The ideal gas equation
The ideal gas equation
From Boyle’s Law we have p x V = constant
and from Charles’s Law V/T = constant
each for a fixed mass of gas and assuming the gases act as an ideal gas.
It follows then that p V = constant x T for a fixed mass of gas where temperature is measured in Kelvin.
The value of the constant will depend on the actual mass/quantity of gas.
If we use one mole of a gas and the pressure is low so that it approximates to an “ideal” gas then the constant is the same for all.
The value of the constant is 8.31 J mol-1K-1 and it is given the symbol R
so pV =RT for one mole
or pV = nRT for n moles
The analysis of the kinetic theory of gases gave us the equation pV = 1/3 Nmc2
For one mole of gas N is Avagadro’s number NA
so pV = 1/3 NA mc2
we can rewrite this as pV = 2/3 NA(1/2mc2) (separating out1/2mc2 as the average kinetic energy of a molecule)
The ideal gas equation is pV = nRT so we can put the two together:
nRT = 2/3 NA(1/2mc2) and then multiply both sides by 3 and dividing both by 2 NA
3nRT/2 NA = (1/2mc2)
now 1/2mc2 is the average kinetic energy of a molecule of the gas. R and NA are both constants so 1/2mc2 = constant x T so the Kelvin temperature of the gas is directly related to the kinetic energy of the molecules.
Boltzmann’s constant
Boltzmann’s constant
The ratio R/NA is called Boltzmann’s constant, given the symbol k, so from the equation:
pV = nRT
n = N (the total number of molecules present)
NA (Avagadro’s number)
so PV = NRT/NA but R/NA is Boltzmann’s constant so:
pV = NkT
Useful pages on the kinetic theory of gases and the gas laws are: