Speed Of Sound In Gases

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A physics demonstration compares how fast sound travels through hydrogen versus oxygen at the same temperature, given that oxygen molecules are sixteen times more massive than hydrogen molecules. By what approximate factor is sound faster in hydrogen?

RankGuru · Accepted Answer

The speed of sound in an ideal gas is given by Laplace's relation $v = \sqrt{\gamma RT/M}$, where $M$ is the molar mass and $\gamma$ is the ratio of specific heats. At the same temperature, and treating both as diatomic gases with the same $\gamma$, the speed depends only on molar mass through $v \propto 1/\sqrt{M}$. Hydrogen and oxygen are both diatomic, so the factor $\gamma$ cancels exactly, and the ratio of the two sound speeds is $\frac{v_H}{v_O} = \sqrt{M_O/M_H} = \sqrt{16} = 4$. The option of 16 times forgets to take the square root of the mass ratio. The option of 2 times would correspond to a mass ratio of only 4, not 16. The option of 8 times has no basis in the inverse-square-root dependence. This applies the NCERT Laplace correction for sound speed in gases, including the role of molar mass. As a plausibility check, lighter molecules move faster at a given temperature and transmit pressure disturbances more quickly, so hydrogen must carry sound faster, and the factor of 4 follows directly from the square root of the sixteen-fold mass difference.

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