Which of the following is not the Maxwell's equation?

Concept:

The equation that relate partial derivatives of properties of p, v, T and s of a compressible fluid are called Maxwell relations.

The four Gibbsian relations for a unit mass are

1) du = Tds – Pdv

2) dh = Tds + vdP

3) df = - Pdv – sdT

4) dg = -sdT + vdP

Since u,h,f and g are the properties thus point functions and the above relations can be expressed as

dz = Mdx + Ndy

with,

\({\left( {\frac{{\partial T}}{{\partial v}}} \right)_s} = \; + {\left( {\frac{{\partial p}}{{\partial s}}} \right)_v}\)

Applying the cyclic relation as

Mdx + Ndy → \({\left( {\frac{{\delta M}}{{\delta y}}} \right)_x} = {\left( {\frac{{\delta N}}{{\delta x}}} \right)_y}\)

Now,

Replacing M,N,y and x by T,p,v,s of each of the Gibbsian equations in cyclic order, we will get the following four relations.

\(1){\left( {\frac{{\partial T}}{{\partial p}}} \right)_s} = {\left( {\frac{{\partial v}}{{\partial s}}} \right)_p}\;\)

\(2){\left( {\frac{{\partial p}}{{\partial T}}} \right)_v} = {\left( {\frac{{\partial s}}{{\partial v}}} \right)_T}\;\)

\(3){\left( {\frac{{\partial T}}{{\partial v}}} \right)_s} = - {\left( {\frac{{\partial p}}{{\partial s}}} \right)_v}\)

\(4){\left( {\frac{{\partial v}}{{\partial T}}} \right)_p} = - {\left( {\frac{{\partial s}}{{\partial p}}} \right)_T}\)