbit. In other words, Bohr's model can be used to predict the exact wavelengths of the four visible lines in hydrogen's atomic spectrum.

A short discussion of atomic spectra and some animation showing the spectra of elements you chose and an animation of electrons changing orbits with the absorption and emission of light can be viewed at Spectral Lines.

What about all of the other transitions that could occur between the allowed hydrogen orbits? For example, where is the line that corresponds to an electron falling from the 4th orbit to the 3rd orbit? It turns out that the energies of all the other transitions don't lie within range of energies included in the visible spectrum. As a result, even though these transitions occur, your eyes can't see the light that's given off as a result. Clearly, then, the predictions from Bohr's model give a perfect fit when compared to the hydrogen atomic spectrum. There's only one small problem…

## Bohr's Model Only Worked for Hydrogen

If you take a look at the periodic table, what you'll notice is that hydrogen is a very special element. Hydrogen is the first element, and thus, it only has one electron. It turns out that Bohr's model of the atom worked very well provided it was used to describe atoms with only one electron. The moment that the Bohr model was applied to an element with more than one electron (which, unfortunately, includes every element except hydrogen), the Bohr model failed miserably.

Bohr's model failed because it treated electrons according to the laws of classical physics. Unfortunately, those laws only apply to fairly large objects. Back when Bohr was developing his model, scientists were only beginning to realize that the laws of classical physics didn't apply to matter as tiny as the electron. Electrons are actually quantum objects, meaning that they can only be described using the laws of quantum physics. Many of the differences between classical physics and quantum physics become particularly important when two or more quantum objects interact. As a result, while Bohr's model worked for hydrogen, it became worse and worse at predicting the atomic spectra for atoms with more and more electrons. Even helium, with two electrons, was something of a disaster!

Bohr's model explained the emission spectrum of hydrogen which previously had no explanation. The invention of precise energy levels for the electrons in an electron cloud and the ability of the electrons to gain and lose energy by moving from one energy level to another offered an explanation for how atoms were able to emit exact frequencies of light. Bohr calculated energies for the energy levels of hydrogen atoms that yielded the exact frequencies found in the hydrogen spectrum. Furthermore, those same energy levels predicted that hydrogen atoms would also emit frequencies of light in the infrared and ultraviolet regions that no one had observed previously. The subsequent discovery that those exact frequencies of infrared and ultraviolet light were present in the hydrogen spectrum provided even greater support for the ideas in the Bohr model.

One of the problems with Bohr's theory was that it was already known that when electrons were accelerated, they emitted radio waves. When you study physics, you will learn that acceleration applies to speeding up, slowing down, and traveling in a curved path. When charged particles are accelerated, they emit radio waves. In fact, that is how we create radio signals—by forcing electrons to accelerated up and down in an antenna. Scientists were creating radio signals in this way since 1895. Since Bohr's electrons were supposedly traveling around the nucleus in a circular path, they MUST emit radio waves, hence lose energy and collapse into the nucleus. Since the electrons in the electron cloud of an atom did not emit radio waves, lose energy, and collapse into the nucleus, there was some immediate doubt that the electrons could be traveling in a curved path around the nucleus. Bohr attempted to deal with this problem by suggesting that the electron cloud contained a certain number of energy levels, that each energy level could hold only a single electron, and that in ground state, all electrons were in the lowest available energy level. Under these conditions, no electron could lose energy because there was no lower energy level available. Electrons could gain energy and go to a higher energy level and then fall back down to the now open energy level thus emitting energy, but once in ground state, no lower positions were open. This explained why electrons circling the nucleus did not emit energy and spiral into the nucleus. Bohr did not, however, offer an explanation for why only the exact energy levels he calculated were present, that is, what is there about electrons in electron clouds that produce only a specific set of energy levels.

Another problem with Bohr's model was the predicted positions of the electrons in the electron cloud. If Bohr's model was correct, the hydrogen atom electron in ground state would always be the same distance from the nucleus. If we could take a series of photographic snapshots of a hydrogen electron cloud that would freeze the position of the electron so we could see exactly where it was located at different times, we still wouldn't know the path the electron followed to get from place to place, but we could see a few positions for the electron. Such an image of electron positions would show the electron could actually be various distances from the nucleus rather that at a constant distance.

If the electron circled the nucleus as suggested by Bohr's model, the electron positions would always be the same distance from the nucleus as shown in Figure 5.17, portion A. In reality, however, the electron is found at many different distances from the nucleus as in Figure 5.17, portion B. To solve all these discrepancies, scientists would need a completely new way of looking at not just energy but at matter as well.

Figure 5.17: The electrons are not a constant distance from the nucleus.

The development of Bohr's model of the atom is a good example of the scientific method. It shows how the observations of atomic spectra lead to the invention of a hypothesis about the nature of matter to explain the observations. The hypothesis also made predictions about spectral lines that should exist in the infrared and ultraviolet regions and when these observations were found to be correct, it provided even more supportive evidence for the theory. Of course, further observations can also provide contradictory evidence that will cause the downfall of the theory which also occurred with Bohr's model. Bohr's model was not, however, a failure. It provided the insights that allowed the next step in the development of our concept of the atom.

## Bohr's Model Unacceptable

You have just seen that the Bohr model applied classical physics to electrons when electrons can only be described using quantum mechanics. In addition, it turns out that Bohr's description of electrons as tiny little objects circling the nucleus along fixed orbits is incorrect as well. Bohr was picturing an atom that looked very much like a small solar system. The nucleus at the center of the atom was like the sun at the center of the solar system, while the electrons circling the nucleus were like the planets circling the sun. However, in quantum mechanics, electrons are thought of more like clouds rather than planets. Rather than "circling" the nucleus confined to orbits, electrons can seem to be everywhere at once, like a fog.

