### 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^{-1}K^{-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 Nmc^{2}

For one mole of gas N is Avagadro’s number *N*_{A}

so pV = 1/3 *N*_{A }mc^{2}

we can rewrite this as pV = 2/3 *N*_{A}(^{1}/_{2}mc^{2}) (separating out^{1}/_{2}mc^{2 } 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 *N*_{A}(^{1}/_{2}mc^{2}) and then multiply both sides by 3 and dividing both by 2* N*_{A}

3nRT/2* N*_{A} = (^{1}/_{2}mc^{2})

**now ^{1}/_{2}mc^{2} is the average kinetic energy of a molecule of the gas. R and N_{A} are both constants so ^{1}/_{2}mc^{2} = 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/ N_{A}**

**is called Boltzmann’s constant**, given the

**symbol**, so from the equation:

*k*pV = nRT

n = N (the total number of molecules present)

*N*_{A} (Avagadro’s number)

so PV = NRT/*N*_{A} but R/*N*_{A }is Boltzmann’s constant so:

** pV = N kT**