Gases

Engineers handle gases constantly: natural gas in pipelines, compressed air in plants, combustion gases in furnaces and engines, oxygen and nitrogen in cylinders, refrigerants in heat pumps, and the air in every ventilation system. Unlike liquids and solids, gases expand to fill their containers and change volume dramatically with pressure and temperature. Getting those changes right is a matter of safety as well as design: a propane cylinder left in the sun, or a compressed-air receiver with a faulty relief valve, stores a great deal of energy.

This chapter develops the ideal gas law and its special cases, shows how to use it for gas stoichiometry and mixtures, explains the molecular picture behind it, and ends with real gases and when the ideal law stops being reliable.

Pressure and temperature

Pressure is force per unit area. The SI unit is the pascal (1 Pa = 1 N/m²), which is small, so engineers use kilopascals and megapascals. Standard atmospheric pressure is 101.325 kPa (1 atm, 1.01325 bar, 14.7 psi).

Two kinds of pressure reading cause frequent mistakes. Absolute pressure is measured from a perfect vacuum. Gauge pressure, what most tire gauges and plant pressure gauges show, is measured relative to the surrounding atmosphere:

Gas laws always use absolute pressure and absolute temperature. Temperature in kelvin is \(T(\text{K}) = T(^\circ\text{C}) + 273.15\). There are no negative temperatures on the kelvin scale: 0 K, absolute zero, is the lowest possible temperature.

The ideal gas law

Experiments in the 1600s to 1800s established three relations for a fixed amount of gas: at constant temperature, \(PV\) is constant (Boyle's law); at constant pressure, \(V \propto T\) (Charles's law); and at constant volume, \(P \propto T\) (Gay-Lussac's law). Avogadro added that equal volumes of different gases at the same temperature and pressure contain the same number of molecules. Together these give the ideal gas law:

With \(P\) in kPa and \(V\) in litres, or \(P\) in Pa and \(V\) in m³, \(R = 8.314\) works directly. With \(P\) in atm and \(V\) in litres, use \(R = 0.08206\) L·atm/(mol·K).

Molar volume

At a given temperature and pressure, one mole of any ideal gas occupies the same volume. At 0 °C and 101.325 kPa it is 22.41 L; at 25 °C and 101.325 kPa it is 24.47 L. Be careful which "standard" conditions a data source uses: chemists, the natural gas industry and ventilation engineers each have their own reference states.

The combined gas law

For a fixed amount of gas whose conditions change, \(nR\) is constant, so

Gas density

Rearranging the ideal gas law with \(n = m/M\) gives the density of a gas:

Dry air (\(M = 28.97\) g/mol) at 20 °C and 101.3 kPa has a density of 1.20 kg/m³. Methane (16.04 g/mol) is 0.67 kg/m³, lighter than air, so natural gas leaks rise and disperse; propane (44.10 g/mol) is 1.83 kg/m³, heavier than air. Hydrogen sulphide (34.08 g/mol), a deadly gas found in sour oil and gas fields, is also heavier than air and pools in low areas and confined spaces.

Gas stoichiometry

Because equal volumes of gases at the same temperature and pressure contain equal numbers of moles, volumes of gases react in the same ratios as the coefficients of the balanced equation. This shortcut makes combustion air calculations quick.

Mixtures and partial pressures

In a mixture of gases, each gas behaves as if it alone filled the container. Its partial pressure is the pressure it would exert on its own, and Dalton's law says the partial pressures add to the total:

where \(y_A\) is the mole fraction of A (moles of A divided by total moles). For ideal gases, mole fraction equals volume fraction, which is why gas compositions are usually quoted by volume.

Partial pressure is what matters for breathing, for how much gas dissolves in a liquid, and for chemical reaction rates. At 5000 m the air is still 20.95% oxygen, but its partial pressure has fallen from 21.2 kPa to 11.3 kPa, which is why climbers need supplemental oxygen and why aircraft cabins are pressurized.

The kinetic-molecular picture

Why do all gases follow the same law? The kinetic theory of gases models a gas as a huge number of tiny molecules in constant random motion, colliding elastically with each other and the walls. Pressure is the average force of those collisions per unit area. Working through the mechanics shows that the average kinetic energy of a molecule depends only on the temperature:

with \(M\) in kg/mol. At 25 °C, nitrogen molecules move at about 515 m/s on average, faster than sound in air. Lighter molecules move faster at the same temperature, which is why hydrogen leaks through seals that hold nitrogen, and why gases with different masses can be separated by diffusion.

Real gases

The ideal gas law assumes the molecules take up no space and do not attract each other. Both assumptions fail at high pressure, where the molecules are crowded together, and at low temperature, near condensation. Engineers measure the deviation with the compressibility factor \(Z = PV/(nRT)\), which is exactly 1 for an ideal gas. Natural gas in high-pressure pipelines typically has \(Z\) around 0.85 to 0.95, and gas meters at custody-transfer points correct for it.

A simple model that captures both effects is the van der Waals equation:

The constant \(b\) accounts for the volume of the molecules themselves, and \(a\) for the attraction between them, which reduces the pressure on the walls.

As a rule of thumb, the ideal gas law is accurate to a few percent for air and similar gases at pressures up to about 1 MPa and temperatures well above their boiling points. Beyond that, use compressibility charts, an equation of state such as Peng–Robinson, or property tables.