Using relative effusion rates to find an unknown molar mass

Question

A gas made up of atoms escapes through a pinhole 0.839 times as fast as N2 gas. Write the chemical formula of the gas.

Answer 1

The formula relating effusion of a gas to its mass is the following:

E = 1/(M)^1/2

To relate the effusion rate (E₁) and mass (M₁) of one gas to the effusion rate (E₂) and mass (M₂) of another gas, we simply write out our equation as a ratio like so:

(E₂/E₁) = (1/(M₂)^1/2)/(1/(M₁)^1/2)

Simplifying this, we then get:

(E₂/E₁) = (M₁^1/2)/(M₂^1/2)

From the given information, we know the ratio of E₂ to E₁ is equal to 0.839 and we know that M₁ is the molar mass of nitrogen gas, which is 28.01. Plugging this into our equation, we get:

.839 = (28.01^1/2)/(M₂^1/2)

.839 = 5.29/M₂^1/2

M₂^1/2 = 5.29/.839

M₂^1/2 = 5.29/.839

M₂^1/2 = 6.31

M₂ = 39.9

Looking at the periodic table, the closest element to our answer is argon. Since there are no diatomic gasses that have a molar mass of half of our answer, we can conclude that the mystery gas is indeed:

Ar

Answer 2

r₂/r₁ = sqrt(M₂/M₁) Graham's Law of diffusion/effusion r = rate, M is molar mass, and 2 is the case you are looking for.

M₂ = M₁(r₁/r₂)² after rearranging

M₂ = 28 *(1/.839)²

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