5.4 Real Gases & Deviations from Ideal Behavior (van der Waals Equation)

5.4 Real Gases & Deviations from Ideal Behavior (van der Waals Equation)

The Ideal Gas Law ($PV = nRT$) describes gases accurately at ambient conditions. However, real molecules possess finite physical volume and exert attractive and repulsive forces, causing deviations under extreme conditions.

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Conditions for Non-Ideal Behavior

Real gases diverge significantly from ideality under two primary conditions:

1. High Pressure: Compresses molecules into close proximity. The volume occupied by the molecules themselves becomes a substantial fraction of total container volume, violating the assumption of negligible particle size.
2. Low Temperature: Slows molecular velocity, reducing kinetic energy relative to intermolecular attractive potential energy. Attractions pull molecules toward one another during close encounters, violating the assumption of zero intermolecular forces.

Conversely, real gases behave most ideally at low pressure (particles widely separated) and high temperature (high kinetic energy overcomes intermolecular attractions).

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The Compressibility Factor ($Z$)

Deviations from ideality are measured by the compressibility factor ($Z$): $Z = \frac{PV}{nRT} = \frac{PV_m}{RT}$

For an ideal gas, $Z = 1.000$ at all temperatures and pressures. Real gases deviate in two distinct regimes:

• $Z < 1.0$ (Attractive Forces Dominate): At moderate pressures ($10$ to $300\text{ atm}$), attractive intermolecular forces draw molecules toward each other, reducing the frequency and momentum of collisions with container walls. The observed pressure is lower than ideal predictions, driving $PV/nRT < 1$.
• $Z > 1.0$ (Excluded Volume Dominates): At very high pressures ($> 400\text{ atm}$), molecules are forced tightly together. Repulsive electron-cloud interactions resist further compression. The actual free space available for motion ($V_{\text{free}}$) is significantly less than container volume, driving observed pressure higher than ideal predictions ($PV/nRT > 1$).

As temperature rises, thermal kinetic energy suppresses intermolecular attractions, flattening the $Z$ curve toward $1.0$ over wider pressure ranges.

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The van der Waals Equation of State

In 1873, Johannes Diderik van der Waals modified the ideal gas equation by introducing two substance-specific corrective constants: $\left(P + \frac{n^2 a}{V^2}\right)(V - nb) = nRT$

On a molar volume basis ($V_m = V/n$): $\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT$

The Attractive Term ($n^2 a / V^2$)

Intermolecular attractions reduce wall collision forces. The frequency of particle-particle interactions scales with concentration squared ($(n/V)^2$). Adding $n^2 a / V^2$ to observed pressure reconstructs hypothetical ideal pressure: $P_{\text{ideal}} = P_{\text{observed}} + \frac{n^2 a}{V^2}$ The constant $a$ quantifies the strength of attractive intermolecular forces ($\text{L}^2\cdot\text{atm}/\text{mol}^2$). Molecules with permanent dipoles or large, polarizable electron clouds have large $a$ values.

The Excluded Volume Term ($nb$)

Real molecules occupy physical space, meaning free volume available for motion is less than container volume: $V_{\text{ideal}} = V_{\text{container}} - nb$ The constant $b$ represents the excluded volume per mole of gas ($\text{L/mol}$). For spherical particles, $b$ is approximately four times actual hard-sphere volume ($b \approx 4 N_A \cdot \frac{4}{3}\pi r^3$) due to mutual spatial exclusion.

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Comparative Table of van der Waals Constants

• Constant $a$ trends: Nonpolar, low-electron species ($\text{He}, \text{H}_2$) have minimal $a$ values, whereas strongly polar molecules capable of hydrogen bonding ($\text{NH}_3, \text{H}_2\text{O}$) display large $a$ constants.
• Constant $b$ trends: Scales with molecular radius and atomic count, being smallest for helium and larger for polyatomic molecules like $\text{CH}_4$ and $\text{CO}_2$.

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Critical Phenomena & Supercritical Fluids

Compressing a gas at constant temperature induces condensation into a liquid if thermal kinetic energy is low enough for attractions to bind molecules.

• Critical Temperature ($T_c$): The highest temperature at which a substance can exist as a liquid, regardless of applied pressure. Above $T_c$, kinetic energy permanently overcomes intermolecular attractions.
• Critical Pressure ($P_c$): The minimum pressure required to liquefy a gas at its critical temperature.
• Supercritical Fluid: The state achieved when $T > T_c$ and $P > P_c$. The liquid-gas boundary vanishes, creating a dense fluid with gas-like diffusion and liquid-like solvent properties (e.g., supercritical $\text{CO}_2$ in decaffeination).

Gases with $T_c$ below room temperature ($298\text{ K}$), such as $\text{N}_2$ ($T_c = 126\text{ K}$) and $\text{CH}_4$ ($T_c = 191\text{ K}$), cannot be liquefied at room temperature by pressure alone.

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Worked Example: Ideal vs van der Waals Pressure

Calculate the pressure exerted by $1.000\text{ mol}$ of chlorine gas ($\text{Cl}_2$) in a $2.000\text{ L}$ vessel at $0.0^\circ\text{C}$ ($273.15\text{ K}$) given $a = 6.49\text{ L}^2\cdot\text{atm}/\text{mol}^2$ and $b = 0.0562\text{ L/mol}$.

1. Ideal Gas Law: $P_{\text{ideal}} = \frac{(1.000)(0.08206)(273.15)}{2.000} = 11.21\text{ atm}$
2. Van der Waals Equation: $P = \frac{nRT}{V - nb} - \frac{n^2 a}{V^2}$
   • $V - nb = 2.000 - (1.000)(0.0562) = 1.9438\text{ L}$
   • Kinetic term: $\frac{(1.000)(0.08206)(273.15)}{1.9438} = 11.53\text{ atm}$
   • Attraction term: $\frac{(1.000)^2(6.49)}{(2.000)^2} = 1.62\text{ atm}$
   • Observed pressure: $P_{\text{vdW}} = 11.53 - 1.62 = 9.91\text{ atm}$

Actual pressure is $1.30\text{ atm}$ lower than ideal ($Z = 0.884$) due to intermolecular attractions between polarizable chlorine molecules.