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Root Mean Square Speed and Graham's Laws

- A certain gas diffuses 2.5 times faster than CO2. What is the molecular mass of the gas?

- A certain gas diffuses 1.7 times slower than N2. What is the molecular mass of the gas?

-H2 effuses through a hole in a container in 4.5hr. How long would it take for CH4 to effuse?
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Answer:

1. 7.0 u; 2. 81 u; 3. 12.7 h.

Step-by-step explanation:

Graham’s Law applies to the diffusion of gases:

The rate of diffusion (r) of a gas is inversely proportional to the square root of its molar mass (M).

r = 1/√M

If you have two gases, the ratio of their rates of diffusion is

r₂/r₁ = √(M₁/M₂)

1. Molecular mass

r₂/r₁ = (2.5r₁)/r₁ = 2.5

2.5 = √(44.01/M₂)

Square both sides:                     6.25 = 44.01/M₂

Multiply each side by M₂:      6.25M₂ = 44.01

Divide each side by 6.25:             M₂ = 7.0 u

The molecular mass of the gas is 7.0 u.

2.  Molecular mass

r₂/r₁ = (r₁/1.7)/r₁ = 1/1.7

1/1.7 = √(28.01/M₂)

Square both sides:                      1/2.89 = 28.01/M₂

Multiply each side by M₂:        M₂/2.89 = 28.01

Multiply each side by 2.89:             M₂ = 81 u

The molecular mass of the gas is 81 u.

3. Time for diffusion

The time for effusion is inversely proportional to the rate. The faster the rate, the slower the time. Thus,

t₂/t₁ = √(M₂/M₁)

Multiply each side by t₁

t₂ = t₁√(M₂/M₁) = 4.5√(16.04/2.016) = 4.5√7.956 = 4.5 × 2.82 = 12.7 h

It would take methane 12.7 h to diffuse.

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