What’s the difference between Maxwell’s equations and Maxwell’s relations?

Maxwell’s equations are fundamental tools for electric and magnetic fields, while Maxwell’s relations focus on thermodynamic quantities like temperature, pressure, volume, and entropy. For more on Maxwell’s equations, see “What are the fundamentals of Maxwell’s equations and how do they relate to TENGs?”

Maxwell’s relations give designers a powerful tool for analyzing thermodynamic systems like internal combustion engines or data center cooling. Like Maxwell’s equations, Maxwell’s relations are based on partial derivatives. The relations describe the symmetry of second-order partial derivatives of thermodynamic quantities.

Maxwell’s relations directly and indirectly cover eight thermodynamic quantities including pressure (P), volume (V), temperature (T), entropy (S) measured in Joules per Kelvin (J/K), internal energy (U) measured in Joules, enthalpy (H), Helmholtz free energy (F), and Gibbs free energy (G) (Figure 1). P, V, and T are simple quantities. Other thermodynamic quantities are more complex, hence the importance of Maxwell’s relations.

Figure 1. The equations that make up Maxwell’s relations. (Image: Metallurgy and Material Science)

For example, enthalpy (H) measures a system’s total heat content, defined as the sum of its internal energy (U) plus the product of pressure (P) and volume (V): H = U + PV. Helmholtz free energy (F) measures the useful work obtainable from a closed system at constant temperature and volume, defined as F = U – TS.

Gibbs free energy (G) is used to predict the spontaneity of a process at a given temperature; it’s defined as G = H – TS, or ∂G = ∂H – T∂S:

When ∂G < 0, the process is spontaneous (favored).
When ∂G > 0, the process is non-spontaneous (not favored).
If ∂G = 0, the process is at equilibrium.

Maxwell relations aren’t just an interesting mathematical exercise; they are practical and enable designers to replace hard-to-measure quantities with easier-to-measure ones when designing thermodynamic systems.

Thermodynamic square

A “thermodynamic square” is a mnemonic diagram that scientists and designers can use to map the relationships between thermodynamic potentials like U, H, F, and G and the corresponding natural quantities like T, P, V, and S. It was originally suggested by German-British physicist Max Born.

In Figure 2(a), natural variables, P, T, V, and S are on the corners, and U, F, G, and H, are located on the sides between their corresponding natural variables, G is between P and T, U is between V and S and so on. In addition, there are two arrows, one pointing from S to T and the other from P to V, that are used to identify the positive and negative signs in the differential relation for each potential.

The square is designed for building Maxwell’s relations. For example, Figure 2(b) shows how to derive (∂T= ∂V)_S and (∂P= ∂S)_V. In this case, one endpoint has an arrow pointing to it, and the other does not. When that happens, the Maxwell relation requires a negative sign: (∂T= ∂V)_S = -(∂P= ∂S)_V.

Figure 2. The thermodynamic square (a) and how it’s used to construct Maxwell’s relations (b). (Image: Wake Forest University)

Cooling data centers

Maxwell’s relations can describe the Carnot cycle used for data center cooling. The Carnot model represents the maximum efficiency possible for converting heat into work or for cooling between two temperatures.

Maxwell’s relations are based on the concept of reversibility, and the Carnot cycle is a reversible process, making it well-suited to be analyzed using Maxwell’s relations. The Carnot cycle consists of four stages (Figure 3).

Isothermal (constant temperature) expansion is when the cooling gas absorbs heat while maintaining a constant temperature and increasing entropy, causing the gas to expand and perform work.
Adiabatic (without heat transfer) expansion follows when the gas continues to expand without exchanging heat with the surroundings, resulting in a drop in the gas’s temperature and constant entropy.
Isothermal compression expels the heat from the gas while decreasing entropy and maintaining a constant temperature.
Adiabatic compression is the final stage, and the gas is compressed without heat transfer. This increases the gas’s temperature while maintaining a constant entropy and returning the system to its starting condition.

Figure 3. The relationships between P & V and between T & S during the four stages of the Carnot cycle. (Image: Paradigm Cooling)

Summary

Maxwell relations enable designers to replace hard-to-measure quantities like enthalpy, Helmholtz free energy, and Gibbs free energy with easier-to-measure ones like temperature, volume, and pressure when designing thermodynamic systems. The relations can be easily remembered using a thermodynamic square, and they are practical tools for designing systems like data center coolers.