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Difficulty: MediumHydrogen: Preparation, Properties, Isotopes, and Water Hardness

Under identical conditions of temperature and pressure, a given volume of deuterium gas (D2\text{D}_2) requires 40 seconds40\text{ seconds} to diffuse through a porous membrane. How long will it take for an equal volume of protium gas (H2\text{H}_2) to diffuse through the same membrane? (Relative atomic masses: H=1.0\text{H} = 1.0, D=2.0\text{D} = 2.0)

  1. 28.3 s28.3\text{ s}Answer
  2. B
    20.0 s20.0\text{ s}
  3. C
    56.6 s56.6\text{ s}
  4. D
    80.0 s80.0\text{ s}

Answer

The time required for an equal volume of protium gas to diffuse is 28.3 s28.3\text{ s}.
According to Graham's Law of diffusion, the time required for a fixed volume of gas to diffuse is directly proportional to the square root of its molar mass (t∝Mt \propto \sqrt{M}). Since protium gas (H2\text{H}_2, M=2.0 g mol−1M = 2.0\text{ g mol}^{-1}) is lighter than deuterium gas (D2\text{D}_2, M=4.0 g mol−1M = 4.0\text{ g mol}^{-1}), it diffuses faster, taking t=40×2/4=28.3 st = 40 \times \sqrt{2/4} = 28.3\text{ s}.

Step-by-Step Solution

1
Calculate the relative molecular masses of deuterium gas (D2\text{D}_2) and protium gas (H2\text{H}_2).
M(D2)=2×2.0=4.0 g mol−1M(\text{D}_2) = 2 \times 2.0 = 4.0\text{ g mol}^{-1} and M(H2)=2×1.0=2.0 g mol−1M(\text{H}_2) = 2 \times 1.0 = 2.0\text{ g mol}^{-1}.
Molecular masses are needed to apply Graham's Law of diffusion.
2
State Graham's Law relating diffusion time to molar mass for a constant volume.
tH2tD2=M(H2)M(D2)\frac{t_{\text{H}_2}}{t_{\text{D}_2}} = \sqrt{\frac{M(\text{H}_2)}{M(\text{D}_2)}}.
The time required for diffusion of a fixed volume is directly proportional to the square root of the molar mass of the gas.
3
Substitute the given values into the formula and solve for tH2t_{\text{H}_2}.
tH2=40×2.04.0=40×0.5≈40×0.7071=28.3 st_{\text{H}_2} = 40 \times \sqrt{\frac{2.0}{4.0}} = 40 \times \sqrt{0.5} \approx 40 \times 0.7071 = 28.3\text{ s}.
Performing the calculation yields the exact diffusion time for protium gas.

Key Concept

Graham's Law of Diffusion applied to hydrogen isotopes
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Topics

Hydrogen: Preparation, Properties, Isotopes, and Water Hardness

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