Answer to Question #89307 in Molecular Physics | Thermodynamics for Shivam Nishad

Question

Using the expression for the partition function of an N-particle ideal gas, obtain the expression for its entropy

Answer 1

According to the definition of entropy, we can write

$$S=\int_{ S_0 }^ {S} dS (1)$$

Using the definition of the heat capacity at constant volume for the first differential and the appropriate Maxwell relation for the second we have:

$$ΔS= \int_ { T_0 }^ {T} \frac {C_V} {dT} + \int_ { V_0 }^ {V} \frac {δ_P} {δT}_V dV (2)$$

Expressing C_V in terms of ĉ_V as developed in the above section, differentiating the ideal gas equation of state, and integrating yields:

$$ΔS= ĉVNk \ln {\frac {T} {T_0}} + Nk \ln {\frac {V} {V_0}} (3)$$

Using (3) we get:

$$S= Nk \ln {\frac {VT ^{C_V}} {f(N)}} (4)$$

where all constants have been incorporated into the logarithm as f(N) which is some function of the particle number N having the same dimensions as VT^ĉV in order that the argument of the logarithm be dimensionless.