Gas Mixtures and Partial Pressure

So far we have only been dealing with pure gases. How do we apply the gas laws to mixtures of two or more gases?

Dalton’s Law of Partial Pressures ® The total pressure of a mixture of gases equals the sum of the pressures that each gas would exert if it were alone.

Mathematically Dalton’s Law of Partial Pressures can be written as:

P_tot = P₁ + P₂ + P₃ + …

This property greatly simplifies our life (actually it probably doesn’t simplify your life that much, but it makes working the homework, quiz and exam problems a lot easier).

When we apply the ideal gas law to mixtures of gases each component gas will have its own P and n, but all of the component gases will have the same T and V.

Example

What is the total pressure exerted by a mixture of 2.00 g of H₂ (MW = 2.016 g/mol), 8.00 g of N₂ (MW = 28.01 g/mol) and 12.0 g of Ar (AW = 39.95 g/mol) at 273 K in a 10.0 L vessel?

First we need to calculate the number of moles of each gas:

2.00 g H₂ ´ [1 mol/2.016 g] = 0.992 mol

8.00 g N₂ ´ [1 mol/28.01 g] = 0.286 mol

12.0 g Ar ´ [1 mol/39.95 g] = 0.300 mol

Now we can calculate the partial pressure of each gas:

P(H₂) = nRT/V = [(0.992)(0.08206)(273)]/(10.0) = 2.22 atm

P(N₂) = nRT/V = [(0.286)(0.08206)(273)]/(10.0) = 0.641 atm

P(Ar) = nRT/V = [(0.300)(0.08206)(273)]/(10.0) = 0.672 atm

We can now use the law of partial pressures to calculate the total pressure:

P_tot = P(H₂) + P(N₂) + P(Ar) = 2.22 + 0.641 + 0.672 = 3.53 atm

A somewhat shorter approach to the same answer would have been to total the number of moles before applying the ideal gas law:

n_tot = n(H₂) + n(N₂) + n(Ar) = 0.922 + 0.286 + 0.300 = 1.578 mol

P_tot = nRT/V = [(1.578)(0.08206)(273)]/(10.0) = 3.53 atm

Another way we can use Dalton’s Law of Partial Pressures is to calculate the partial pressure of any gas in a mixture if we know the total pressure and the number of moles of each gas (and vice versa).

P₁/P_tot = n₁/n_tot

Example

If a gas which is 5 mol% H₂ and 95 mol% N₂ is present in a container at 200 atm, what is the partial pressure of each gas?

P(H₂) = P_totn_H2/n_tot = [(200 atm)(0.05n_tot)]/( n_tot) = 10 atm

P(N₂) = P_totn_N2/n_tot = [(200 atm)(0.95n_tot)]/( n_tot) = 190 atm