Question

2. Imagine that the partition function of a system is given by Z), where A is a 3N A V2 constant. Find average energy, pressure, C, equation of state, compressibility, and thermal expansion coefficient. Find the entropy of this system. Is it free of Gibbs paradox?

Answer 1

z = (A v^2/beta N^2)^3 N beta = 1/K T E bar = -1/z partial differential z/partial differential beta = -1/(A v^2/beta N^2)^3 N . 3 N (A v^2/beta N^2)^3 N - 1 - A v^2/N^2 beta^2 E bar = 3 N beta N^2/A v^2 A v^2/N^2 beta^2 E bar = 3 N/beta or E bar = 3 N K T c v = d E bar/dT = > C_v = 3 N K p = -partial differential F/partial differential v = K T partial differential/partial differential v ln z = K T partial differential/partial differential v (3 N ln A v/beta N^2) = k T 3 N 1/A v^2/beta N^2 2 A v/beta N^2 p = 6 N k T/v = > p = 6 N K T/v rho v = 6 N K T rightarrow v = 6 N K T/p Compressibility K_T = -1/v (partial differential v/partial differential p)_T = -1/v partial differential/partial differential p
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