Question

verify the maxwell relations associated with the helmholtz free energy F and the Gibbs free energy F by using the equaiton of state an ideal gas and the expressions for S(T,V) and S(P,V) when c_v is constant.

Answer 1

Using the Helmholtz Free Energy Expression,

  

Differentiating above equation,

Using 1st law of thermodynamics in above :

From above equation we can see that Helmholtz free energy is the function of T and V, then we can write

Comparing above two equations we'll get

  ............................................(1)

and ..............................................(2)

Now we can write Helmholtz free energy F as

  

  

Using equation 1 and 2 in above,

  

    FIRST MAXWELL REALTION.

Similarly, using the Gibbs Free Energy Expression,

  

  

Differentiating above equation,

Using 1st law of thermodynamics in above :

From above equation we can see that Helmholtz free energy is the function of T and P, then we can write

Comparing above two equations we'll get

  ............................................(3)

and ..............................................(4)

Now we can write Gibbs free energy G as

  

  

Using equation 3 and 4 in above,

  

  SECOND MAXWELL REALTION.