Real gases and real solutions

Compressibility factor. The deviation of a real gas from ideal behaviour is measured directly by comparing the actual $PV$ product to the ideal one:

where: $P$ is pressure, $V$ is volume, $n$ is moles, $R$ is the gas constant, $T$ is temperature. $Z=1$ for an ideal gas at every condition. Real gases show $Z<1$ at moderate pressure, because net intermolecular attraction pulls the gas in and makes it more compressible than ideal, and $Z>1$ at high pressure, where the finite size of the molecules (excluded volume) dominates and resists further compression.

Van der Waals equation. The simplest model that corrects for both effects at once adds an attractive-pressure term and subtracts an excluded-volume term from the ideal-gas law:

where: $a$ is a substance-specific constant capturing the strength of intermolecular attraction (it reduces the effective pressure the container walls feel), and $b$ is a substance-specific constant capturing the volume molecules physically occupy (it reduces the volume actually available for the gas to move in). Both are empirical fits, not universal constants — different for $\text{CO}_2$, $\text{N}_2$, $\text{He}$, and so on. The Boyle temperature $T_B=a/(Rb)$ is the special temperature at which these two corrections cancel over a range of pressures, so the gas behaves ideally there even though $a,b\ne0$.

Principle of corresponding states. Normalising pressure, volume and temperature by the critical constants ($P_r=P/P_c$, $V_r=V/V_c$, $T_r=T/T_c$) puts different real gases on approximately the same footing: at equal reduced variables, most gases show approximately the same $Z$. This is why a single reduced equation of state can (approximately) describe many different substances at once.

Fugacity. Instead of discarding the ideal-gas chemical-potential formula, real-gas non-ideality is folded into a corrected effective pressure called fugacity, $f=\gamma P$, where $\gamma$ is the fugacity coefficient and $\gamma\to1$ as $P\to0$:

where: $\mu$ is the chemical potential, $\mu^\circ$ the standard chemical potential, $R$ the gas constant, $T$ the temperature, and $p^\circ$ the standard-state pressure. This is formally identical to the ideal-gas formula with $f$ standing in for $p$, so every equilibrium relation built earlier ($K$, $\Delta_rG^\circ=-RT\ln K$) survives unchanged if fugacities/activities are used consistently throughout.

Real solutions. Analogously, deviations from Raoult's law are absorbed into an activity coefficient $\gamma_i$:

Two reference states are used: the Raoult's-law reference ($\gamma_i\to1$ as $x_i\to1$, natural for the majority solvent) and the Henry's-law reference ($\gamma_i\to1$ as $x_i\to0$, natural for a dilute solute, where $p_i=K_Hx_i$ and $K_H\ne p_i^*$). The excess Gibbs energy $G^E=G^{real}-G^{ideal}$ (and analogous $H^E$, $S^E$, $V^E$) measures, introductorily, how far a real mixture departs from ideal-solution behaviour.

One mole of $\text{CO}_2$ gas at 400 K occupies 5.0 L. Using van der Waals constants $a=3.640\ \text{L}^2\text{atm mol}^{-2}$, $b=0.04267\ \text{L mol}^{-1}$, find the pressure predicted by the van der Waals equation, compare it with the ideal-gas prediction, and compute $Z$.

Step 1. Ideal-gas prediction first, as a baseline: $P_{ideal}=\dfrac{nRT}{V}=\dfrac{(1)(0.08206)(400)}{5.0}=6.565\ \text{atm}$, using $R=0.08206\ \text{L atm K}^{-1}\text{mol}^{-1}$ to match the given units.

Step 2. Apply the van der Waals formula for $P$ derived above, $P=\dfrac{nRT}{V-nb}-\dfrac{an^2}{V^2}$. The first term: $\dfrac{(1)(0.08206)(400)}{5.0-(1)(0.04267)}=\dfrac{32.824}{4.9573}=6.621\ \text{atm}$. The second term: $\dfrac{(3.640)(1)^2}{(5.0)^2}=\dfrac{3.640}{25.0}=0.1456\ \text{atm}$.

Step 3. Combine: $P_{vdW}=6.621-0.1456=6.475\ \text{atm}$. This is lower than the ideal prediction of $6.565\ \text{atm}$ — at these moderate conditions the attractive term outweighs the excluded-volume term for $\text{CO}_2$.

Step 4. Compute $Z$ using the van der Waals pressure: $Z=\dfrac{PV}{nRT}=\dfrac{(6.475)(5.0)}{(1)(0.08206)(400)}=\dfrac{32.375}{32.824}=0.9863$. Since $Z<1$, real $\text{CO}_2$ here is slightly more compressible than an ideal gas, with actual pressure a little below the ideal-gas prediction — exactly the direction expected for a gas with significant intermolecular attraction.

• Applying the fugacity or activity correction and then plugging the original uncorrected $p$ or $x_i$ back into an equilibrium formula — once $f$ or $a_i$ is introduced, every downstream formula must use it consistently.
• Confusing the Raoult's-law reference ($\gamma_i\to1$ as $x_i\to1$) with the Henry's-law reference ($\gamma_i\to1$ as $x_i\to0$) — the numerical $\gamma_i$ for the same real solution differs depending on which convention is used.
• Treating van der Waals $a$ and $b$ as universal constants rather than substance-specific empirical fits, and forgetting the equation is only a first-order correction that breaks down near the critical point.
• Reading $Z\ne1$ as "the gas disobeys thermodynamics" — non-ideality is fully absorbed by using $f$ (or $Z$) consistently inside the ideal-gas-shaped formulas; the framework itself remains exact.

Ask the tutor: if the Raoult's-law versus Henry's-law reference-state choice still feels arbitrary, ask the AI tutor to compute $\gamma_i$ for the same chloroform data under both conventions side by side.