8.3 Thermodynamic equations of state

Equations of State

Concept of an Equation of State

An equation of state is a mathematical relationship between state variables (pressure, volume, temperature) that describes the thermodynamic behavior of a system. With one of these equations in hand, you can calculate unknown thermodynamic properties from a limited set of known variables.

Different equations of state apply depending on the system and the range of conditions you're working in. The simplest is the ideal gas equation; more sophisticated models like the van der Waals, Redlich-Kwong, and Peng-Robinson equations handle real-gas behavior with increasing accuracy.

Derivation from the Fundamental Equation

The starting point is the fundamental relation, which connects internal energy $U$ to entropy $S$, volume $V$, and particle numbers $N_i$:

$dU = TdS - PdV + \sum_i \mu_i dN_i$

where $T$ is temperature, $P$ is pressure, and $\mu_i$ is the chemical potential of species $i$.

For a simple, single-component system at constant particle number, this reduces to:

$dU = TdS - PdV$

From this expression, you can read off the natural variables of $U$ as $S$ and $V$. The partial derivatives give you the thermal equation of state and the mechanical equation of state:

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

These two relations are the thermodynamic equations of state. They express $T$ and $P$ as functions of the natural variables $S$ and $V$, and together they contain all the thermodynamic information in the fundamental equation.

A Maxwell relation follows directly from the equality of mixed second partial derivatives of $U$:

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

This connects how temperature changes with volume (at constant entropy) to how pressure changes with entropy (at constant volume). You'll use relations like this one to derive measurable consequences from equations of state.

Connecting to Measurable Properties

The real power of thermodynamic equations of state is that they let you compute properties that are hard to measure directly from properties that are easy to measure (like $P$, $V$, and $T$).

Simple example with the ideal gas equation ($PV = nRT$):

• Temperature from known pressure and volume: $T = PV/nR$
• Pressure from known temperature and volume: $P = nRT/V$

Going beyond $P$, $V$, $T$: Using Maxwell relations and the equation of state together, you can derive expressions for enthalpy, entropy, heat capacities, and Gibbs free energy. For instance, the internal pressure of a gas is given by:

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

This result comes from combining the fundamental relation with a Maxwell relation. For an ideal gas, you can verify that the right-hand side equals zero, confirming that the internal energy of an ideal gas depends only on temperature, not volume.

Analysis of Thermodynamic Systems

Equations of state let you predict how a system behaves under changing conditions. The van der Waals equation is a good example of what a more realistic model can do:

$\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT$

Here $a$ and $b$ are substance-specific constants, and $V_m$ is the molar volume. The $a/V_m^2$ term corrects for intermolecular attractions, while $b$ corrects for the finite volume occupied by the molecules themselves.

With an equation like this, you can analyze:

• Phase transitions (gas-to-liquid condensation), which the ideal gas law cannot predict at all
• Critical points, where the distinction between liquid and gas phases vanishes
• Compressibility factors ($Z = PV_m/RT$), which quantify how far a real gas deviates from ideal behavior

Comparing Equations of State

Ideal Gas vs. Van der Waals

The ideal gas equation works well when molecules are far apart and moving fast, so their size and attractions don't matter much. As pressure increases or temperature drops, those assumptions break down, and the van der Waals corrections become significant.

Beyond Van der Waals

More advanced equations of state build on the same idea of correcting for molecular interactions but do so with greater accuracy:

• Redlich-Kwong modifies the attractive term to include temperature dependence, improving predictions over a wider range.
• Peng-Robinson further refines the attractive term and is widely used in chemical engineering because it handles liquid densities and vapor-liquid equilibria better.

Each successive model introduces additional parameters or functional forms to better represent specific substances or mixtures, but the underlying logic is the same: start from the ideal gas law and add corrections for the physics that the ideal model ignores.