Niels Bohr: How He Derived His Model Mathematically

I just made a video about the personal history of Niels Bohr and how he made his model and when I was done, I was asked to make a video showing mathematically in detail how Bohr derived the position, energy, and, most importantly, the frequency of light created for an electron in an atom.  So here you go.  Ready? Let’s go!

Before I start, I would like to give a tiny backstory.  I often use the word quantize, which is defined as “to restrict (a variable quantity) to discrete values rather than to a continuous set of values.”  This whole idea of restricting things to discrete values started in 1900 when Max Planck restricted the energy of light to be in little energy packets where the energy equaled a constant (h) times the frequency of light (given the Greek letter v).  11 years later Ernest Rutherford had a theory that the majority of an atom is smashed into a tiny nucleus which was assumed to be positive to counter the electrons which are negative.  Rutherford didn’t know what that meant about the electrons, maybe they were evenly distributed in the atom, maybe they were in a big ring around the atom-like Saturn, maybe they were in many rings.  Rutherford didn’t know and wasn’t particularly concerned about it.  Niels Bohr was attempting to mesh these two ideas when he was reminded of the equation for the possible frequencies of light produced by burning hydrogen, called the Balmer formula and, as he recalled years later, “as soon as I saw Balmer’s formula, the whole thing was immediately clear to me.[i]”  He also found that his new formula could be used to explain strange shadows in a star as being from ionized helium.  So, let me explain how he accomplished this.  

Niels Bohr
Niels Bohr

What’s in here?

1) Tiny Backstory which I just did

2) Bohr’s assumptions,

3) why Bohr made his assumptions, and

4) the math

5) Conclusions for Hydrogen and Helium +!

Part 2: Bohr’s assumptions

Bohr made 6 assumptions in his July 1913 paper

  1. In his first paper, he only dealt with systems with one electron
  2. The Electron spins around a nucleus in a stable orbit which assumed to be a circle
  3. The electron can only circle around the nucleus at set positions, which Bohr called “stationary states” although it can “magically” jump between the rings in a quantum leap (Bohr did not use the word magic)
  4. The energy of light produced is equal to the change in energy of the electron as it jumps between states
  5. All Classical laws of physics apply with three exceptions: 1) he knew that in classical physics objects just don’t jump between positions, 2) in classical physics a rotating charged particle will radiate energy and spiral into the nucleus but Bohr just basically said, “not for my electrons” and 3) in classical physics vibrating electrons create light due to their vibrations, in this theory the light created is due to the change in energy!
  6. The position of the electron is limited because its energy is quantized with the rule that

the work needed to remove the electron (i.e. the energy binding the electron to the atom) equals an integer times Planck’s constant times the frequency of the electron’s spin divided by 2! 

Part 3: Why Did Bohr Make These Assumptions?

1-2)

Bohr started with a single electron moving in a circle because he wanted to be as simple as possible.  He also wanted to derive Balmer’s formula from basic principles, and hydrogen only has one electron.  Note that he did not assume that the nucleus only had one proton so that his theories would work for hydrogen and for helium + (helium missing one electron) or Li ++ (lithium missing two electrons) etcetera.

3-5)

 Bohr knew from reading the debates and talks from the 1911 Solvay Conference that, at this time, quantum mechanics consisted of following the physics rules for almost everything but then adding a quantum twist so that, say, the energy was quantized.   Therefore, it was not controversial to quantize the energy which quantized the position of the electron.  What Bohr did was say, “Hey, if I can break one rule, why not break a few more?” so he also made a rotating electron not produce light and he separated the energy produced by the vibrating light from the energy of the electron.  Stating that the energy of radiation is unconnected to the energy of the electron was more radical, and was what shocked several people including Einstein.  However, this is also what has endured from Bohr’s model, we still think that electrons radiate light with an energy equal to the *change* in the energy of the electron!

