In a molecular-beam experiment

CHEMISTRY · Multi-part · AP · Chemistry · Properties of Substances and Mixtures · Properties of Substances and Mixtures

Problem

In a molecular-beam experiment, a collimated stream of hydrogen molecules (

\text{H}_2

) passes through a small aperture. A detector records the time of flight of molecules over a fixed distance, allowing the experimenters to determine the molecules’ root-mean-square speed. At

280\,\text{K}

the measured

v_{\text{rms}}

1.60 \times 10^3\,\text{m s}^{-1}

. The experiment is then repeated at

560\,\text{K}

. Later, the same apparatus is used with an unknown diatomic gas (assumed ideal) at

280\,\text{K}

. The detector reports a

v_{\text{rms}}

equal to the value previously observed for hydrogen at

560\,\text{K}

. (The molar mass of

\text{H}_2

2.0\,\text{g mol}^{-1}

. Universal gas constant

R = 8.31\,\text{J mol}^{-1}\,\text{K}^{-1}

.) (a) Derive an expression for the ratio of the root-mean-square speeds of the same gas at two different absolute temperatures. (b) Using your result from part (a), calculate the expected

v_{\text{rms}}

of hydrogen molecules at

560\,\text{K}

. (c) Determine the molar mass of the unknown diatomic gas, given that its

v_{\text{rms}}

280\,\text{K}

equals the value obtained in part (b). (d) The rate at which a gas effuses through a small opening is proportional to its

v_{\text{rms}}

. Explain qualitatively how the effusion rate of the unknown gas at

280\,\text{K}

compares with that of hydrogen at the same temperature.

Related theory

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