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Translation Partition Functions (Worksheet)

Translation Partition Functions (Worksheet)

Last updated

Jul 3, 2021
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Name: ______________________________

Section: _____________________________

Student ID#:__________________________

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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 $Br_2$

Q1

Q2

Q3

Q4

Q5

Q6

Q7

Q8

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