   # Ideal gas equation

Combining the equations PV = constant, P/T = constant and V/T = constant gives:

PV/T = constant

If we use 1 mole of gas the constant is known as the molar gas constant (R).
So for one mole of the ideal gas equation is:

Ideal Gas equation (one mole):          PV = RT

For a change from P1, V1 and T1 to P2, V2 and T2 the equation can be written:

P1V1/T1 = P2V2/T2

(This version is not restricted to one mole)

Example problem
0.3 m3 of an ideal gas are heated at constant pressure from 27o C to 127o C.
What is the new volume of the gas?

V2 = [V1T2]/T1 = [0.3 x 400]/300 = 0.4 m3 (Notice that the temperatures used are always in Kelvin!)

Now the volume of one mole of an ideal gas at Standard Temperature and Pressure (STP) (1.014x105 Pa and 273.15 K) is 0.0224m3 and so

1.014x105 x 0.0224 = 1 x R x 273.15 and therefore R = 8.314 JK-1mol-1.

For n moles this equation becomes:

Ideal Gas equation (n moles):         PV = nRT

Example problems
1. If 600g of argon have a pressure of 1.5x105 Pa and a volume of 0.3m3 what is the temperature of the gas?

Molar mass of argon = 40g and so we have 15 moles.
Therefore: T = PV/nR = [1.5x105x0.3]/[15x8.31] = 361 K = 88oC

We have of course assumed that argon behaves as an ideal gas.

2. A petrol - air mixture in the piston of a car engine initially has volume of 50cc, a temperature of 27o C and is at a pressure of 2x105 Pa. When it is ignited by the spark plug the volume increases to 450cc and the pressure drops to 8x104 Pa. What is the final temperature of the mixture. (assume that it behaves as an ideal gas!)

2x105x50)/300 = 8x104x450/T2
Therefore final temperature (T2) = [8x104x450x300]/[2x105x50] = 1080 K = 807 oC

The equation of state for 1 kg of the gas is:

PV = [R/M]T =rT

where M is the molar mass in kg (2 x 10-3 for hydrogen and 32 x 10-3 for oxygen, for example) and r is a further constant that depends on the gas under consideration. Therefore for m kg of the gas we have:

PV = mRT/M = mrT

For a fixed mass of gas whose conditions are changed from P1, V1 and T1 to P2, V2 and T2 the equation of state can be written:

P1V1/T1 = P2 V2/T2

Note that the temperature must always be measured in kelvins. 