6)) Back in 1900 Planck said that the energy from light was quantized so that E = nhv (where n is an integer, h is Planck’s constant, and v is the frequency of light).  Bohr disassociated the energy of the light from the energy of the electron so he could make any limitation he wanted on the energy of the electron.  He then set the limit that the energy needed to remove an electron from the atom W, to be =nhw/2 (where w is the frequency of rotation of the electron).  Which begs the question, why divide by 2?  Well, in the paper Bohr made an argument about averaging the work from the state of no motion which leads to the 2.  He also might have divided by 2 because then the equations work out well for hydrogen.  It turns out that for a circular orbit, this limitation is equivalent to quantizing the angular momentum, so that mvr = nh/2Pi, but this was a conclusion, not his initial proposal as I will show you right now. 

The Bohr Model

Part 4: OK Let’s Get to the Math!

Let’s start with an electron of mass m, and charge e moving in a circle around a nucleus (of charge Ze), at a radius r and constant speed v.  The only significant force on the electron is electrical, so using Coulomb’s law we get that the electrical force is Coulomb’s constant, k, times the charge of the electron, e, times the charge of the nucleus, Ze, divided by the distance squared, which can be simplified to be Z times Ke squared divided by r squared.  Now, according to Newton’s law force equals mass times acceleration and an object moving in a circle at a constant speed has an acceleration is v2/r.  Let’s call this equation (1). 

The work needed to remove the electron from the atom, which we will label W, is the potential electrical energy holding the electron in the atom (which is Coulomb’s constant times the two charges divided by the distance) minus the electron’s kinetic energy (1/2 mv2).  If you multiply equation (1) by r, you get that Zke2/r = mv2 or that the potential energy equals twice the kinetic energy.  Since 1 mv2 – ½ mv2 = ½ mv2, W can be reduced to 1/2mv2 which is the same as ½ the potential energy or 1/2kZe2/r.  Let’s call this equation (2)

These two equations are as far as you can go without quantum restrictions.  As I stated earlier, Bohr restricted his atom so that the work needed to remove an electron was quantized to equal nhw/2, where n is an integer, h is Planck’s constant, and w is the frequency of rotation.  Plugging that into equation 2 and multiplying by 2, we get that mv2 = nhw.  Now we need an expression for w, the frequency of rotation.  If the electron is moving in a circle at a constant rate, then the velocity is the distance over time, where the distance is the circumference or 2 pi times the radius, and the frequency is 1 over time.  Dividing by 2 pi r, we get that the frequency = v/2pr, plugging that into the equation above and dividing one v and multiplying by r and you get that the classic angular momentum: mvr = nh/2pi (this is where most derivations start and let’s call it equation 3).  Now we are finally ready to solve the problem!

Part 5: Get to the Results!

First, we are going to solve for the possible position of the electron, i.e. the allowed radii.  Let us start with equation 1.  Multiply both sides by r squared and then divide both sides by Zke2.  Finally, multiply the right side by m/m.  The numerator is the angular momentum, mvr, squared, which, using equation 3 with nh/2pi to get r = n2h2/(4pi2kZe2m). 

I would like to rearrange this equation by taking n and Z out, so we get r = n2/z (h2/4pi2e2mk)

Or r = n2/Z * ao where a0 = h2/4 pi2e2mK.  Since a0 is a constant, we can plug and chug and it turns out that ao = 0.53 Angstroms.  This, by the way, is called the Bohr radius (let’s call this equation 4).  For example, in the hydrogen atom Z = 1, so the possible radii are 1a0 or 4a0 or 9a0 or 16a0 or… you get the idea.

[Side note, I use the word naught to represent the zero next to a, I am using naught as the old fashioned English word for nothing as in “he is naught but a worthless fool” not NOT as in “I am NOT giving you candy” or knot as in tying a knot.  That was what I was taught and I figured you might want to know where it comes from as you might hear it from other teachers]

We are now in a position to find the Work needed to remove an electron from an energy level, n.  Plugging our new definition of the radius into equation 2, we get W = k Ze2*Z/(2n2a0) (rearranging so that the Zs and ns are in front gives us, Z2/n2 (ke2/2a0) = Z2/n2 (2pi2k2e4m/h2) plugging numbers in Bohr got that the Work is Z2/n2 times 13.6 eV which is still the prediction today.  This means that for hydrogen where Z = 1, the lowest state (n=1), the energy is -13.6 eV (or it takes 13.6 eV to remove an electron) and the next state is -13.6/4 = -3.4 eV and the next is -13.6/9 = 1.51 eV and so on. 

