Thermodynamic Property Relations: Maxwell Equations

In thermodynamics, we often need to determine properties that are difficult to measure directly, such as entropy or internal energy. The Maxwell relations provide a powerful mathematical bridge, allowing engineers to calculate these elusive quantities using easily measurable properties like pressure, temperature, and volume. By leveraging the fundamental structure of thermodynamic potentials, these identities transform abstract theory into practical tools for designing engines, refrigeration systems, and chemical processes.

The Foundation: Thermodynamic Potentials and Exact Differentials

To understand Maxwell relations, you must first be comfortable with thermodynamic potentials. These are state functions—like internal energy $U$, enthalpy $H$, Helmholtz free energy $A$, and Gibbs free energy $G$—whose changes are path-independent. Their definitions are:

• Internal Energy: $U$
• Enthalpy: $H=U+pV$
• Helmholtz Free Energy: $A=U-TS$
• Gibbs Free Energy: $G=H-TS=U+pV-TS$

When a function is a state function, its differential is exact. This is the critical mathematical property. For example, the combined first and second law for a closed, simple compressible system gives the fundamental relation for internal energy: $dU=TdS-pdV$. Here, $U$ is expressed as a function of entropy $S$ and volume $V$. Because $dU$ is an exact differential, a special condition applies: the order of taking second partial derivatives does not matter. This is the exactness condition, also known as the symmetry of second derivatives or Schwarz's theorem.

Deriving the Maxwell Relations

The exactness condition is the engine that generates the Maxwell relations. Let's see how it works for internal energy. If $U=U(S,V)$, then its differential is: $$dU={\left(\frac{\partial U}{\partial S}\right)}_{V}dS+{\left(\frac{\partial U}{\partial V}\right)}_{S}dV$$ Comparing this to the fundamental relation $dU=TdS-pdV$, we can identify the conjugate variables: $${\left(\frac{\partial U}{\partial S}\right)}_V=T\text{and}{\left(\frac{\partial U}{\partial V}\right)}_S=-p$$

Now, apply the exactness condition: the mixed second partial derivatives of $U$ must be equal. That is: $$\frac{\partial }{\partial V}{\left(\frac{\partial U}{\partial S}\right)}_V=\frac{\partial }{\partial S}{\left(\frac{\partial U}{\partial V}\right)}_S$$ Substituting the identities for $T$ and $-p$, we get our first Maxwell relation: $${\left(\frac{\partial T}{\partial V}\right)}_S=-{\left(\frac{\partial p}{\partial S}\right)}_V$$

This process is repeated for the other three primary potentials, each starting from their fundamental differential form:

1. From $dH=TdS+Vdp$ (where $H=H(S,p)$): $${\left(\frac{\partial T}{\partial p}\right)}_S={\left(\frac{\partial V}{\partial S}\right)}_p$$
2. From $dA=-SdT-pdV$ (where $A=A(T,V)$): $${\left(\frac{\partial S}{\partial V}\right)}_T={\left(\frac{\partial p}{\partial T}\right)}_V$$
3. From $dG=-SdT+Vdp$ (where $G=G(T,p)$): $${\left(\frac{\partial S}{\partial p}\right)}_T=-{\left(\frac{\partial V}{\partial T}\right)}_p$$

These four equations are the core Maxwell relations. Each one connects a partial derivative involving entropy (hard to measure) to a partial derivative involving $p$, $V$, and $T$ (easy to measure).

Applying Maxwell Relations to Solve Engineering Problems

The true power of these relations lies in their application. They allow you to compute changes in entropy, internal energy, and enthalpy using only $p-V-T$ data and specific heat information.

Example: Calculating Entropy Change with Respect to Volume Suppose you need ${\left(\frac{\partial S}{\partial V}\right)}_T$ for a gas in an isothermal process. Measuring entropy directly is impractical. However, the Maxwell relation derived from the Helmholtz free energy states: ${\left(\frac{\partial S}{\partial V}\right)}_T={\left(\frac{\partial p}{\partial T}\right)}_V$. The right-hand side is far easier to find. If you have an equation of state for the gas, you can compute it directly. For an ideal gas, $pV=RT$, so ${\left(\frac{\partial p}{\partial T}\right)}_V=\frac{R}{V}$. Therefore, the entropy change becomes $dS=\frac{R}{V}dV$ at constant $T$, which integrates to the familiar $\Delta S=R\ln (V_2/V_1)$ for an isothermal ideal gas process.

Example: Relating Thermal Expansion to an Adiabatic Process Consider the third relation: ${\left(\frac{\partial T}{\partial p}\right)}_S={\left(\frac{\partial V}{\partial S}\right)}_p$. The left side describes how temperature changes with pressure during an isentropic (adiabatic and reversible) compression—a key process in turbomachinery. The right side can be manipulated further using chain rules and definitions like the coefficient of thermal expansion. This shows how a macroscopic, measurable property like thermal expansion is fundamentally linked to the behavior of a system under adiabatic compression.

Common Pitfalls

1. Confusing Which Variables Are Held Constant: The subscripts on the partial derivatives are not decorative; they are essential. Using ${\left(\frac{\partial p}{\partial T}\right)}_V$ is completely different from using ${\left(\frac{\partial p}{\partial T}\right)}_p$ (which is zero). Always double-check the constant property for each derivative before applying a Maxwell relation. A helpful mnemonic is the "Thermodynamic Square," but understanding the derivation is the best safeguard.

1. Applying Relations to the Wrong System: The standard Maxwell relations are derived for simple compressible systems (where only $p-V$ work is considered). Applying them directly to systems with electrical, magnetic, or surface tension work, or to open systems, without proper modification will lead to errors. Always confirm your system matches the assumptions behind the potentials you are using.

1. Mixing Up the Sign: The signs in the Maxwell relations are systematic but easy to misplace. A reliable method is to remember the pattern from the derivation: potentials with $S$ and $V$ as natural variables ($U$, $A$) yield a negative sign when the derivatives involve one extensive ($S$, $V$) and one intensive ($T$, $p$) variable from different pairs. Conversely, potentials with $S$ and $p$ ($H$, $G$) yield a positive sign for the analogous cross-derivative. Writing the fundamental differentials correctly is the first step to getting the sign right.

Summary

• Maxwell relations are four mathematical identities derived from the exactness condition of the differentials of thermodynamic potentials ($U$, $H$, $A$, $G$).
• They provide critical links between partial derivatives of abstract properties (like entropy $S$) and derivatives involving only measurable properties (pressure $p$, volume $V$, temperature $T$).
• Their primary engineering utility is to calculate entropy changes and other non-measurable quantities using an equation of state and specific heat data, which are readily obtainable.
• Correct application requires meticulous attention to which variables are held constant in each partial derivative.
• Mastering these relations is essential for advancing from basic thermodynamic cycles to the analysis of real substances and complex processes in mechanical and chemical engineering.