Boltzmann Partition Function of H_2

[transmini]

Homework Statement

In the real world, most oscillators are not perfectly harmonic. For a quantum oscillator, this means that the spacing between energy levels is not exactly uniform. The vibration levels of an $H_2$ molecule, for example, are more accurately described by the approximate formula
$$ E_n \approx \epsilon(1.03n-0.03n^2), \space\space n=0, 1, 2, ...,$$
where $\epsilon$ is the spacing between the two lowest levels. Thus, the levels get closer together with increasing energy. (This formula is reasonably accurate only up to about $n=15$; for slightly higher $n$ it would say that $E_n$ decreases with increasing $n$. In fact, the molecule dissociates and there are no more discrete levels beyond $n\approx15$.) Use a computer to calculate the partition function, average, energy, and heat capacity of a system with this set of energy levels. Include all levels through $n=15$, but check to see how the results change when you include fewer levels. Plot the heat capacity as a function of $\frac{kT}{\epsilon}$. Compare to the case of a perfectly harmonic oscillator with evenly spaced levels, and also to the vibrational portion of the graph in Figure 1.13.

Homework Equations

Partition Function: $Z = \sum_n \space e^\frac{-E_n}{kT}$
Average Energy: $\bar{E} = \sum_n \space E_ne^\frac{-E_n}{kT}$
Heat Capacity: $\frac{\partial\bar{E}}{\partial T}$

The Attempt at a Solution

First I tried to determine what $\epsilon$ was but just found $\epsilon=E_1-E_0=1-0=1$ but I don't feel like this could be right, because if $\epsilon$ was just $1$, why would it be there at all. But rolling with this, I wrote some code in Matlab to find $Z \approx 1$ and $\bar{E} \approx 1.61x10^{-17} eV$, but the average energy doesn't make sense either, because it's 17 orders of magnitude lower than the first nonzero energy level. Then I didn't type anything for the heat capacity yet, but tried to derive another formula:
$$C = \frac{\partial\bar{E}}{\partial T} = \frac{\partial}{\partial T}(\frac{1}{Z}\sum_n \space E_ne^{-\beta E_n})$$
$$C = \frac{-1}{Z^2}\frac{\partial Z}{\partial T} \sum_n \space E_ne^{-\beta E_n} + \frac{1}{Z}\sum_n \space (-E_n^2e^{-\beta E_n} \frac{\partial\beta}{\partial T})$$
$$\frac{\partial Z}{\partial T} = \frac{\partial}{\partial T} (\sum_n \space e^{-\beta E_n}) = \sum_n \space (-E_ne^{-\beta E_n} \frac{\partial\beta}{\partial T})$$
$$\frac{\partial\beta}{\partial T} = -kT^2$$
$$C = \frac{-1}{Z^2}kT^2(\sum_n \space E_n e^{-\beta E_n})^2 + \frac{1}{Z}kT^2 \sum_n \space E_n^2 e^{-\beta E_n}$$
I haven't tried coding this out yet, since I'm not even sure if the formula is correct.

So my questions here are:
How do I find $\epsilon$ if I do not have it correct already
If my values for $Z$ and $\bar{E}$ are incorrect, is it something with my formulas, $\epsilon$, or should I double check my code?
Is the formula I have for $C$ a correct derivation from the starting equation?

 

[kuruman]

It's probably easier to start with$$<E>=-\frac{\partial \ln(Z)}{\partial \beta}$$ then use $$C=\frac{\partial <E>}{\partial \beta}\frac{\partial \beta}{\partial T}$$
As far as ε is concerned, remember that the problem asks you to plot the specific heat as a function of kT/ε which is a dimensionless quantity. Therefore, you don't need a value for it. That should also resolve your 17 orders of magnitude issue.

Edit: Corrected erroneous expression for $<E>$.

 

[transmini]

It's probably easier to start with$$<E>=-\frac{\partial Z}{\partial \beta}$$ then use $$C=\frac{\partial <E>}{\partial \beta}\frac{\partial \beta}{\partial T}$$
As far as ε is concerned, remember that the problem asks you to plot the specific heat as a function of kT/ε which is a dimensionless quantity. Therefore, you don't need a value for it. That should also resolve your 17 orders of magnitude issue.

But for the $\epsilon$, along with the plot, it also was asked to calculate the partition function. Doesn't this imply you need a specific value of $\epsilon$ and $T$? Or is it asking just for the general form of the function here, in which the values do not matter? Also, isn't $<E> = -\frac{\partial}{\partial\beta}ln(Z)$?

 

[kuruman]

But for the ϵ\epsilon, along with the plot, it also was asked to calculate the partition function. Doesn't this imply you need a specific value of ϵ\epsilon and TT? Or is it asking just for the general form of the function here, in which the values do not matter?

You are not given any numbers. You are expected to find expressions for Z and <E> and C in terms of parameter y = kT/ε which is dimensionless therefore works for any ε and T.

Also, isn't $$<E>= -\frac{\partial}{\partial\beta}ln(Z)?$$

Yes, sorry, I forgot the "ln". I edited the post to correct the error. Thanks for pointing out.

 

[transmini]

You are not given any numbers. You are expected to find expressions for Z and <E> and C in terms of parameter y = kT/ε which is dimensionless therefore works for any ε and T.

Yes, sorry, I forgot the "ln". I edited the post to correct the error. Thanks for pointing out.

Okay, thank you. That's just poor wording in my opinion then since it says to "calculate with a computer" when there's nothing to technically calculate. I'll leave this open until next week when I get the solutions back.

 

[kuruman]

Okay, thank you. That's just poor wording in my opinion then since it says to "calculate with a computer" when there's nothing to technically calculate.

Sure there is. Note that the partition function is dimensionless and can be cast in terms of a single parameter y = kT/ε. See if you can find an algebraic expression for the specific heat in terms of y only. You can then use a computer to tabulate C for a set of y values and plot as a function of y.