# Workgroup 5: Translation Partition Function

• • Contributed by Delmar Larsen
• Founder and Director at Libretexts

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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 dm3), 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 $$Br_2$$