Ideal Gas vs Real Gas

Most students assume the ideal gas law works everywhere. It doesn't. The ideal gas vs real gas distinction matters because PV = nRT is a model, not a fact. It describes a hypothetical gas that has no intermolecular forces and occupies no volume. Real gases, every actual gas you will measure, deviate from this model. The question is not whether they deviate, but by how much and under what conditions. The ideal gas law (PV = nRT) is accurate at high temperatures and low pressures, roughly above 1.5 times a gas's boiling point and below about 5 atm for most diatomic gases. Outside that range, you need a real-gas equation.

The kinetic-molecular theory rests on four assumptions: gas particles have negligible volume compared to the container; they exert no forces on each other except during elastic collisions; their average kinetic energy is proportional to absolute temperature; and collisions with the container walls are perfectly elastic. These assumptions let you derive PV = nRT directly. At ordinary conditions, room temperature and one atmosphere, many gases come close enough that the error is under 1%. Oxygen at 298 K and 1 atm, for example, has a deviation of about 0.3%. But push to 10 atm or drop the temperature near the boiling point, and those small errors become large.

The breakdown happens in two ways. First, intermolecular attractions become significant at low temperatures. Molecules spend more time near each other, reducing the measured pressure compared to the ideal prediction. Second, at high pressures the volume of the molecules themselves is no longer negligible. The container's volume is partly occupied by the molecules, so the available empty space is smaller than the measured volume. These two effects act in opposite directions: attraction reduces pressure, molecular volume increases it. Which one dominates depends on the gas and the conditions.

The compressibility factor Z = PV/nRT quantifies the deviation. For an ideal gas, Z = 1 exactly. For a real gas, Z < 1 when attraction dominates (low temperature, moderate pressure) and Z > 1 when molecular volume dominates (very high pressure). Nitrogen at 0 °C and 200 atm has Z ≈ 1.02. Carbon dioxide at 50 °C and 200 atm has Z ≈ 0.60. The closer Z is to 1, the more trustworthy your ideal-gas calculation is.

The compressibility factor Z is the ratio of the actual molar volume to the ideal molar volume at the same temperature and pressure. Z = V_actual / V_ideal. In practice, you calculate Z as PV/nRT and compare it to 1. If Z is between 0.95 and 1.05, the ideal gas law is within 5% and is usually good enough for coursework. If Z is outside that range, you need a real-gas equation. Engineering textbooks provide generalized compressibility charts that plot Z against reduced pressure and reduced temperature. For a quick classroom check, remember: most diatomic gases stay within 1% of ideal behavior below 5 atm and above 200 K.

The van der Waals equation modifies PV = nRT with two constants that correct for intermolecular forces and molecular size. The equation is (P + a(n/V)²)(V - nb) = nRT. Parameter a (units: L²·atm/mol²) corrects for attractive forces between molecules. Larger a means stronger attraction. Parameter b (units: L/mol) accounts for the volume occupied by one mole of molecules. Larger b means larger molecules. The constants are determined experimentally and vary widely between gases. Helium, with very weak attractions, has a = 0.03412 L²·atm/mol² and b = 0.02370 L/mol. Carbon dioxide, with strong attractions, has a = 3.592 L²·atm/mol² and b = 0.04267 L/mol. Water vapor has a = 5.464 L²·atm/mol² and b = 0.03049 L/mol. The CRC Handbook of Chemistry and Physics publishes a full table of these constants.

How to Use the Van Der Waals Equation

To solve for pressure given n, V, and T, rearrange the equation: P = nRT/(V - nb) - a(n/V)². Use the same units as your constants. The van der Waals equation is cubic in V, so solving for volume requires a numerical method or a calculator with a solver. For coursework, you will usually be given n, T, and P and asked to find V, or given n, V, and T and asked to find P. The equation works reasonably well for gases near their critical point but fails near the condensation point, where more advanced formulas like Peng-Robinson are needed.

Van Der Waals Constants for Common Gases (CRC Handbook)

Below are values for a and b from the CRC Handbook. The constants vary slightly between editions; always use the edition your instructor specifies. Helium: a = 0.03412, b = 0.02370. Neon: a = 0.211, b = 0.0171. Argon: a = 1.345, b = 0.03219. Krypton: a = 2.318, b = 0.03978. Xenon: a = 4.194, b = 0.05105. Hydrogen: a = 0.2444, b = 0.02661. Nitrogen: a = 1.390, b = 0.03913. Oxygen: a = 1.358, b = 0.03183. Carbon dioxide: a = 3.592, b = 0.04267. Water: a = 5.464, b = 0.03049. Methane: a = 2.273, b = 0.04303. Ammonia: a = 4.170, b = 0.03707. Chlorine: a = 6.493, b = 0.05622. Sulfur dioxide: a = 6.714, b = 0.05636. Ethane: a = 5.489, b = 0.06380. Propane: a = 8.664, b = 0.08445. Butane: a = 14.66, b = 0.1226. Pentane: a = 19.09, b = 0.1460. Benzene: a = 18.24, b = 0.1154. Ethanol: a = 12.18, b = 0.08407. Chloroform: a = 15.17, b = 0.1022. Carbon tetrachloride: a = 20.39, b = 0.1383.

Compare the pressure predicted by the ideal gas law and the van der Waals equation for 1.00 mole of CO₂ at 298.15 K in a 10.0 L container. Use R = 0.082057 L·atm/(mol·K).Van der Waals: P = nRT/(V - nb) - a(n/V)². For CO₂, a = 3.592 L²·atm/mol² and b = 0.04267 L/mol. V - nb = 10.0 - (1.00)(0.04267) = 9.95733 L.a(n/V)² = 3.592(1.00/10.0)² = 3.592(0.01) = 0.03592 atm. So P = 2.456 - 0.03592 = 2.420 atm. The ideal-gas pressure (2.446 atm) is 1.1% higher than the van der Waals pressure (2.420 atm). At 1.00 atm and room temperature, the difference is smaller. At 50 atm, the difference would exceed 10%. The lesson: at moderate pressures and temperatures, the ideal gas law is fine; at high pressures, use the van der Waals equation.

For high school and first-year college chemistry, the ideal gas law covers the vast majority of problems. Use it when pressure is below 5 atm and temperature is above 200 K. Check whether your textbook uses the old STP (273.15 K, 1 atm, molar volume 22.414 L/mol) or the IUPAC STP (273.15 K, 1 bar, molar volume 22.710 L/mol). The 1.3% difference matters when significant figures are tight. For physics problems, use R = 8.314 J/(mol·K) because energy units are required. For engineering problems, you will encounter the specific gas constant R_specific = R_universal / M, and you must distinguish between per-mole and per-pound-mole values. The most common student error is using the wrong R value. The second most common is entering temperature in °C instead of K. The third is using the combined gas law when the number of moles changes. If you need to calculate gas density or molar mass, those topics have their own equations derived from PV = nRT. The van der Waals equation is the fallback when your instructor says 'use the real-gas correction.' For gases near condensation, do not trust the ideal gas law at all.