Chap19 solutions

Physical Chemistry

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19 Statistical thermodynamics: the concepts Solutions to exercises Discussion questions E19.1(b) Consider the value of the partition function at the extremes of temperature. The limit of q as T approaches zero, is simply g 0 , the degeneracy of the ground state. As T approaches infinity, each term in the sum is simply the degeneracy of the energy level. If the number of levels is infinite, the partition function is infinite as well. In some special cases where we can effectively limit the number of states, the upper limit of the partition function is just the number of states. In general, we see that the molecular partition function gives an indication of the average number of states thermally accessible to a molecule at the temperature of the system. E19.2(b) The statistical entropy may be defined in terms of the Boltzmann formula, S = k ln W , where W is the statistical weight of the most probable configuration of the system. The relation between the entropy and the partition function is developed in two stages. In the first stage, we justify Boltzmann’s formula, in the second, we express W in terms of the partition function. The justification for Boltzmann’s formula is presented in Justification 19.6. Without repeating the details of this justification, we can see that the entropy defined through the formula has the properties we expect of the entropy. W can be thought of as a measure of disorder, hence the greater W , the greater the entropy; and the logarithmic form is consistent with the additive properties of the entropy. We expect the total disorder of a combined system to be the product of the individual disorders and S = k ln W = k ln W 1 W 2 = k ln W 1 + k ln W 2 = S 1 + S 2 . In the second stage the formula relating entropy and the partition function is derived. This derivation is presented in Justification 19.7. The expression for W , eqn 19.1, is recast in terms of probabilities, which in turn are expressed in terms of the partition function through eqn 10. The final expression which is eqn 19.34 then follows immediately. E19.3(b) Since β and temperature are inversely related, strictly speaking one can never replace the other. The concept of temperature is useful in indicating the direction of the spontaneous transfer of energy in the form of heat. It seems natural to us to think of the spontaneous direction for this transfer to be from a body at high T to one at low T . In terms of β , the spontaneous direction would be from low to high and this has an unnatural feel. On the other hand, β has a direct connection to the energy level pattern of systems of atoms and molecules. It arises in a natural, purely mathematical, manner from our knowledge of how energy is distributed amongst the particles of our atomic/molecular system. We would not have to invoke the abstract laws of thermodynamics, namely the zeroth and second laws in order to define our concept of temperature if we used β as the property to indicate the natural direction of heat flow.
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