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Groupwork 2 Translational Partition Functions

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    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 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?

    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.