More dramatically, we are also in a position to determine the possible energies of light produced by an electron when it jumps from one shell to another.  Bohr predicted that the light gets energy from the electron losing energy when it falls from one shell to another, na -> nb.  Which would be 13.6 eV Z^2 (1/nb2-1/na2)!  If, as Planck postulated, the light has an energy of hv, where v is the frequency of the light then the possible frequencies of light would be 13.6 eV*Z^2/h (1/nb2-1/na2) we can rewrite this as Z2 times R (1/nb2-1/na2) where R = 13.6 eV/h.  Now, for the hydrogen atom, Z = 1 and this is exactly, EXACTLY, the empirical equation (or the equation from the experiment without theoretical backing) called the Balmer series for the frequencies of light from burning hydrogen, where R is called the Rydberg constant.  Bohr used the values for e, h, m, and k as they were known at the time and found a Rydberg constant of 3.1 x 10^15 Hz which was within error of the accepted value at the time of 3.29 x 10^15 Hz.  

But Bohr had another trick up his sleeve.  After he learned about the Balmer series he started to read about different equations for different spectrums and he found this unusual spectrum called the Pickering series named after the boss of the woman (Williamina Fleming) that discovered that a star had an unusual pattern where it looked like it had half of the hydrogen spectrum that looked like the Balmer series with half integers.  In 1912, the year before Bohr published his paper, a scientist named Alfred Fowler reproduced these lines with a mixture of hydrogen and helium in a laboratory which he attributed to hydrogen or “protohydrogen” with half integers.  Niels Bohr thought that maybe his model could solve the mystery.  He realized that the Z2 in the equation for the possible frequencies meant that the resulting frequencies from a helium+ atom (or a helium atom with two protons but only one electron) would look exactly like a hydrogen atom with half integers!  This was the fact that really convinced people that Bohr’s model was saying something important.

Interestingly, Alfred Fowler, the scientist who had made the experiments with hydrogen and helium, was not convinced and wrote that his new lines were not exactly equivalent to multiplying the Rydberg constant by 4, they were equivalent to multiplying it by 4.0016.  This objection actually made Bohr realize that he had made a small mistake in his calculations.  He had assumed that the nucleus was completely stationary, however, even though the nucleus is far heavier than the electron, even for a hydrogen atom, the nucleus should still wobble a little bit which changes the energy.  Bohr knew from planetary physics that you need to deal with the effective mass defined as m = m1m2/(m1 +m2).  He then showed that with the new effective mass the Rydberg constant is multiplied by 4.00163!          

Bohr atom model

Currently, we *define* the Rydberg constant from the constants that Bohr derived to be 2 pi2 k2 e4 m/h3.  (by the way, lots of scientists like using the permittivity of free space epsilon 0 instead of Coulomb’s constant, k, where k = 1/(4pi epsi0), so that the Rydberg constant would be e4m/(8 epsi02h3)).  Because of the order that Bohr derived everything, we often derive the Rydberg constant for an infinitely big nucleus with no wabble, and then adjust it with the effective mass for the nucleus we are dealing with.    

So, there you go, that is how Bohr derived his equations.  If you are interested in the personal history of why Bohr wrote his paper, check out my first video on the subject, far less math but more context.  Thanks!

Bohr mostly used modern notation with a few exemptions.  He used the Greek letter t for the quantum number and I prefer n, as do most textbooks.  He used “a” for the radius and I prefer r.  But the biggest difference is that Bohr used capital E for the charge of the nucleus and modern textbooks use Z times the charge of an electron or proton, e, where Z is the atomic number meaning the number of protons in the nucleus.  Remember that at the time, it was just 2 years after the idea of a nucleus, and no one knew about atomic numbers although Bohr was pretty sure that hydrogen’s nucleus had a charge of 1e and helium had a charge of 2e. 


[i] Niels Bohr Oral History (October 31, 1962) American Institute of Physics – Session III

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