# Workgroup 5: Translation Partition Function

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- 57470

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The partition function for a single atom or molecule can be expressed

\[q(V ) = \sum_i e^{−\epsilon_i/k_BT}\]

where \(\epsilon _i\) describes the energy of the atom or molecule (e.g., derived from solving the Schrödinger equation for a specific system).

## Q1

From out discussions of objects in three dimensional space, how many degrees of free does a single atom have? What degrees of freedom does a molecule have (there are two limiting cases)? List type and number in both cases. Each degree has a characteristic energy. Atoms and molecules also have electronic energy. How do the types you wrote and electronic energies compare for atoms and molecules?

## Q2

We can model translational motion as a particle in a box quantum problem. Recall that the energy of a particle in a 1-D box (PIB) is

\[\epsilon_n = \dfrac{n^2h^2}{8mL^2}\]

What is \(n\) and what is \(L\) in the PIB energy expression (and possible limitations on the vlaues of either)?

## Q3

This energy expression for the PIB given in Q2 is not appropriate to describe the atom in a volume, V, because it is one dimensional. What is the expression for a particle in a 3D box? Make it easy for yourself and make the box a cube.

## Q4

Substitute your expression for the particle in a cubic box into the definition of the molecular partition function given above. Remembering that \(e^{x+y} = e^xe^y\), how can you simplify your expression?

## Q5

Now consider that you cannot tell motion in the x direction from that in the y direction from that in the z direction. How can this simplify your expression for \(q\)?

## Q6

Now you should have an expression that contains a summation over values of the quantum states for the atom in the box. If the box is a one liter in volume (1 dm^{3}), what is the difference in energy between the lowest two energy levels of the atom in the box? How big is this number compared to \(h\), Planck's constant?

## Q7

Convert your summation expression for \(q\) into an integral. Given that

\[ \int_o^{\infty} e^{-an^2} dn = \sqrt{\dfrac{\pi}{4a}}\]

what is the value of \(q\) in this case after solving the integral. This is the general form for the translation partition function (hint: the limits of integration may need to be approximated to different values).

## Q8

Calculate the translational partition function of an \(Br_2\)

molecule at 300 K. Assume the V is 1 liter.