Ideal Gas Law

The equation

P · V = n · R · T

where:

• P = pressure (Pa or atm).
• V = volume (m³ or L).
• n = number of moles of gas.
• R = universal gas constant.
• T = absolute temperature (K).

R values:

• R = 8.314 J/(mol·K) — for SI units (P in Pa, V in m³).
• R = 0.08206 L·atm/(mol·K) — for chemistry (P in atm, V in L).

Or in molecular form:

P · V = N · k_B · T

where N is the number of molecules and k_B = R/N_A = 1.381 × 10⁻²³ J/K.

Historical sub-laws

Law	Year	Statement
Boyle's law	1662	P·V = constant (at fixed T, n)
Charles' law	1787	V/T = constant (at fixed P, n)
Gay-Lussac's law	1809	P/T = constant (at fixed V, n)
Avogadro's law	1811	V/n = constant (at fixed P, T)
Combined gas law	1834 (Clapeyron)	P·V/T = constant for fixed n
Ideal gas law	1834	P·V = n·R·T

Standard temperature and pressure

Old IUPAC definition (used in most textbooks):

• T = 0°C = 273.15 K.
• P = 1 atm = 101,325 Pa.
• 1 mol of any ideal gas occupies 22.41 L.

New IUPAC (1982):

• T = 273.15 K, P = 100 kPa.
• V = 22.71 L per mole.

NIST:

• T = 20°C = 293.15 K, P = 1 atm.
• V = 24.06 L per mole.

Processes for ideal gases

Process	Constraint	P-V-T relation
Isobaric (constant P)	P = const	V/T = const
Isochoric (constant V)	V = const	P/T = const
Isothermal (constant T)	T = const	P·V = const (Boyle)
Adiabatic (no heat)	Q = 0	P·V^γ = const, T·V^(γ-1) = const
γ (adiabatic index)	—	= C_p/C_v; ~1.4 for diatomic gases (air); ~1.67 for monoatomic

From kinetic theory

Consider a gas of N molecules of mass m in volume V at temperature T. Molecules bounce around with random velocities; their collisions with walls produce pressure.

Statistical analysis gives:

P · V = (1/3) · N · m · <v²> = (2/3) · N · <½ · m · v²>

The average translational kinetic energy is:

<KE> = (3/2) · k_B · T

(Equipartition theorem — ½k_B·T per degree of freedom.)

Substituting:

P · V = (2/3) · N · (3/2 · k_B · T) = N · k_B · T = n · R · T

Recovers the ideal gas law from microscopic mechanics. Beautiful unification.

JavaScript — ideal gas calculations

const R = 8.314;  // J/(mol·K)
const k_B = 1.380649e-23;
const N_A = 6.02214e23;

// Solve PV = nRT for any unknown
function pressure(n, V, T) { return n * R * T / V; }
function volume(n, P, T) { return n * R * T / P; }
function temperature(P, V, n) { return P * V / (n * R); }
function moles(P, V, T) { return P * V / (R * T); }

// 1 mole at STP
console.log(`1 mole at STP: V = ${volume(1, 101325, 273.15).toFixed(4)} m³`);  // 0.0224

// Inflate a balloon at room temp: 1 atm pressure, 1 L volume
const balloon = moles(101325, 0.001, 293);  // 0.0416 mol
console.log(`1 L at 20°C, 1 atm: ${balloon.toFixed(4)} mol = ${(balloon * 6.022e23).toExponential(2)} molecules`);

// Air pressure at altitude (using PV=nRT with T variation, density variation)
// At constant T (rough approx), P ∝ density ∝ n/V
function densityAtAltitude(P_alt, T_alt, M = 0.029) {  // M = 0.029 kg/mol for air
  // PV = nRT  →  ρ = nM/V = PM/(RT)
  return P_alt * M / (R * T_alt);
}

console.log(`Sea level air density: ${densityAtAltitude(101325, 293).toFixed(3)} kg/m³`); // ~1.20

// Charles' law: balloon expansion when heated
function newVolumeAtNewTemp(V_initial, T_initial, T_final) {
  return V_initial * (T_final / T_initial);  // at constant P
}

console.log(`1 L balloon, 25°C → 100°C: ${newVolumeAtNewTemp(1, 298, 373).toFixed(2)} L`); // 1.25 L

// Adiabatic process for diatomic gas
function adiabaticTemp(T_initial, V_initial, V_final, gamma = 1.4) {
  // T·V^(γ-1) = constant
  return T_initial * Math.pow(V_initial / V_final, gamma - 1);
}

console.log(`Adiabatic compression to half volume: T 300K → ${adiabaticTemp(300, 2, 1).toFixed(0)} K`);
// ~396 K (123°C — bike pump heats up!)

// Average molecular speed (rms)
function rmsSpeed(T, M_kg_per_mol) {
  // KE_total = (3/2)RT, KE per molecule = (3/2)k_B·T
  // ½ m v² = (3/2)k_B·T, so v = √(3k_B·T/m) = √(3RT/M)
  return Math.sqrt(3 * R * T / M_kg_per_mol);
}

console.log(`Air at 20°C, rms speed: ${rmsSpeed(293, 0.029).toFixed(0)} m/s`);  // ~503
console.log(`H₂ at 20°C, rms speed: ${rmsSpeed(293, 0.002).toFixed(0)} m/s`);   // ~1900 (light molecules fast)

Where ideal gas law shows up

• Engineering. Engine cycles (Otto, Diesel, Brayton), HVAC design, gas turbines, compressors, refrigeration.
• Chemistry. Stoichiometry of gas-phase reactions, gas chromatography, vapor pressure calculations.
• Atmospheric science. Pressure variation with altitude, adiabatic lapse rates, weather modeling.
• Aviation. Cabin pressurization, jet engine cycles, atmospheric properties at altitude.
• Astrophysics. Stellar structure (interior pressures), exoplanet atmospheres, interstellar medium.
• Industrial gas. Storage, transport, scuba tanks, medical gas supply.
• Education. Foundational thermodynamics topic; introduces molar quantities, kinetic theory.

Common mistakes

• Using Celsius in PV=nRT. MUST use Kelvin. Celsius gives 0 = 0 nonsense at freezing.
• Mixing units of R. R = 8.314 J/(mol·K) requires P in Pa, V in m³. R = 0.08206 L·atm/(mol·K) for P in atm, V in L. Don't mix.
• Forgetting "ideal" assumptions. At high P, low T, near condensation, real gases deviate. Use van der Waals or other real-gas equation when accuracy matters.
• Treating PV=nRT as exact. It's an approximation. Within its valid range (low to moderate P, T well above boiling), it's very good (< 1% error). Outside, deviations grow.
• Confusing n (moles) with mass. n = mass / molar_mass. Plugging mass directly gives wrong answer. Always convert mass to moles first.
• Forgetting that 22.4 L/mol is at STP only. At room temp (~293 K), it's 24.06 L/mol. Always check conditions.