An equation of state is a mathematical formula that describes how the physical properties of a substance, like its pressure, volume, and temperature, relate to each other. If you know two of those properties, an equation of state lets you calculate the third. These equations are foundational tools in physics, chemistry, and engineering, used for everything from designing chemical plants to modeling the interior of planets.
The Core Idea
Every substance exists in some thermodynamic “state” defined by measurable properties: how hot it is, how much space it occupies, and how much pressure it exerts. These measurable properties are called state variables. An equation of state is simply the mathematical relationship connecting them.
The simplest and most familiar example is the ideal gas law: PV = nRT. Here, P is pressure, V is volume, T is temperature, n is the amount of gas (measured in moles), and R is the universal gas constant, which has an exact value of 8.314 joules per mole per kelvin. This single equation lets you predict what happens when you heat a sealed container of gas (pressure rises), compress it (temperature rises), or let it expand (temperature drops). It was derived from experiments and later confirmed by the kinetic theory of gases, which treats gas molecules as tiny independent particles bouncing around in space.
Why the Ideal Gas Law Has Limits
The ideal gas law works well for gases at low pressures and high temperatures, but it rests on two assumptions that break down in real-world conditions. First, it assumes gas molecules are so small relative to their container that their own volume is essentially zero. Second, it assumes molecules don’t attract or repel each other at all. Neither assumption holds when gas is compressed to high pressures or cooled close to the point where it condenses into a liquid. Under those conditions, molecules are packed closely enough that their physical size matters, and the attractive forces between them start pulling them together in ways the ideal gas law can’t account for.
Equations for Real Gases
In 1873, Johannes van der Waals proposed a modified equation that corrects for both of those shortcomings. His equation looks like this: (P + a/V²)(V − b) = nRT. The two new terms, a and b, are constants specific to each gas, determined by experiment.
The constant b represents the physical volume of the molecules themselves. By subtracting b from the total volume, the equation accounts for the fact that not all the space in a container is available for molecules to move through; some of it is taken up by the molecules’ own bulk. The constant a corrects for the attractive forces between molecules. Those attractions effectively reduce the pressure a gas exerts on its container walls, because molecules heading toward the wall get tugged back slightly by their neighbors. The a/V² term adds that “missing” pressure back into the equation.
The van der Waals equation was a major conceptual leap, but it’s still a simplified model. More refined equations have been developed for industrial use. The Peng-Robinson equation, introduced in 1976, has become the most widely used equation of state in the petroleum and natural gas industry. It performs especially well near the critical point, the specific temperature and pressure where liquid and gas phases become indistinguishable. That makes it particularly useful for modeling gas condensate systems, where hydrocarbons hover right at the boundary between liquid and vapor. The Soave-Redlich-Kwong equation is another common choice, performing comparably in most situations.
The Critical Point
One of the most powerful applications of an equation of state is predicting a substance’s critical point. Every pure substance has a specific temperature and density at which the distinction between its liquid and gas phases vanishes entirely. Above this critical temperature and pressure, the substance becomes a “supercritical fluid” with properties of both a liquid and a gas.
Mathematically, the critical point is found by looking for the temperature and density where the first and second derivatives of pressure with respect to density both equal zero. In practical terms, this means the equation’s pressure-volume curve flattens out completely at that point, with no bump or dip to distinguish liquid from gas. Engineers use numerical methods to solve for these critical conditions, which are essential for designing processes that involve phase changes, like natural gas processing or refrigeration.
Beyond Gases: Solids and Liquids
Equations of state aren’t limited to gases. Scientists also use them to describe liquids and solids, though these are considerably more complex. Most classical equations of state were designed for gas-liquid systems and don’t independently describe the solid phase. The Peng-Robinson equation, for instance, works well for gases and liquids but is not applicable when one phase is a high-pressure gas and the other is a solid.
Developing a single unified equation that handles all three phases, solid, liquid, and gas, remains a significant challenge. The physics of the solid-liquid transition is fundamentally different from the gas-liquid transition. Gas and liquid phases share a critical point where they merge continuously into each other, but there is no such critical point between a solid and a liquid. That discontinuity makes it difficult to capture both transitions in one smooth mathematical expression. Modified equations have been proposed that attempt this, but they require additional parameters and careful calibration against experimental data.
Equations of State in Earth Science
Geophysicists use a specialized equation of state called the Birch-Murnaghan equation to describe how minerals behave under the extreme pressures found deep inside Earth and other planets. At the pressures in Earth’s core, reaching hundreds of gigapascals (millions of times atmospheric pressure), the relationship between pressure, volume, and temperature looks nothing like the ideal gas law. The Birch-Murnaghan equation instead relates pressure to the compression of a solid material, using parameters like the material’s bulk modulus (a measure of how resistant it is to compression) and how that resistance changes as pressure increases.
Researchers at institutions like the Carnegie Institution for Science use this equation to analyze experiments where they shock iron and iron-silicon alloys to extreme conditions, replicating what happens inside Earth’s core. By fitting their pressure-density data to a third-order Birch-Murnaghan equation, they found that iron with about 8.6% silicon by weight matches the density profile of Earth’s outer core, while iron with roughly 3.8% silicon matches the inner core. This kind of work helps scientists determine what the planet’s interior is actually made of, without ever being able to sample it directly.
Intensive vs. Extensive Variables
One detail worth understanding is that the variables in an equation of state come in two types. Intensive variables, like temperature and pressure, are independent of how much material you have. A cup of boiling water and a bathtub of boiling water are both at 100°C. Extensive variables, like volume and the number of molecules, scale with the size of the system. Double the amount of gas and you double its volume (at the same temperature and pressure).
Equations of state typically relate intensive variables to each other, or mix intensive and extensive variables together. The ideal gas law, for example, connects two intensive variables (pressure and temperature) with two extensive ones (volume and the number of moles). Many advanced equations are written in terms of “molar volume,” the volume per mole of substance, which converts the extensive variable of total volume into an intensive one. This makes the equation applicable regardless of how much material you’re working with.