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Differential forms of thermodynamic potentials are:

dU = TdS - PdV

dH =TdS +VdP

dA= - SdT - PdV

dG = - SdT +VdP

Four Maxwell relations are derived from each of these four thermodynamic potentials.

*1st Relation:*

Differential form of internal energy U is:

dU = TdS - PdV

Differentiating this with respect to S whereas V is kept constant gives:

(∂U/∂S)V = T and with respect to V whereas S is kept constant gives:

(∂U/∂V)S = -P

Also differentiating given equations with respect to V at constant S and with respect to S at constant V respectively will provide:

∂^{2}U/∂V∂S = (∂T/∂V) S and ∂^{2}U/∂S∂V = - (∂P/∂S)V

Mixed second order derivatives of U that means:

(∂T/∂V)S = - (∂P/∂S)V

This equation is one of the four Maxwell relations.

*2nd Relation:*

From differential form of Enthalpy H (i.e. dH = TdS +VdP)

(∂H/∂S)P = T and (∂H/∂P)S = P

Second derivative provides:

(∂^{2}H/∂P∂S) = (∂T/∂P)S and ∂^{2}H/∂S∂P = (∂P/∂S)P

Mixed second order derivatives of H are equal, therefore:

(∂T/∂P)S = (∂P/∂S)P

This equation is one of four Maxwell's relations.

*3rd Relation:*

From differential of Helmholtz free energy A (i.e. dA = - SdT - PdV)

(∂A/∂T)V = -S and (∂A/∂V)T = -P

Second derivative provides:

(∂^{2}A/∂V∂T) = - (∂S/∂V)T and ∂^{2}A/∂T∂V = - (∂P/∂T)V

Mixed second derivatives of A are equal, hence

- (∂S/∂V)T = - (∂P/∂T)V i.e. (∂S/∂V)T = (∂P/∂T)V

(∂P/∂T)V in equation can be achieved from equation of state. This means that variation of entropy S with respect to volume V at constant temperature T for the system can be achieved from equation of state for system.

*4th Relation:*

From differential of Gibb's free energy G (i.e. dG = - SdT +VdP), partial derivative provides

(∂G/∂T)P = -S and (∂G/∂P)T = V

Differentiating both equations with respect to P at constant T, and with respect to T at constant P respectively provide

(∂^{2}G/∂P∂T)P = - (∂S/∂P)T and (∂^{2}G/∂T∂P) = (∂V/∂T)P respectively.

Mixed second derivatives of G are equal, therefore

(∂S/∂P)T = - (∂V/∂T)P

(∂V/∂T)P in given equation can be attained from equation of state.

*Maxwell Relations:*

Four Maxwell Relations of thermodynamics are given below:

(∂T/∂V)S = - (∂P/∂S)V

(∂T/∂P)S = (∂P/∂S)P

(∂S/∂V)T = (∂P/∂T)V

(∂S/∂P)T = - (∂V/∂T)P

*Usefulness:*

These relations (that is Maxwell relations) are helpful in thermodynamics calculations. For instance, one can replace the derivative of entropy with the derivative of simple thermodynamic variation like temperature. Equations give derivative of entropy in terms of temperature T, volume V and pressure P.

(∂S/∂V)T = (∂P/∂T)V and (∂S/∂P)T = - (∂V/∂T)P

For the ideal gas, PV = nRT

(∂P/∂T)V = nR/V = P/T and (∂V/∂T)P = nR/P = V/T

For the ideal gas, equation becomes:

(∂S/∂V)T = P/T

(∂S/∂P)T = -V/T

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