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# Maxwell’s Relations

The Maxwell relations are derived from Euler’s reciprocity relation. The relations are expressed in partial differential form. The Maxwell relations consists of the characteristic functions: internal energy U, enthalpy H, Helmholtz free energy F, and Gibbs free energy G and thermodynamic parameters: entropy S, pressure P, volume V, and temperature T. Following is the table of Maxwell relations for secondary derivatives:

 $$\begin{array}{l}+(\frac{\partial T}{\partial V})_{S}=-(\frac{\partial P}{\partial S})_{V}=\frac{\partial^2 U}{\partial S\partial V}\end{array}$$ $$\begin{array}{l}+(\frac{\partial T}{\partial P})_{S}=+(\frac{\partial V}{\partial S})_{P}=\frac{\partial^2 H}{\partial S\partial P}\end{array}$$ $$\begin{array}{l}+(\frac{\partial S}{\partial V})_{T}=+(\frac{\partial P}{\partial T})_{V}=\frac{\partial^2 F}{\partial T\partial V}\end{array}$$ $$\begin{array}{l}-(\frac{\partial S}{\partial P})_{T}=+(\frac{\partial V}{\partial T})_{V}=\frac{\partial^2 G}{\partial T\partial P}\end{array}$$

## What are Maxwell’s relations?

These are the set of thermodynamics equations derived from a symmetry of secondary derivatives and from thermodynamic potentials. These relations are named after James Clerk Maxwell, who was a 19th-century physicist.

### Derivation of Maxwell’s relations

Maxwell’s relations can be derived as:

$$\begin{array}{l}dU=TdS-PdV\end{array}$$
(differential form of internal energy)

$$\begin{array}{l}dU=(\frac{\partial z}{\partial x})_{y}dx+(\frac{\partial z}{\partial y})_{x}dy\end{array}$$
(total differential form)

$$\begin{array}{l}dz=Mdx+Ndy\end{array}$$
(other way of showing the equation)

$$\begin{array}{l}M=(\frac{\partial z}{\partial x})_{y}\end{array}$$

And

$$\begin{array}{l}N=(\frac{\partial z}{\partial y})_{x}\end{array}$$

From

$$\begin{array}{l}dU=TdS-PdV\end{array}$$

T=

$$\begin{array}{l}(\frac{\partial U}{\partial S})_{V}\end{array}$$

And

$$\begin{array}{l}-P=(\frac{\partial U}{\partial V})_{S}\end{array}$$

$$\begin{array}{l}\frac{\partial }{\partial y}(\frac{\partial z}{\partial x})_{y}=\frac{\partial }{\partial x}(\frac{\partial z}{\partial y})_{x}=\frac{\partial^2 z}{\partial y\partial x}=\frac{\partial^2 z}{\partial x\partial y}\end{array}$$
(symmetry of second derivatives)

$$\begin{array}{l}\frac{\partial }{\partial V}(\frac{\partial U}{\partial S})_{V}=\frac{\partial }{\partial S}(\frac{\partial U}{\partial V})_{S}\end{array}$$

$$\begin{array}{l}(\frac{\partial T}{\partial V})_{S}=-(\frac{\partial P}{\partial S})_{V}\end{array}$$

### Common forms of Maxwell’s relations

 Function Differential Natural variables Maxwell Relation U dU = TdS – PdV S, V $$\begin{array}{l}(\frac{\partial T}{\partial V})_{S}=-(\frac{\partial P}{\partial S})_{V}\end{array}$$ H dH = TdS + VdP S, P $$\begin{array}{l}(\frac{\partial T}{\partial P})_{S}=(\frac{\partial V}{\partial S})_{P}\end{array}$$ F dF = -PdV – SdT V, T $$\begin{array}{l}(\frac{\partial P}{\partial T})_{V}=(\frac{\partial S}{\partial V})_{T}\end{array}$$ G dG = VdP – SdT P, T $$\begin{array}{l}(\frac{\partial V}{\partial T})_{P}=-(\frac{\partial S}{\partial P})_{T}\end{array}$$

Where,

T is the temperature

S is the entropy

P is the pressure

V is the volume

U is the internal energy

H is the entropy

G is the Gibbs free energy

F is the Helmholtz free energy

With respect to pressure and particle number, enthalpy and Maxwell’s relation can be written as:

$$\begin{array}{l}(\frac{\partial \mu }{\partial P})_{S,N} = (\frac{\partial V}{\partial N})_{S,P} = (\frac{\partial^2 H}{\partial P\partial N})\end{array}$$

### Solved Examples

Example 1:

Prove that

$$\begin{array}{l}(\frac{\partial V}{\partial T})_{p}=T\frac{\alpha }{\kappa _{T}}-p\end{array}$$
.

Solution:

Combining first and second laws:

dU = TdS – pdV

Diving both the sides by dV

$$\begin{array}{l}\frac{\mathrm{d} U}{\mathrm{d} V}|_{T}=\frac{T\mathrm{d} S}{\mathrm{d} V}|_{T}-p\frac{\mathrm{d} V}{\mathrm{d} V}|_{T}\end{array}$$

$$\begin{array}{l}\frac{\mathrm{d} U}{\mathrm{d} V}|_{T}=(\frac{\partial U}{\partial V})_{T}\end{array}$$

$$\begin{array}{l}\frac{T\mathrm{d} S}{\mathrm{d} V}|_{T}=(\frac{\partial S}{\partial V})_{T}\end{array}$$

$$\begin{array}{l}\frac{\mathrm{d} V}{\mathrm{d} V}|_{T}=1\end{array}$$

$$\begin{array}{l}(\frac{\partial U}{\partial V})_{T}=T(\frac{\partial S}{\partial V})_{T}-p\end{array}$$

$$\begin{array}{l}(\frac{\partial p}{\partial T})_{V}=(\frac{\partial S}{\partial V})_{T}\end{array}$$

$$\begin{array}{l}(\frac{\partial U}{\partial V})_{T}=T(\frac{\partial p}{\partial T})_{V}-p\end{array}$$

$$\begin{array}{l}(\frac{\partial p}{\partial T})_{V}=\frac{\alpha }{\kappa _{T}}\end{array}$$

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## Frequently Asked Questions – FAQs

### Maxwell’s relations are named after which scientist?

It is named after Scientist James Clerk Maxwell.

### What are Maxwell’s relations?

These are set of thermodynamics equations derived from a symmetry of secondary derivatives and from thermodynamic potentials.

### What is enthalpy?

Enthalpy (H) is the sum of the internal energy (U) and the product of pressure (P) and volume(V).

### Define Gibbs free energy.

Gibbs free energy can be defined as the maximum amount of work that can be extracted from a closed system.

### Name three thermodynamic parameters.

Thermodynamic parameters are: Pressure P, volume V, and temperature T.

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