The Gaseous State — Notes

This chapter is for formative assessment only (not in the annual theory paper). A gas is described by $p$, $V$, $T$ and $n$. The state of a substance depends on the balance between thermal energy (which keeps molecules apart) and intermolecular forces (which pull them together): in a gas thermal energy dominates. Forces in brief: dispersion (all molecules, weak, grows with size), dipole–dipole (polar molecules), hydrogen bonding (H on N, O, F), ion–dipole. Units: 1 atm = 101.325 kPa = 760 mmHg = 1.01325 bar; $T = t + 273.15$.

Boyle: at constant $n$ and $T$, $p \propto 1/V$, so $p_1V_1 = p_2V_2$. $p$ against $V$ is a hyperbola; $p$ against $1/V$ is a straight line of slope $nRT$.

Charles: at constant $n$ and $p$, $V \propto T$ (in kelvin), so $V_1/T_1 = V_2/T_2$. $V$ against $t$ in °C is a straight line meeting $V = 0$ at −273.15 °C.

Combined: $\dfrac{p_1V_1}{T_1} = \dfrac{p_2V_2}{T_2}$.

Gay-Lussac (pressure law): at constant $n$ and $V$, $p \propto T$. Avogadro: at the same $T$ and $p$, $V \propto n$; equal volumes of gases have equal numbers of molecules. Together: $pV = nRT$.

R depends on units: 0.08314 L bar K⁻¹ mol⁻¹; 0.0821 L atm K⁻¹ mol⁻¹; 8.314 J K⁻¹ mol⁻¹; 62.36 L mmHg K⁻¹ mol⁻¹. (1 L bar = 100 J.)

Density and molar mass: $d = \dfrac{pM}{RT}$ and $M = \dfrac{dRT}{p}$; at fixed $p$ and $T$ density is proportional to molar mass.

Dalton’s law: $p_\text{total} = p_1 + p_2 + \dots$, with $p_i = x_i\,p_\text{total}$ and $x_i = n_i/n$.

Gas collected over water: $p_\text{gas} = p_\text{total} - p_{\ce{H2O}}$. Vapour pressure of water (mmHg): 17.5 at 20 °C, 23.8 at 25 °C, 26.7 at 27 °C, 31.8 at 30 °C, 55.3 at 40 °C.

Real gases deviate because (i) molecules have a finite volume (important at high $p$) and (ii) there are attractions (important at low $T$ and moderate $p$). A real gas is closest to ideal at low pressure and high temperature.

$Z = \dfrac{pV}{nRT}$. $Z = 1$ ideal; $Z < 1$ attractions dominate; $Z > 1$ molecular volume dominates. Z against p: H₂ and He at 273 K start at 1 and rise; N₂ dips slightly then rises; CO₂ dips far below 1 then rises.

$\left(p + \dfrac{an^2}{V^2}\right)(V - nb) = nRT$. a (L² bar mol⁻²) corrects for attractions (larger a: easier to liquefy); b (L mol⁻¹) is the excluded volume per mole, about four times the actual volume of the molecules. Examples: He a = 0.0346, b = 0.0238; CO₂ a = 3.640, b = 0.0427.

Liquefaction: above the critical temperature Tc no pressure liquefies a gas; the critical pressure pc is the pressure needed at Tc. Tc: CO₂ 304.1 K, NH₃ 405.5 K, Cl₂ 417 K, O₂ 154.6 K, N₂ 126.2 K, H₂ 33.2 K. Gases with large a have high Tc.