Ehrenfest equations

Quantitative consideration

Ehrenfest equations are the consequence of continuity of specific entropy s and specific volume v , which are first derivatives of specific Gibbs free energy – in second-order phase transitions. If one considers specific entropy s as a function of temperature and pressure, then its differential is: d s = ( ∂ s ∂ T ) P d T + ( ∂ s ∂ P ) T d P . As ( ∂ s ∂ T ) P = c P T , ( ∂ s ∂ P ) T = − ( ∂ v ∂ T ) P , then the differential of specific entropy also is:

d s i = c i P T d T − ( ∂ v i ∂ T ) P d P ,

where i = 1 and i = 2 are the two phases which transit one into other. Due to continuity of specific entropy, the following holds in second-order phase transitions: d s 1 = d s 2 . So,

( c 2 P − c 1 P ) d T T = [ ( ∂ v 2 ∂ T ) P − ( ∂ v 1 ∂ T ) P ] d P

Therefore, the first Ehrenfest equation is:

Δ c P = T ⋅ Δ ( ( ∂ v ∂ T ) P ) ⋅ d P d T .

The second Ehrenfest equation is got in a like manner, but specific entropy is considered as a function of temperature and specific volume:

Δ c V = − T ⋅ Δ ( ( ∂ P ∂ T ) v ) ⋅ d v d T

The third Ehrenfest equation is got in a like manner, but specific entropy is considered as a function of v и P :

Δ ( ∂ v ∂ T ) P = Δ ( ( ∂ P ∂ T ) v ) ⋅ d v d P .

Continuity of specific volume as a function of T and P gives the fourth Ehrenfest equation:

Δ ( ∂ v ∂ T ) P = − Δ ( ( ∂ v ∂ P ) T ) ⋅ d P d T .

Application

Derivatives of Gibbs free energy are not always finite. Transitions between different magnetic states of metals can't be described by Ehrenfest equations.