The Bohr Theory of the Hydrogen Atom: Part II

This post will focus on some equation manipulation regarding how to derive the equations for the radius of the hydrogen atom and the velocity of an electron according to Bohr Theory. Also, we will examine the significance of these findings on the properties of the hydrogen atom. This will be done for the level of a first year university student, and the objective is to make these equations as clear to follow as possible.

The last post discussed how classical physics lacked the ability to describe the discrete spectrum of the atom as well as its set radius. Our man Bohr equated the classical physics equation for a force in circular motion (assuming the electron in the atom maintained a circular orbit), with the force of electric charge, also known as Coulomb's Law. 

             (A)

As an aside, let’s recall some formulae from circular motion, a favourite topic from first year physics. Knowing from Newton’s second law that  and that acceleration in circular motion is given by , clearly .  Also, taking the charge squared per atomic mass unit is defined by Coulomb's Law.

Moving onward, we see from equation (A) that the r cancels. This provides us with:

   (B)

Note that already we can tell some pretty neat things about the energy of the atom from equation (B). Remember that and .  So the kinetic energy is equal to twice the potential energy with opposite sign. Cool!

Now, we factor momentum into the equation. Recall regular good old momentum is defined as  Angular momentum is defined as angular momentum= where r is the position from the origin of the circle. In the last post, we discussed that Bohr quantized angular momentum as  for discrete values of n. You guessed it! We are going to equate the two definitions of angular momentum and see where this leads us.

             (C)

Next, we multiply both sides of the equation (B) byto make the equation into a more workable form. I am going to show all of the steps so things don’t get too hairy.

         (D)

But aha! We recognize equation (D) to be equation (C) squared! We substitute for  obtaining

           (E)

And then we solve for the radius of the atom, r, obtaining

         (F)

Note that  which is a nicer way to express equation (F), however as I can’t figure out how to write many symbols on my mac (not that I would be able to even find them on a windows, not going to lie), you’ll have to write it out for yourself and save a few extra characters.

At last, plug in the values for the the mass of the electron, the electric charge, Planck’s constant, and pi and you obtain the constant 0.530 A. Which means you can compute your problem sets with a much simpler version of equation (F) to punch into the calculator.

                (G)

Sometimes it is rather useful to calculate the velocity of an electron. We do this under Bohr Theory by referring way back to equation (C) and isolating for velocity.

            (H)

Then we substitute in equation (F) for the radius.

But this looks dreadfully ugly, so we take pity and clean it up a bit. First, we take the reciprocal of the denominator and then we collect like terms to see what cancels easily.

And finally, we have simplified for the velocity of the electron of the atom:

           (I)

HURRAH!

Another comment about Bohr’s assumptions is that he assumed the nucleus of the atom was stationary. It is not. To account for the motion of the nucleus something called the reduced mass of the electron is used in the equation. It is given by , but at the first year university level, no one is ever going to expect you to know that. However, no one is ever going to expect you to derive the above either, so just the exposure to it is good.

FIN