The fact that Bohr's model worked as well as it did for hydrogen is actually quite remarkable! Of course, Bohr's fictional "solar system atom" wasn't a random guess. Bohr had actually thought quite a bit about what might be going on inside the atom, and his work marked the first major step towards understanding where electrons are found in the atom. Therefore, despite the fact that the Bohr model wasn't entirely correct, Niels Bohr was awarded a Nobel Prize for his theory in 1922. It turns out that a complete description of the atom and atomic spectra, requires an understanding of quantum physics. Quantum physics describes a bizarre world that behaves according to rules which only apply to very, very small objects like electrons. Back when Bohr developed his model of the atom, scientists had never heard of quantum physics. In fact, most scientists at the beginning of the 20th century thought that everything could be described using classical physics. Even today, scientists still don't fully understand quantum physics and the world that quantum physics describes.

## Lesson Summary

• Niels Bohr suggested that electrons in an atom were restricted to specific orbits around the atom's nucleus.
• Bohr argued that an electron in a given orbit has a constant energy, thus he named these orbits energy levels.
• When an electron gains energy (from an electric current or an electric spark), it can use this energy to jump from a lower energy orbit (closer to the nucleus) to a higher energy orbit (farther from the nucleus).
• When an electron falls from a higher energy orbit (farther from the nucleus) to a lower energy orbit (closer to the nucleus) it releases energy in the form of light.
• White is not a color of light itself, but rather, results when light of every other color is mixed together
• In Bohr's model, electrons can only exist in certain orbits and thus, can only have certain energies. As a result, we say that the energies of the electrons are quantized.
• Bohr used the formula $E_n = \tfrac{-Rhc}{n^2}$ to predict he energy level of an electron in the nth energy level (or orbit) of a hydrogen atom.
• Because, the electron is only allowed to exist at certain energy levels according to the Bohr model, there are only a few possible energies of light which can be released when electrons fall from one energy level to another. As a result, the Bohr model explains why atomic spectra are discontinuous.
• The Bohr model successfully predicts the four colored lines in hydrogen's atomic spectrum, but it fails miserably when applied to any atom with more than one electron. This is due to the differences between the laws of classical physics and the laws of quantum physics.
• The Bohr model is no longer accepted as a valid model of the atom.

## Review Questions

1. Decide whether each of the following statements is true or false:
(a) Niels Bohr suggested that the electrons in an atom were restricted to specific orbits and thus could only have certain energies.
(b) Bohr's model of the atom can be used to accurately predict the emission spectrum of hydrogen.
(c) Bohr's model of the atom can be used to accurately predict the emission spectrum of neon.
(d) According to the Bohr model, electrons have more or less energy depending on how far around an orbit they have traveled.
2. According to the Bohr model, electrons in an atom can only have certain, allowable energies. As a result, we say that the energies of these electrons are _______.
3. The Bohr model accurately predicts the emission spectra of atoms with…
(a) less than 1 electron.
(b) less than 2 electrons.
(c) less than 3 electrons.
(d) less than 4 electrons.
4. Consider an He+ atom. Like the hydrogen atom, the He+ atom only contains 1 electron, and thus can be described by the Bohr model. Fill in the blanks in the following statements.
(a) An electron falling from the n = 2 orbit of He+ to the n = 1 orbit of He+ releases ______ energy than an electron falling from the n = 3 orbit of He+ to the n = 1 orbit of He+.
(b) An electron falling from the n = 2 orbit of He+ to the n = 1 orbit of He+ produces light with a ______ wavelength than the light produced by an electron falling from the n = 3 orbit of He+ to the n = 1 orbit of He+.
(c) An electron falling from the n = 2 orbit of He+ to the n = 1 orbit of He+ produces light with a ______ frequency than the light produced by an electron falling from the n = 3 orbit of He+ to the n = 1 orbit of He+.
5. According to the Bohr model, higher energy orbits are located (closer to/further from) the atomic nucleus. This makes sense since negative electrons are (attracted to/repelled from) the positive protons in the nucleus, meaning it must take energy to move the electrons (away from/towards) the nucleus of the atom.
6. According to the Bohr model, what is the energy of an electron in the first Bohr orbit of hydrogen?
7. According to the Bohr model, what is the energy of an electron in the tenth Bohr orbit of hydrogen?
8. According to the Bohr model, what is the energy of an electron in the seventh Bohr orbit of hydrogen?
9. If an electron in a hydrogen atom has an energy of −6.06×10−20 J, which Bohr orbit is it in?
10. If an electron in a hydrogen atom has an energy of −2.69×10−20 J, which Bohr orbit is it in?
11. If an electron falls from the 5th Bohr orbital of hydrogen to the 3rd Bohr orbital of hydrogen, how much energy is released (you can give the energy as a positive number)?
12. If an electron falls from the 6th Bohr orbital of hydrogen to the 3rd Bohr orbital of hydrogen, what wavelength of light is emitted? Is this in the visible light range?

## Vocabulary

Bohr energy level
Distinct energies corresponding to the orbits (or circular paths) of electrons around the atomic nucleus, according to Bohr's model of the atom.
Bohr model of the atom
Bohr's explanation of why elements produced discontinuous atomic spectra when struck by an electric current. According to this model, electrons were restricted to specific orbits around the nucleus of the atom in a solar system like manner.
classical physics
The laws of physics that describe the interactions of large objects.
quantum mechanics
The laws of physics that describe the interactions of very small (atomic or subatomic) objects. Also known as "wave mechanics" and "quantum physics".