Real Gases & the Limits of the Ideal Model

Divide the measured pV by nRT and you have Z, the compressibility factor. For an ideal gas Z = 1 at every pressure and temperature; for a real gas it is a function Z(p, T) whose shape is the whole story of this topic. Two facts hold for every gas. First, Z → 1 as p → 0, so the ideal gas is the low-density limit of every real one, not an approximation to some gases but the common limit of all of them, which is why R is universal. Second, at fixed T the curve leaves 1 with a slope whose sign depends on temperature: for a cool gas Z first falls below 1, reaches a minimum and then climbs; for a hot gas it rises from the start. CO₂ at STP has Vₘ = 22.26 L mol⁻¹ against the ideal 22.41 L mol⁻¹, so Z = 0.993, a 0.7% departure at one atmosphere. Compress the same gas to 40 bar at 300 K and Z ≈ 0.77, a 23% shortfall in volume. Z < 1 says the gas is easier to compress than an ideal one, which only attraction between molecules can explain; Z > 1, reached by every gas at high enough pressure, says it is harder, which is the molecules' own volume refusing to be squeezed out.

Treat each molecule as a hard sphere of diameter d. Two centres cannot come closer than d, so around any molecule there is a sphere of radius d, volume (4/3)πd³ = 8 × (π/6)d³, that the centre of another molecule cannot enter. That is eight molecular volumes, but the exclusion belongs to a pair and is counted once per pair, so the volume denied to each molecule is four times its own volume, and per mole b = 4N_A(π/6)d³. The volume actually available to the gas is V − nb, and the ideal law applies to that: p(V − nb) = nRT, which raises p above nRT/V. Run the argument backwards on nitrogen: b = 3.87 × 10⁻⁵ m³ mol⁻¹ gives a molecular volume b/(4N_A) = 1.61 × 10⁻²⁹ m³ and d = (6 × 1.61 × 10⁻²⁹/π)¹⁄³ = 3.1 × 10⁻¹⁰ m, a sensible 0.31 nm for N₂. Check the size of the correction: at 1 atm and 300 K, Vₘ = 0.0246 m³ mol⁻¹ and nb/V = 0.16%; at 100 atm it is 16%, no longer a correction but a main term.

A molecule deep inside the gas is pulled equally in all directions by its neighbours. One about to strike the wall has neighbours on only one side, so it feels a net inward pull that slows it just before impact, and it delivers less momentum than the kinetic model assumed. The reduction in wall pressure has two factors. The inward force on each arriving molecule is proportional to how many neighbours are within reach, so to the molar density n/V; and the rate at which molecules arrive at the wall is also proportional to n/V. Their product gives Δp = a(n/V)², so the pressure the gas actually exerts is p = nRT/(V − nb) − an²/V². Dimensions fix the constant: an²/V² is a pressure, so a is in Pa m⁶ mol⁻². Energetically, a/Vₘ is the attractive potential energy per mole that the gas has given up, and comparing it with RT tells you whether attraction matters. For CO₂ at 1 atm and 300 K, a/Vₘ = 0.364/0.0246 = 14.8 J mol⁻¹ against RT = 2494 J mol⁻¹, a 0.6% effect. At 40 bar, with Vₘ = 4.8 × 10⁻⁴ m³ mol⁻¹, it is 758 J mol⁻¹, 30% of RT, and the gas is well on its way to condensing.

Write the equation per mole, p = RT/(Vₘ − b) − a/Vₘ², multiply by Vₘ/RT and expand 1/(1 − b/Vₘ) as a geometric series, valid because b ≪ Vₘ: Z = 1 + (b − a/RT)/Vₘ + b²/Vₘ² + … The first term is the ideal gas, exactly, so the model passes its correspondence check: whatever a and b are, Z → 1 as Vₘ → ∞. The second term is the second virial coefficient B(T) = b − a/RT, and its sign decides the low-pressure behaviour of Z. When RT < a/b, attraction wins, B < 0 and Z dips below 1; when RT > a/b, size wins, B > 0 and Z rises from the start. The crossover, T_B = a/(Rb), is the Boyle temperature, where the two corrections cancel to first order and the gas obeys Boyle's law to higher pressure than at any other temperature. For nitrogen the model gives T_B = 0.137/(8.314 × 3.87 × 10⁻⁵) = 426 K; the measured value is about 327 K. The existence of a Boyle temperature and its order of magnitude are right; the number is 30% out, the first sign that two constants cannot carry the whole intermolecular force law.

Plot p against Vₘ at fixed T. For large T the term RT/(Vₘ − b) dominates and the isotherm is a displaced hyperbola. As T falls the −a/Vₘ² term carves a dent into it, and at one temperature the dent flattens into a horizontal inflection: (∂p/∂Vₘ)_T = 0 and (∂²p/∂Vₘ²)_T = 0 at the same point. Differentiate: −RT/(Vₘ − b)² + 2a/Vₘ³ = 0 and 2RT/(Vₘ − b)³ − 6a/Vₘ⁴ = 0. Divide the second by the first to eliminate RT and a at once: 2/(Vₘ − b) = 3/Vₘ, so V_c = 3b. Substitute back: RT_c = 2a(2b)²/(3b)³ = 8a/(27b), then p_c = RT_c/(2b) − a/(9b²) = a/(27b²). Two constants have produced three critical data, so one combination of them must be parameter-free: Z_c = p_cV_c/(RT_c) = (a/27b²)(3b)(27b/8a) = 3/8 = 0.375 for every substance. Measured values: argon 0.291, nitrogen 0.289, CO₂ 0.274, water 0.229. Fit a and b of CO₂ to its measured T_c = 304 K and p_c = 73.8 bar and the model predicts V_c = 3b = 128 cm³ mol⁻¹; the measured value is 94 cm³ mol⁻¹, 27% smaller. The model interpolates between two limits it gets right, and the critical region is where it is stretched thinnest.

For T < T_c the isotherm has a minimum and a maximum, and between them (∂p/∂V)_T > 0: compress the fluid and its pressure falls. No homogeneous substance can do that: a region slightly denser than its neighbours would have lower pressure, be squeezed further and run away, so the loop is never observed. Nature replaces it by a horizontal line at the vapour pressure pₛₐₜ(T), placed so that it cuts equal areas from the loop above and below (Maxwell's construction, from equal Gibbs energy of liquid and vapour). Along that line the fluid is a mixture: compress it and the volume falls at constant pressure as vapour turns to liquid, until only liquid is left and the steep liquid branch takes over. An ideal gas can do none of this: p = RT/Vₘ has (∂p/∂V)_T = −RT/Vₘ² < 0 at every point and every temperature, so no critical point, no vapour pressure, no liquid. Hence the laboratory cylinders: CO₂ (T_c = 304 K) sits as liquid under its own vapour pressure at room temperature, while nitrogen (T_c = 126 K) cannot be liquefied at 300 K by any pressure and is stored as gas at 200 bar.