Discuss the below:
Q: We consider a classical ideal gas of N particles of mass m which are independent of each other and are in a volume V. The particles are in an external potential which has a value ε in a part volume v < V and the value 0 in the remaining area V - v. During compression only the volume V changes, i.e. v is constant.
a) Calculate the canonical partition function
Z(T,V,N) = (1/(h^3N)N!) ∫ ... ∫ dx1 ... dxN dp1 ... dpN exp(-βE(x1,...,xN;p1,...,pN))
From this derive expressions for the free energy F and the internal energy E.
b) Calculate the pressure p(T,V,N). What is the equation of state of the system?
c) Calculate the grand canonical partition function Zg(T,V,μ) = Σ Z(T,V,N)exp(-μN), where the sum ranges over N from 0 to infinity
d) Calculate the average particle number as a function of T and μ. From this determine the pressure as a function of , T, V and v. Compare this result wíth the result from c)