Chap20 solutions

Physical Chemistry

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20 Statistical thermodynamics: the machinery Solutions to exercises Discussion questions E20.1(b) The symmetry number, σ , is a correction factor to prevent the over-counting of rotational states when computing the high temperature form of the rotational partition function. An elementary interpretation of σ is that it recognizes that in a homonuclear diatomic molecule AA the orientations AA and A A are indistinguishable, and should not be counted twice, so the quantity q = kT /hcB is replaced by q = kT /σhcB with σ = 2. A more sophisticated interpretation is that the Pauli principle allows only certain rotational states to be occupied, and the symmetry factor adjusts the high temperature form of the partition function (which is derived by taking a sum over all states), to account for this restriction. In either case the symmetry number is equal to the number of indistinguishable orientations of the molecule. More formally, it is equal to the order of the rotational subgroup of the molecule. E20.2(b) The temperature is always high enough (provided the gas is above its condensation temperature) for the mean translational energy to be 3 2 kT . The equipartition value. Therefore, the molar constant- volume heat capacity for translation is C T V, m = 3 2 R . Translation is the only mode of motion for a monatomic gas, so for such a gas C V, m = 3 2 R = 12 . 47 J K 1 mol 1 : This result is very reliable: helium, for example has this value over a range of 2000 K. When the temperature is high enough for the rotations of the molecules to be highly excited (when T θ R ) we can use the equipartition value kT for the mean rotational energy (for a linear rotor) to obtain C V, m = R . For nonlinear molecules, the mean rotational energy rises to 3 2 kT , so the molar rotational heat capacity rises to 3 2 R when T θ R . Only the lowest rotational state is occupied when the temperature is very low, and then rotation does not contribute to the heat capacity. We can calculate the rotational heat capacity at intermediate temperatures by differentiating the equation for the mean rotational energy (eqn 20.29). The resulting expression, which is plotted in Fig. 20.9 of the text shows that the contribution rises from zero (when T = 0) to the equipartition value (when T θ R ). Because the translational contribution is always present, we can expect the molar heat capacity of a gas of diatomic molecules ( C T V, m + C R V, m ) to rise from 3 2 R to 5 2 R as the temperature is increased above θ R . Molecular vibrations contribute to the heat capacity, but only when the temperature is high enough for them to be significantly excited. The equipartition mean energy is kT for each mode, so the maximum contribution to the molar heat capacity is R . However, it is very unusual for the vibrations to be so highly excited that equipartition is valid and it is more appropriate to use the full expression for the vibrational heat capacity which is obtained by differentiating eqn 20.32. The curve in Fig. 20.10 of the text shows how the vibrational heat capacity depends on temperature. Note that even when
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