Physics

Physics

Problem

For accurate tuning of a musical instrument the speed of sound in hydrogen has to match the speed of sound in nitrogen at 300 K300\,\text{K}. Assume that both gases behave ideally and remain at the same pressure. Take molar masses MH2=2.0 g mol−1M_{\text{H}_{2}}=2.0\,\text{g\,mol}^{-1} and MN2=28.0 g mol−1M_{\text{N}_{2}}=28.0\,\text{g\,mol}^{-1}. For a rough estimate, use γ=1.40\gamma =1.40 for both gases. a. (i) Write the expression for the speed of sound in an ideal gas, clearly identifying each symbol used. b. (ii) Calculate the temperature (in ∘ ⁣C^\circ\!\text{C}) at which hydrogen must be kept so that its speed of sound equals that of nitrogen at 300 K300\,\text{K}. c. (iii) Comment briefly on whether such a temperature is practically attainable under ordinary laboratory conditions.
Intuitive explanation

Speed of sound in a gas depends on both temperature and molecular mass through the relation v=γRT/Mv = \sqrt{\gamma RT/M}. Since both gases are at the same pressure and temperature, you need to find what happens when you adjust one gas's temperature so their sound speeds match—this requires recognizing that the lighter hydrogen molecules move faster at any given temperature, so hydrogen must be cooled relative to nitrogen to equalize their speeds.

Intuitive explanation

Speed of sound in a gas depends on both temperature and molecular mass through the relation v=γRT/Mv = \sqrt{\gamma RT/M}. Since both gases are at the same pressure and temperature, you need to find what happens when you adjust one gas's temperature so their sound speeds match—this requires recognizing that the lighter hydrogen molecules move faster at any given temperature, so hydrogen must be cooled relative to nitrogen to equalize their speeds.

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