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.