Differential Forms of the State Functions

In thermodynamics, state functions describe the system's macroscopic properties and depend only on the system's current state, not on the process used to reach that state. Examples include internal energy (\( U \)), enthalpy (\( H \)), Helmholtz free energy (\( A \)), and Gibbs free energy (\( G \)).

Total Differentials

State functions can be expressed in terms of their natural variables (\( T, p, V \) through total differentials. For instance, from the first law of thermodynamics \( U = Q + W \), the differential of internal energy (\( U \)) is:

\[ dU = T dS - p dV \]

The differential of enthalpy \( H \), defined as \( H = U + pV \), is given by:

\[ dH = T dS + V dp \]

The differential of Helmholtz free energy \( A \), defined as \( A = U - TS \), is:

\[ dA = -S dT - p dV \]

Lastly, the differential of Gibbs free energy \( G \), defined as \( G = H - TS \), is:

\[ dG = -S dT + V dp \]

 

Maxwell Relations

Maxwell relations are a set of thermodynamic equations derived from the symmetry of second derivatives, according to Schwarz’s theorem. This theorem states that the order of partial differentiation of a state function does not affect the result. These relations provide a way to relate different thermodynamic properties to measurable quantities.

 

From the differentials of state functions:

\[ dU = T dS - p dV = \left( \frac{\partial U}{\partial S} \right)_V dS + \left( \frac{\partial U}{\partial V} \right)_S dV \]

\[ dH = T dS + V dp = \left( \frac{\partial H}{\partial S} \right)_p dS + \left( \frac{\partial H}{\partial p} \right)_S dp \]

\[ dA = -S dT - p dV = \left( \frac{\partial A}{\partial T} \right)_V dT + \left( \frac{\partial A}{\partial V} \right)_T dV \]

\[ dG = -S dT + V dp = \left( \frac{\partial G}{\partial T} \right)_p dT + \left( \frac{\partial G}{\partial p} \right)_T dp \]

we get the dependence of the state functions on their natural variables:

$ \left( \frac{\partial U}{\partial S} \right)_V = T $           $ \left( \frac{\partial U}{\partial V} \right)_S = -p $

$ \left( \frac{\partial H}{\partial S} \right)_p = T $           $ \left( \frac{\partial H}{\partial p} \right)_S = V $

$ \left( \frac{\partial A}{\partial T} \right)_V = -S $          $ \left( \frac{\partial A}{\partial V} \right)_T = -p $

$ \left( \frac{\partial G}{\partial T} \right)_p = -S $          $ \left( \frac{\partial G}{\partial p} \right)_T = V $

 

Using the Schwarz's rule:

\[ \frac{\partial}{\partial x} \left( \frac{\partial f(x, y)}{\partial y} \right) = \frac{\partial}{\partial y} \left( \frac{\partial f(x, y)}{\partial x} \right) \]

we derive the Maxwell relations:

  • From internal energy \( U \)

\[ \left( \frac{\partial T}{\partial V} \right)_S = - \left( \frac{\partial p}{\partial S} \right)_V \]

  • From the differential of enthalpy \( H \):

\[ \left( \frac{\partial T}{\partial p} \right)_S = \left( \frac{\partial V}{\partial S} \right)_p \]

  • Starting from Helmholtz free energy \( A \):

\[ \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial p}{\partial T} \right)_V \]

  • Finally, for Gibbs free energy \( G \):

\[ \left( \frac{\partial S}{\partial p} \right)_T = - \left( \frac{\partial V}{\partial T} \right)_p \]

 

Maxwell relations allow us to express less accessible quantities, like entropy and internal energy, in terms of more easily measured variables like temperature, pressure, and volume. These additional relations help thermodynamicists relate macroscopic quantities and make experimental measurements that can be used to calculate changes in state functions.