But first, let's actually take a more careful look at this sum.
So, It's relatively easy to write down, there's a few imposing Greek letters and
Roman letters, but otherwise not, not so complicated.
However, in order to evaluate that sum, well there's no closed form that we can
just jot down an answer for what that sums to as n goes from one to infinity.
However, the translational energy levels, you'll recall, are very closely spaced
one to another. And calculus as a field was basically
born, when people tried to figure out how to do sums over densely spaced well in
this case I'll call them levels. So, given that kind of dense spacing that
sum can be transformed to an integral. Alright, that is a continuous function
instead of a sum. So, if I do that I would write my
partition function, and this is equivalent to moving from a quantum
mechanical system to a classical system. In a classical system of course you would
view all translational energies are possible.
Its a little odd actually to think that only certain translational energies are
possible. That's quantum mechanics for you its
always a little bit odd. But they're so dense that its very nearly
classical, and we can make this approximation.
So, then what I want to evaluate is the cube of not the sum, but the integral
from 1 to infinity. The index is n, so I'll go over an
infinite test mode dn, e to the minus beta h squared n squared over 8ma squared
right the, the same argument but now inside an integral instead of a sum.
Well, it turns out that that integral, this one here on the left, that's
actually not any easier to solve than the sum itself.
However, if all we do is make a pretty small change, let's change the bottom
index on the integral, this is a definite integral, let's change it from 1 to 0.
In that case if you look in a integral table you will discover that integrals
from 0 to infinity of the form dne to the minus alpha m squared, and that's called
a gaussian function. So, gaussians appear in many places in
science, and it turns out that integral has a nice analytic solution.
It is the square root of pi over 4 Alpha. Right, so, nice and straightforward, easy
to write down. So, in our case, that that integral table
alpha is equal to everything that multiplies n squared.
That is, h squared divided by 8ma squared k t, and so when we plug in for that
integral. We get 8 pi ma squared k t all divided by
4 h squared to the 1/2 power. And so just in case I, I went a little
fast here on something, I did transform along the way.
Here's beta, remember beta's 1 over k t. So, I just write k t down the
denominator, I'm going to find it a little more convenient to look at the k
and the t. And so if this was alpha, I need to have
alpha in the denominator of this expression.
So, I take this whole thing and everything that was in the denominator
here will go up to the numerator, and sure enough there's the 8 and the m and
the a squared and the k t and the pie sticks around from this.
Meanwhile this h squared went into the denominator, and here's the 4, alright.
So, just fatefully plugging in the appropriate values given our integral.
So, in order then to get the translational partition function that was
simply the cube of this integral. And so, I'll cube this expression, and
when I do that I'll get the whole thing to the 3/2 power instead of the 1/2
power. And I'll take an 8 divided by a 4, and
I'll just replace that with a 2. And the last thing I'll do, is I'll
notice, here I had a squared, all to the 1/2 power, so that's just a.
So, when I cube it I'll get a cubed. And what is a cubed?
a cubed is the volume of the box we were solving, the particle in a box equation
for. So, I'll just pull that out to really
emphasize, here's where the volume dependence comes in to the translational
partition function. So, it's this expression, which depends
on the mass of the atom, boltzmann's constant, temperature, plung's constant,
and volume. So, just keep in mind then, that the
reason there is a volume is that the side of the box dictates the allowed energy
levels. The actual choice of this volume is part
of a so called standard-state convention. So, we would get different values for the
translational partition function if we chose different volumes for the particle
in a box. And so, thermodynamicists just get
together every few years in a meeting somewhere on earth, I suppose.
And they decide that listen, I'll report all my values for a certain size box, if
you do the same size box for your values. And that way we'll always be comparing
apples to apples. And so that is a standard state
convention to choose a particular size for your box.