The ideal gas law, $PV = nRT$ , rests on two key assumptions: gas particles have zero volume, and they exert no forces on each other. Neither assumption holds for real molecules.

Intermolecular forces (both attractive and repulsive) become significant when molecules are close together, which happens at high pressures and low temperatures.
Finite molecular volume matters when the total gas volume is small enough that the space molecules physically occupy is no longer negligible.
Near the critical point or at high pressures, these effects compound, and the ideal gas law can give predictions that are wildly off from measured values.

Need for Accurate Equations of State

Because real gases deviate from ideal behavior, we need equations of state that build in corrections for intermolecular forces and molecular size. The van der Waals equation is the most well-known example, but others (Redlich-Kwong, Peng-Robinson) exist for greater accuracy in specific regimes.

These corrected equations matter in practice: designing industrial reactors, sizing compressors, engineering refrigeration cycles, and working with gases at high pressure all require predictions that the ideal gas law simply can't deliver.

Van der Waals Equation of State

Modifying the Ideal Gas Law

The van der Waals equation introduces two substance-specific parameters to patch the ideal gas law:

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

Here $V_m$ is the molar volume (volume per mole), $R$ is the gas constant, and $T$ is temperature. For $n$ moles, the equation is often written as:

$\left(P + \frac{n^2 a}{V^2}\right)(V - nb) = nRT$

What do the parameters mean?

$a$ corrects for attractive intermolecular forces. A larger $a$ means stronger attractions between molecules (e.g., $a$ is much larger for water vapor than for helium).
$b$ corrects for finite molecular volume. It represents the excluded volume per mole, the space molecules themselves take up. Larger molecules have larger $b$ values.

Think of it this way: the $(V_m - b)$ term says "the effective free space is less than the total volume," and the $(P + a/V_m^2)$ term says "the effective pressure driving molecules apart is higher than the measured pressure, because attractions pull molecules inward."

Calculating Pressure, Volume, or Temperature

To solve for any one variable, rearrange the equation with the other quantities known.

Solving for pressure is the most straightforward case:

$P = \frac{nRT}{V - nb} - \frac{n^2 a}{V^2}$

Solving for volume is trickier because the equation is cubic in $V$ . In practice you'll either:

Rearrange into a standard cubic polynomial and solve numerically (or with the cubic formula).
Use an iterative/graphical approach.
Start from the ideal gas volume as an initial guess and refine.

Solving for temperature is again straightforward algebra:

$T = \frac{(P + n^2a/V^2)(V - nb)}{nR}$

Real Gases vs. Ideal Gases

Behavior at Different Conditions

At low pressures and high temperatures, molecules are far apart and moving fast. Intermolecular forces are negligible, molecular volume is tiny compared to the container, and real gases behave nearly ideally.
As pressure increases or temperature decreases, molecules spend more time close together. Attractive forces pull them inward, and their finite size starts to matter. Deviations from ideality grow.

A useful rule of thumb: the higher the reduced temperature ( $T/T_c$ ) and the lower the reduced pressure ( $P/P_c$ ), the more ideal the behavior.

Condensation and Liquefaction

Ideal gases, by definition, never condense. Real gases do.

Condensation occurs when attractive intermolecular forces overcome the kinetic energy of the molecules, causing them to collapse into a liquid phase.
Liquefaction is achieved by increasing pressure, decreasing temperature, or both. Below the critical temperature, there exists a pressure at which the gas will liquefy. Above the critical temperature, no amount of pressure will produce a distinct liquid phase.

Volume Deviations

At high pressures, the measured molar volume of a real gas is larger than the ideal gas law predicts. Why? The ideal gas law assumes molecules are point particles, so it underestimates how much space the gas occupies when molecules are packed closely. The van der Waals $b$ parameter corrects for exactly this effect.

At moderate pressures, attractive forces can actually cause the real gas volume to be smaller than ideal, because attractions pull molecules closer together. So the direction of deviation depends on which effect dominates.

Compressibility Factor for Real Gases

Assumptions and Deviations, Non-Ideal Gas Behavior | Chemistry: Atoms First

Definition and Deviation from Unity

The compressibility factor $Z$ quantifies how much a real gas deviates from ideal behavior:

$Z = \frac{P V_m}{RT}$

where $V_m$ is the measured molar volume. Equivalently, $Z = V_{\text{actual}} / V_{\text{ideal}}$ .

For an ideal gas, $Z = 1$ exactly, at all conditions.
For a real gas:
- $Z < 1$ : Attractive forces dominate. The gas is more compressible than an ideal gas (molecules are being "pulled" together).
- $Z > 1$ : Repulsive forces and finite molecular volume dominate. The gas resists compression more than an ideal gas would.

Measuring Deviation from Ideal Behavior

The farther $Z$ is from 1, the more the gas deviates from ideality. You can calculate $Z$ from experimental $P$ , $V$ , $T$ data, or estimate it from an equation of state like van der Waals.

At very low pressures, $Z \to 1$ for all gases. At intermediate pressures, most gases dip below 1 (attractions win), then rise above 1 at high pressures (repulsions and molecular volume win). The Boyle temperature is the temperature at which $Z \approx 1$ over a range of moderate pressures, meaning the attractive and repulsive corrections roughly cancel.

Compressibility Factor Diagrams

Z vs. P diagrams (at constant $T$ ) are the most common. They show the characteristic dip-then-rise pattern described above. Different isotherms reveal how temperature affects the deviation.
Z vs. P diagrams using reduced variables ( $P/P_c$ , $T/T_c$ ) allow different gases to collapse onto roughly the same curves, which is the basis of the principle of corresponding states.

These plots are a quick visual tool for assessing how "non-ideal" a gas is under given conditions.

Critical Properties of Real Gases

Definition of Critical Point

The critical point is the temperature and pressure at which the boundary between liquid and gas phases vanishes. Above the critical point, the substance exists as a supercritical fluid with properties intermediate between liquid and gas.

The critical point is defined by three quantities:

Critical temperature $T_c$
Critical pressure $P_c$
Critical molar volume $V_c$

Critical Temperature, Pressure, and Volume

$T_c$ is the highest temperature at which a gas can be liquefied by compression alone. Above $T_c$ , no amount of pressure produces a distinct liquid phase.
$P_c$ is the pressure required to liquefy the gas at exactly $T_c$ .
$V_c$ is the molar volume at the critical point.

For example, $\text{CO}_2$ has $T_c = 304.2 \text{ K}$ and $P_c = 73.8 \text{ atm}$ , which is why supercritical $\text{CO}_2$ extraction is feasible near room temperature but requires substantial pressure.

Calculating Critical Properties Using Equations of State

At the critical point, the $P$ vs. $V_m$ isotherm has an inflection point. Mathematically, both the first and second derivatives of pressure with respect to molar volume are zero:

$\left(\frac{\partial P}{\partial V_m}\right)_{T_c} = 0 \quad \text{and} \quad \left(\frac{\partial^2 P}{\partial V_m^2}\right)_{T_c} = 0$

Applying these conditions to the van der Waals equation yields:

$T_c = \frac{8a}{27Rb}, \qquad P_c = \frac{a}{27b^2}, \qquad V_c = 3b$

From these, you can derive the critical compressibility factor for a van der Waals gas:

$Z_c = \frac{P_c V_c}{R T_c} = \frac{3}{8} = 0.375$

Real gases typically have $Z_c$ values between 0.2 and 0.3, which tells you the van der Waals equation overestimates $Z_c$ . More sophisticated equations of state (Redlich-Kwong, Peng-Robinson) give better predictions of critical behavior by using different functional forms for the attractive and repulsive terms.

2,589 studying →