Chemistry

Chemistry

Problem

A mixture in thermal equilibrium consists of equal numbers of hydrogen molecules containing normal hydrogen (H2\text{H}_2) and deuterium molecules (D2\text{D}_2). The mixture is held at constant temperature in a rigid vessel that has a very small pinhole leak to vacuum. Over an extended period, particles escape only through this opening; the vessel otherwise remains isolated and at the initial temperature. The molar masses are 2.02 g mol−12.02\,\text{g\,mol}^{-1} for H2\text{H}_2 and 4.04 g mol−14.04\,\text{g\,mol}^{-1} for D2\text{D}_2. Assume ideal‐gas behavior. (a) Calculate the ratio of the root‐mean‐square speeds vrms,H2/vrms,D2v_{\text{rms,H}_2}/v_{\text{rms,D}_2}. (b) Obtain the corresponding ratio of effusion rates for the two molecular species through the pinhole and state which species is lost more rapidly. (c) Suppose that after some time 20%20\% of the original H2\text{H}_2 molecules have effused from the vessel. Using your result from part (b) and assuming the relative effusion rates remain constant during this interval, estimate the percentage of the original D2\text{D}_2 molecules that have been lost.
Intuitive explanation

The key insight is that effusion rate depends on molecular speed, which is inversely related to mass. Lighter H2\text{H}_2 molecules move faster on average than heavier D2\text{D}_2 molecules at the same temperature, so they escape through the pinhole more frequently. Over time, this preferential loss of the lighter isotope shifts the composition of the remaining gas toward D2\text{D}_2 enrichment—a separation driven purely by kinetic differences, not chemical reactivity.

Intuitive explanation

The key insight is that effusion rate depends on molecular speed, which is inversely related to mass. Lighter H2\text{H}_2 molecules move faster on average than heavier D2\text{D}_2 molecules at the same temperature, so they escape through the pinhole more frequently. Over time, this preferential loss of the lighter isotope shifts the composition of the remaining gas toward D2\text{D}_2 enrichment—a separation driven purely by kinetic differences, not chemical reactivity.

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