For a compressible system, V, S, P and T are specific volume, specific entropy, pressure and temperature respectively. As per Maxwell's relation

Explanation:

If there is a relation between x, y & z, then z may be expressed as a function of x & y.

An equation of type dz = Mdx + Ndy  is exact differentials, it can be written as \({\left( {\frac{{\partial M}}{{\partial y}}} \right)_x} = {\left( {\frac{{\partial N}}{{\partial x}}} \right)_y}\)

For a pure substance undergoing a reversible process, these equations hold true:

1. dU = T.ds – p.dv
2. dH = T.ds + v.dp
3. dF = -p.dV – s.dT
4. dG = v.dP –s.dT

They can be written as

1. \({\left( {\frac{{\partial T}}{{\partial v}}} \right)_s} = - {\left( {\frac{{\partial p}}{{\partial s}}} \right)_v}\)
2. \({\left( {\frac{{\partial T}}{{\partial p}}} \right)_s} = \;{\left( {\frac{{\partial v}}{{\partial s}}} \right)_p}\)
3. \({\left( {\frac{{\partial p}}{{\partial T}}} \right)_v} = \;{\left( {\frac{{\partial s}}{{\partial v}}} \right)_T}\)
4. \({\left( {\frac{{\partial v}}{{\partial T}}} \right)_p} = - {\left( {\frac{{\partial s}}{{\partial p}}} \right)_T}\)

These are known as Maxwell’s Equation.

Out of these options, 4 does not match with Maxwell’s Equation.

Other Maxwell relations are:

if x =  ϕ(y, z)

\(dx = {\left( {\frac{{\partial x}}{{\partial y}}} \right)_z}dy + {\left( {\frac{{\partial x}}{{\partial z}}} \right)_y}dz\)

\(or,\;{\left( {\frac{{\partial x}}{{\partial y}}} \right)_z}{\left( {\frac{{\partial z}}{{\partial x}}} \right)_y}{\left( {\frac{{\partial y}}{{\partial z}}} \right)_x} = - 1\)

If F = ϕ(x, y, z)

\({\left( {\frac{{\partial x}}{{\partial y}}} \right)_F}{\left( {\frac{{\partial y}}{{\partial z}}} \right)_F}{\left( {\frac{{\partial z}}{{\partial x}}} \right)_F} = 1\)