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.
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!
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.
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) by
to make the
equation into a more workable form. I am going to show all of the steps so
things don’t get too hairy.
to 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
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.
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.
, 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
