Ideal gases and the gas laws

Three empirical laws

The gas laws were found by experiment, squeezing and warming and measuring, which is what empirical means. Each of them holds one variable fixed. Boyle's law says that at constant temperature, pV is constant.

Charles's law says that at constant pressure, V/T is constant. The pressure law says that at constant volume, p/T is constant. Both of those demand kelvin, and in Celsius they are simply false. Required practical 8 investigates Boyle's and Charles's laws directly, and the mass of gas stays fixed throughout each one.

A gas thermometer works on exactly this principle, that any physical property changing steadily with temperature can be used to measure it. Three are standard, each reliable over its own range. The volume of a trapped gas. The resistance of a thermistor. The e.m.f. of a thermocouple, whose tiny junction makes it the pick for fast-changing or very high temperatures.

Outranking them all is the thermodynamic scale, which depends on no substance at all. It starts at absolute zero, and the size of its unit comes from physics rather than from any thermometer's habits. Since 2019 the kelvin has been fixed by giving the Boltzmann constant the exact value k = 1.380 649 × 10⁻²³ J K⁻¹, which pins one kelvin to a definite amount of particle energy. The triple point of water used to do that job and no longer does, so a textbook defining the kelvin by it is out of date.

Absolute zero

Plot volume against Celsius temperature for any gas and the line comes out straight. Extrapolate it backwards and it reaches zero volume at −273 °C. Every gas's line meets the axis at the same place. The meeting is an extrapolation from the dilute-gas region where gases behave nearly ideally, since real gases condense long before the axis; that shared intercept defines absolute zero, 0 K, the temperature at which particle energy sits at its minimum. 0 °C is the freezing point of water, which is 273 degrees above absolute zero and has no special status on the thermodynamic scale.

One equation of state

Set the three laws side by side and a single proportionality covers all of them, pV ∝ T$pV \propto T$pV ∝ T, with T the thermodynamic temperature in kelvin. A gas that obeys that at every pressure is, by definition, an ideal gas. Nothing real manages it perfectly. Laboratory gases come close over ordinary ranges of pressure and temperature. They depart furthest from it when compressed hard or cooled towards the temperature at which they would liquefy, which is where the assumptions of kinetic theory begin to fail.

Turn the proportionality into an equation and the constant records how much gas is there. That gives the ideal gas equation, written in two forms. For n moles,

with R = 8.31 J K⁻¹ mol⁻¹, the molar gas constant. For N molecules,

with k = 1.38 × 10⁻²³ J K⁻¹, the Boltzmann constant.

The Avogadro constant N_A = 6.02 × 10²³ mol⁻¹ is the exchange rate between the two, since N = nN_A and k = R/N_A. The same bridge links molar mass in kilograms per mole to molecular mass in kilograms per molecule, through M = N_Am. Use pV = nRT when the amount is given in moles and pV = NkT when it is given as a number of molecules.

One more tool. When a gas expands or is compressed at constant pressure, the work done is

which is not printed with the thermal equations (the booklet's only W = pΔV sits on the Engineering physics option page), and is how “work” from mechanics enters gas problems.