Prove that a reversible adiabatic process cannot also be iso

Prove that a reversible adiabatic process cannot also be isothermal for T > 0.

Solution

When a system with two independent thermodynamic co-ordinates undergoes a reversible adiabatic change of state then we can specify one thermodynamic co-ordinate for the final state but not two , because change in entropy = S₂-S₁= 0, i.e. S is fixed . Thus a reversible adiabatic process can\'t be isothermal because such a process will fix both T and S and it is impossible.