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NameMay 8, 2012Chemistry 120B Final ExaminationUseful formulasEntropy,S, as a function of energy,E, number of particles,Nand volume,V,S(E, V, N) =kBlnW(E, N, V).Probability ofνthmicrostate for system at temperatureT= (kBβ)-1:Pν= e-βEν/Q,Q=Xνe-βEν= e-βA,where in macroscopic thermodynamicsA=E-TS, withE=hEi=∑νEνPνanddE=TdS-pdV+XiμidNi,pis pressure,μiis chemical potential of speciesi, anddA=-SdT-pdV+XiμidNi.Maxwell-Boltzmann velocity distribution:φ(~v)∝exp(-βm|~v|2/2).Gibbs-Duhem equation:0 =SdT-Vdp+XiNidμiIdeal gas equation of state and chemical potential:βp=ρ=N/V,andμi=kBTln(a3iNi/V),withai= microscopic length.Integrals:Z∞-∞dx x2nexp(-αx2) = (-1)ndndαnrπαZ∞0dx x2n+1exp(-αx2) = (-1)ndndαn12α1
Note about this examinationThis Final Examination attempts to test your knowledge of the central principles of Chem-istry 120B, a course which could be entitled “The Boltzmann distribution and its impli-cations.”This distribution is the center piece for all we have discussed, from the laws ofthermodynamics, to the meaning of equilibrium, mean values, fluctuations and stability, torates of relaxation and phase transformations. If you’ve learned the material, you will becomfortable thinking about the behaviors of systems with many atoms in terms of statistics,starting with the fact that the equilibrium probability for something to occur is proportionalto the Boltzmann weighted sum over all the micro-states consistent with that occurrence.This partition function, as it is called, is the exponential of-1/kBTtimes the free energy(i.e., reversible work) to create that occurrence.Macroscopic thermodynamics describes the mean values of the Boltzmann distribution.Mean values of fluctuating extensive variables are controlled by the conjugate intensive vari-ables (e.g., considering the entropy differential,p/Tis conjugate toV, and 1/Tis conjugatetoE). For a large enough system, typical fluctuations from the mean of any extensive quan-tity are negligible in comparison to the mean, except at conditions of phase equilibrium.More precisely, ifhXiis the mean of an extensive quantity of a system withNmolecules,then this mean is of orderN, and the root-mean-square fluctuation,h(X- hXi)2i1/2is gen-erally of order√N. For example,h(V- hVi)2i=-kBT(∂hVi/∂p)T,N>0.