12.1 Maxwell relations

Maxwell relations connect partial derivatives of thermodynamic properties that would otherwise seem unrelated. They let you replace quantities that are hard to measure (like how entropy changes with pressure) with quantities that are straightforward to measure in a lab (like how volume changes with temperature). For a Thermo I course, these relations are essential tools for working with real substances whose equations of state you know.

Maxwell Relations

Derivation from Fundamental Thermodynamic Equations

Maxwell relations come from the four fundamental property relations (also called Gibbs equations). Each one expresses a thermodynamic potential in terms of its natural variables:

• $dU = TdS - PdV$ → natural variables: $S, V$
• $dH = TdS + VdP$ → natural variables: $S, P$
• $dF = -SdT - PdV$ → natural variables: $T, V$
• $dG = -SdT + VdP$ → natural variables: $T, P$

Here $U$ is internal energy, $H$ is enthalpy, $F$ is Helmholtz free energy, and $G$ is Gibbs free energy.

The key mathematical fact behind Maxwell relations is the exactness condition: for any exact differential $dz = Mdx + Ndy$, the mixed second partial derivatives are equal, so $(\partial M / \partial y)_x = (\partial N / \partial x)_y$. Since each Gibbs equation is an exact differential, you can apply this condition directly.

For example, starting from $dU = TdS - PdV$, you identify $M = T$ and $N = -P$. The exactness condition gives:

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

Repeating this for all four potentials yields the four Maxwell relations:

Physical Interpretation

Consider the last relation from the table above. The left side, $(\partial S / \partial P)_T$, asks how entropy changes when you compress something isothermally. That's not something you can stick a probe into and measure. But the right side, $-(\partial V / \partial T)_P$, is just the negative of the isobaric thermal expansion, which you can measure with a thermometer and a graduated cylinder.

This is the core idea: Maxwell relations translate between quantities that describe the same underlying physics but from different experimental perspectives. They also reveal a deep symmetry in thermodynamic relationships, confirming that the four potentials are all consistent with each other.

These relations are particularly useful for:

• Evaluating entropy changes using $P$-$V$-$T$ data
• Connecting heat capacities, compressibilities, and thermal expansion coefficients
• Analyzing phase transitions where direct measurement of certain derivatives is impractical

Applications of Maxwell Relations

Determining Changes in Thermodynamic Properties

The typical workflow for using a Maxwell relation looks like this:

1. Identify what you need. Determine which partial derivative or property change the problem asks for.
2. Check if it's directly measurable. If the derivative involves entropy as a function of $P$ or $V$, it probably isn't.
3. Substitute using a Maxwell relation. Replace the hard-to-measure derivative with its Maxwell equivalent.
4. Integrate if needed. To find a finite change (not just a derivative), integrate both sides over the appropriate variable between the initial and final states.
5. Plug in the equation of state. Use the given $P$-$V$-$T$ relationship (ideal gas, van der Waals, etc.) to evaluate the integral.

Examples of Applications

Change in entropy during isothermal compression:

Suppose you need $\Delta S$ as an ideal gas is compressed isothermally from $V_1$ to $V_2$. Start with the Maxwell relation from $dF$:

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

For an ideal gas, $P = nRT/V$, so $(\partial P / \partial T)_V = nR/V$. Integrating at constant $T$:

$\Delta S = \int_{V_1}^{V_2} \frac{nR}{V} \, dV = nR \ln\frac{V_2}{V_1}$

Since $V_2 < V_1$ for compression, $\Delta S < 0$, which makes physical sense: isothermal compression of an ideal gas reduces entropy.

Relating $C_P - C_V$ to measurable quantities:

Maxwell relations, combined with other property relations, let you derive the important result:

$C_P - C_V = -T \frac{\left[(\partial V / \partial T)_P\right]^2}{(\partial V / \partial P)_T}$

The right side contains only $P$-$V$-$T$ derivatives, which are measurable. This is how you'd determine $C_P - C_V$ for a real substance without calorimetry at every state point.

Choosing Maxwell Relations

Identifying the Appropriate Maxwell Relation

The right Maxwell relation to use depends on two things: which variables are held constant in your process, and which derivative you're trying to evaluate.

A practical way to choose:

• Constant $S$ and $V$ (isentropic, isochoric) → use the relation from $dU$
• Constant $S$ and $P$ (isentropic, isobaric) → use the relation from $dH$
• Constant $T$ and $V$ (isothermal, isochoric) → use the relation from $dF$
• Constant $T$ and $P$ (isothermal, isobaric) → use the relation from $dG$

The last two (from $dF$ and $dG$) come up most often in practice because $T$ and $P$ are the variables you typically control in experiments.

Strategy for Selecting Maxwell Relations

1. Write down what's given and what you need to find.
2. Identify which variables are held constant in the process.
3. Match those constant variables to the natural variables of one of the four potentials.
4. Write the corresponding Maxwell relation and check that it contains the derivative you need.
5. If a single Maxwell relation doesn't get you there, combine it with other property relations (like the definitions of $C_P$, $C_V$, or the equation of state) to build up the expression you need.

Physical Significance of Maxwell Relations

Interconnectedness of Thermodynamic Properties

Maxwell relations are not just mathematical tricks. They reflect the fact that thermodynamic potentials are state functions with well-behaved (exact) differentials. Because of this, the order in which you change two variables doesn't matter, and that constraint forces specific equalities between seemingly different derivatives.

This symmetry means that the entire thermodynamic behavior of a simple compressible substance can, in principle, be determined from its equation of state plus one heat capacity. Maxwell relations are the bridge that makes this possible.

Experimental Verification

You can verify Maxwell relations experimentally by measuring both sides independently and comparing. For example:

• Measure $(\partial V / \partial T)_P$ by heating a substance at constant pressure and tracking volume change.
• Independently measure $(\partial S / \partial P)_T$ using calorimetric data at constant temperature.
• The Maxwell relation from $dG$ predicts these should be equal in magnitude and opposite in sign.

The consistent agreement between such measurements across different substances and conditions confirms that the thermodynamic framework built on the first and second laws is internally consistent and universally applicable.