Final_Template - Name May 8 2012 Chemistry 120B Final Examination Useful formulas Entropy S as a function of energy E number of particles N and

Final_Template - Name May 8 2012 Chemistry 120B Final...

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Name May 8, 2012 Chemistry 120B Final Examination Useful formulas Entropy, S , as a function of energy, E , number of particles, N and volume, V , S ( E, V, N ) = k B ln W ( E, N, V ) . Probability of ν th microstate for system at temperature T = ( k B β ) - 1 : P ν = e - βE ν /Q, Q = X ν e - βE ν = e - βA , where in macroscopic thermodynamics A = E - TS , with E = h E i = ν E ν P ν and d E = T d S - p d V + X i μ i d N i , p is pressure, μ i is chemical potential of species i , and d A = - S d T - p d V + X i μ i d N i . Maxwell-Boltzmann velocity distribution: φ ( ~v ) exp( - βm | ~v | 2 / 2) . Gibbs-Duhem equation: 0 = S d T - V d p + X i N i d μ i Ideal gas equation of state and chemical potential: βp = ρ = N/V , and μ i = k B T ln ( a 3 i N i /V ) , with a i = microscopic length. Integrals: Z -∞ d x x 2 n exp( - αx 2 ) = ( - 1) n d n d α n r π α Z 0 d x x 2 n +1 exp( - αx 2 ) = ( - 1) n d n d α n 1 2 α 1
Note about this examination This 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 of thermodynamics, to the meaning of equilibrium, mean values, fluctuations and stability, to rates of relaxation and phase transformations. If you’ve learned the material, you will be comfortable 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 proportional to 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 /k B T times 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/T is conjugate to V , and 1 /T is conjugate to E ). 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, if h X i is the mean of an extensive quantity of a system with N molecules, then this mean is of order N , and the root-mean-square fluctuation, h ( X - h X i ) 2 i 1 / 2 is gen- erally of order N . For example, h ( V - h V i ) 2 i = - k B T ( h V i /∂p ) T,N > 0.

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