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124 Chapter 4 translational states can be quite different (the energy transfer between vibrational states is usually orders of magnitude slower than that between translational states). 4-38. Consider a system of independent diatomic molecules constrained to move in a plane, that is, a two-dimensional ideal diatomic gas. How many degrees of freedom does a two-dimensional diatomic molecule have? Given that the energy eigenvalues of a two-dimensional rigid rotator are (where I is the moment of inertia of the molecule) with a degeneracy = 2 for all J except J = 0, = derive an expression for the rotational partition function. The vibrational partition function is the same as for a three-dimensional diatomic gas. Write out and derive an expression for the average energy of this two-dimensional ideal diatomic gas. A two-dimensional diatomic molecule has two translational degrees of freedom, one vibrational degree of freedom, and one rotational degree of freedom. We know that So We are told that is the same for a two-dimensional gas as it is for a three-dimensional gas, and we know for an ideal two-dimensional gas from Problem 4.3. We can now obtain an expression for the average energy of this gas: = We now wish to find the temperature-dependent terms of In q: In q = In + + terms not containing T Now, as in Example 4.5, we can take = 3RT 2 + RO 2 +