Thermodynamics, Statistical Physics, and Quantum Mechanics

136 where we have expressedGas a function of P. The chemical potential so we obtain, from (S.4.39.3), Thisapproximation isvalid ...

137 The equation of state istherefore analogous to thegas in a containerwith rigid walls, where c) The entropy, energy, and heat ...

138 The freeenergy associatedwith thecoordinate-dependentpart of the par- tition function is The averagepotentialenergy is given ...

139 The averageproceeds in precisely the sameway,yielding b) The heat capacity, at constantvolume is equal to Frompart (a), we h ...

140 numeratorequals zero. If is one of the then by theassumption ofU infinite, the term still equals zero. Finally, if then by l ...

141 4.4 3 Diatomic Molecules in Two Dimensions (Columbia) a) The partitionfunction may be calculated in theusual way by multi- p ...

142 A diatomicrotor confined to aplane would havethreedegrees offreedom, two translational andone rotational. Hence, The quantiz ...

143 yielding Then (S.4.44.3)becomes with b) In order to find the conjugatemomenta we must compute the Lagrangian Expressing thro ...

144 Now the free energy FforNsuch classical molecules may be found from The entropy isthen and the energyEand heat capacityCare ...

The energyEand heat capacityCare then So, at low temperatures the heatcapacitycorresponding to the rotational degrees of freedom ...

146 We see that this is the same asfound in (S.4.44.12).Since we expecta heat capacity per degree of freedom of 1/2, we see that ...

goes to 1/2 when goes toinfinity;hence, at b) The entropy may be found from (S.4.45.2)–(S.4.45.4): The entropy per particle, isg ...

148 b) The averagenumber of open links isgiven by whichdoes notdepend onN. It is alsozipped uptight! Hanging Chain (Boston) a) L ...

149 where Therefore, The averagelength is We can check that, if Molecular Chain (MIT, Princeton, Colorado) a) Consider onelink o ...

150 Therefore, as it should, since (for the specified direction of the tensile force ) it corresponds to a thermodynamicinequali ...

THERMODYNAMICS AND STATISTICAL PHYSICS Nonideal Gas Heat Capacities (Princeton) From the definition of for a gas, 151 Since we a ...

(^152) SOLUTIONS b) Let us write the van derWaalsequation for onemole of the gas in the form from which we obtain Substituting f ...

THERMODYNAMICS AND STATISTICAL PHYSICS 153 where we used So, we obtain Substituting into (S.4.50.2) yields b,c) We cannot determ ...

154 SOLUTIONS (since implies Integrating (S.4.50.6) and (S.4.50.7), weobtain where and aresome functions of temperature. Since w ...

THERMODYNAMICS AND STATISTICAL PHYSICS 155 van der Waals (MIT) a) The heat capacity isdefined as By using the Maxwell relation w ...