## Pathria chapter 4 diatomic molecule problem

Posted by peeterjoot on March 18, 2013

## Question: Diatomic molecule ([1] pr 4.7)

Consider a classical system of non-interacting, diatomic molecules enclosed in a box of volume at temperature . The Hamiltonian of a single molecule is given by

Study the thermodynamics of this system, including the dependence of the quantity on .

## Answer

**Partition function**

First consider the partition function for a single diatomic pair

Now we can make a change of variables to simplify the exponential. Let’s write

or

Our volume element is

It wasn’t obvious to me that this change of variables preserves the volume element, but a quick Jacobian calculation shows this to be the case

Our remaining integral can now be evaluated

Our partition function is now completely evaluated

As a function of and as in the text, we write

**Gibbs sum**

Our Gibbs sum, summing over the number of molecules (not atoms), is

or

The fact that we can sum this as an exponential series so nicely looks like it’s one of the main advantages to this grand partition function (Gibbs sum). We can avoid any of the large approximations that we have to use when the number of particles is explicitly fixed.

**Pressure**

The pressure follows

**Average energy**

or

**Average occupancy**

but this is just , or

**Free energy**

**Entropy**

**Expectation of atomic separation**

The momentum portions of the average will just cancel out, leaving just

Expanding the numerator by parts we have

This gives us

# References

[1] RK Pathria. *Statistical mechanics*. Butterworth Heinemann, Oxford, UK, 1996.

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