PV=nRT

The ideal gas Law

PV = nRT

Where does this come from?

Robert Boyle found

PV = a constant

That is, the product of the pressure of a gas times the volume of a gas is a constant for a given sample of gas. In Boyle's experiments the Temperature (T) did not change, nor did the number of moles (n) of gas present. So Boyle found

PV = (nRT)

but did not explore the effect the temperature, or the number of moles would have on pressure and volume.

Jaques Charles found

V = (a constant) T

That is, the volume of a given sample of gas increases linearly with the temperature if the pressure (P) and the amount of the gas (n) is kept constant. So Charles found

V = (nR/P) T

Avagadro's Postulate

At the same temperature and pressure equal volumes of all gasses contain the same number of molecules.

V = n (a constant)

V = n (RT/P)

Guy Lussac found that 1 volume of Cl₂ combined with 1 volume of H₂ to make 2 volumes of HCl. The equation for the reaction is

With this example we can clearly see the relationship between the number of moles of a gas, and the volume of a gas.

At constant temperature and pressure the volume of a gas is directly proportional to the number of moles of gas.

Not so coincidentally if V is constant instead of P then

P = n (RT/V)

At constant temperature and volume the pressure of a gas is directly proportional to the number of moles of gas.

You could remember all the different gas laws,

P₁V₁ = P₂V₂

P₁/T₁ = P₂/T₂

V₁/T₁ = V₂/T₂

and so on...

Or you could think about the problem a bit and use PV=nRT.

N₂O is placed in a piston. Initially the volume of the piston is 3.0 L, and the pressure of the gas is 5.0 atm. The piston is used to compress the gas to a volume of 1.5 L; determine the pressure of the N₂O.

well, before the compression

P₁V₁ = n₁R₁T₁

after expansion

P₂V₂ = n₂R₂T₂

since n, R, and T do not change

substituting

P = 1.0 x 10 L

See, if you forget all those different relationships you can just use PV=nRT.

A PV = nRT problem

What is the volume of 1 mole of an ideal gas at STP (Standard Temperature and Pressure = 0 °C, 1 atm)?

PV = nRT

(1) V = 1(0.08206)(273.15)

V = 22.41 L

So, the volume of an ideal gas is 22.41 L/mol at STP.

This, 22.4 L, is probably the most remembered and least useful number in chemistry.

Another example

What is the volume of 5.0 g NH3 at 25 °C and 1 atm. of pressure?

Well we just found that the volume of 1 mole of an ideal gas is 22.41 L so we can use this as a conversion factor...right?

Everyone remembers that 1 mol of an ideal gas occupies a volume of 22.4 L, but this is probably the least useful number in chemistry. Alot of people forget that this relationship is only true at STP (0 °C and 1 atm.).

So, use

PV=nRT

To use PV=nRT we need to have moles of NH₃.

It is not practical to use PV=nRT as a conversion in a factor label problem so we will just solve for V.

V = 7.18 = 7.2 L NH₃

Another Problem

Seltzer water is made by dissolving CO₂ in water. Seltzer can be made at home using small containers of pressurized CO₂. If one of the cartridges contains 20.00 mL CO₂ at 55.00 atm at 23.0 °C and it expands into an empty seltzer bottle with a volume of 1.000 L and the resulting pressure is 1.000 atm what is the temperature of the gas.

Before the gas expands...