# Groupwork 2 Translational Partition Functions

- Page ID
- 63418

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

*Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.*

The partition function for a single atom or molecule can be expressed \(q(V)=\sum_{i}e^{-\epsilon_{i}/k_{B}T}\) , where *ε* describes the energy of the atom or molecule.

From quantum mechanics, what different kinds of energy can an atom have? What different kinds of energies can a molecule have?

If we consider only translational motion for an atom in a volume, *V*, we could model this as a particle in a box. Recall that the energy of a particle in a box (PIAB) is \(\epsilon_n=\frac{n^2h^2}{8ma^2}\). What is *n* and what is *a* in the PIAB energy expression?

This energy expression 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.

Substitute your expression for the particle in a cubic box into \(q(V)=\sum_{i}e^{-\epsilon_{i}/k_{B}T}\) . Remembering that \(e^{x+y}=e^xe^y\), how can you simplify your expression for \(q(V)=\sum_{i}e^{-\epsilon_{i}/k_{B}T}\)?

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*?

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?

BONUS: Convert your summation expression for *q* into an integral. Given that \(\int_{0}^{\inf}e^{-an^2}dn=(\frac{\pi}{4a})^{\frac{1}{2}}\), what is the value of *q*? This is the translational partition function.