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 out1/2mc 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