The Maxwell relation derived from the differential expression for the Helmholtz free energy (dA) is

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The Maxwell relation derived from the differential expression for the Helmholtz free energy (𝑑𝐴dA) relates the second derivative of the Helmholtz free energy with respect to two different thermodynamic variables. It can be expressed as: (∂2𝐴∂𝑋∂𝑌)𝑇=−(∂2𝐴∂𝑌∂𝑋)𝑇(∂X∂Y∂2A​)T​=−(∂Y∂X∂2A​)T​ Where:

• 𝐴A is the Helmholtz free energy.
• 𝑋X and 𝑌Y are two thermodynamic variables (e.g., pressure 𝑃P, volume 𝑉V, temperature 𝑇T, etc.).
• 𝑇T is the temperature.

This relation states that the second partial derivatives of the Helmholtz free energy with respect to two different variables, 𝑋X and 𝑌Y, are equal but opposite in sign when taken at constant temperature (𝑇T). This relationship is a consequence of the equality of mixed partial derivatives and plays a crucial role in thermodynamics for deriving relationships between different thermodynamic properties.