Describing a gas
A gas is described by four variables: pressure P, volume V, temperature T and amount n. Pressure units include atmospheres (atm), kilopascals and millimetres of mercury, with 1 atm = 101.325 kPa = 760 mmHg. In every gas law the temperature must be in kelvin; using degrees Celsius is the most common error in this topic.
The individual laws
- Boyle's law: at constant n and T, P × V is constant, so P1 V1 = P2 V2.
- Charles's law: at constant n and P, V / T is constant.
- Avogadro's law: at constant P and T, volume is proportional to moles.
- Combined gas law: P1 V1 / T1 = P2 V2 / T2 when the amount is fixed.
The ideal gas equation
Combining them gives PV = nRT. With pressure in atm and volume in litres, R = 0.08206 L·atm/(mol·K). Rearranged for density d = mass/V, the molar mass is M = dRT / P. At 273.15 K and 1 atm (the definition of standard temperature and pressure used here) one mole of ideal gas occupies about 22.41 L.
Mixtures and motion
Dalton's law: the total pressure is the sum of partial pressures, and each gas's partial pressure is its mole fraction times the total, P_A = x_A P_total. Graham's law: the rate of effusion is inversely proportional to the square root of molar mass, so lighter gases escape faster: rate1 / rate2 = sqrt(M2 / M1).
The kinetic molecular theory explains these laws by picturing tiny particles in constant random motion, far apart, with no attractions. The average kinetic energy is proportional to absolute temperature only: doubling T in kelvin doubles the average energy. Real gases deviate from ideality at high pressure (particle volume matters) and low temperature (attractions matter).
Worked example
A rigid 5.00 L tank holds 0.250 mol of an ideal gas at 310 K. Find the pressure, then the pressure if the tank is cooled to 155 K.
- P = nRT / V = (0.250)(0.08206)(310) / 5.00 = 1.27 atm.
- Because n and V stay fixed, pressure is proportional to T. Halving the kelvin temperature from 310 K to 155 K halves the pressure: 1.27 / 2 = 0.636 atm.
- Check with the gas law: (0.250)(0.08206)(155) / 5.00 = 0.636 atm. It agrees.
Notice that if you had halved the Celsius temperature instead (37 °C to 18.5 °C), you would have found an answer that is far too large. Whenever a problem lists a temperature in °C, convert first by adding 273.15, then solve.