Again

Second order terms

(2.119)

When at equilibrium

T1 = T2, p1 = p2

(2.120)

Conditions of stability
At equilibrium,

(2.121)

these are necessary but not sufficient conditions for equilibrium. To prove that S is a maximum, and G or F a minimum, it must satisfy
If the system is perturbed, and for any infinitesimal change of the system

(2.122)

It represents the stability of the system. The system must revert to the original state.
For a spontaneous change, from Equation

(2.123)

For stability

(2.124)

Let us choose U=U(S, V) and represents in powers of

(2.125)   (2.126)

The third order and higher order terms are neglected.

Since it must satisfy the condition given bellow

(2.127)

These inequalities indicate how to signs of some important physical quantities become for a system to be stable.

Since

(2.128)

(2.129)

Eq (2.129) is the condition for thermal stability.