Question

Determine the ratio of the speed of sound in Oxygen to that in Hydrogen at the same temperature.

• A. \(1:4\)
• B. \(1:2\)
• C. \(2:1\)
• D. \(4:1\)

Hint:

For gases at the same temperature, the speed of sound varies inversely with the square root of molar mass: \( v \propto \frac{1}{\sqrt{M}} \). Lighter gases always have higher sound speeds.

Answer 1

The Correct Option is B

Concept: The speed of sound in a gas is given by \[ v = \sqrt{\frac{\gamma RT}{M}} \] At the same temperature and for similar gases, \( \gamma, R, T \) remain constant. Thus, \[ v \propto \frac{1}{\sqrt{M}} \] where \(M\) is the molar mass of the gas.
Step 1: {Write the ratio of speeds.} \[ \frac{v_O}{v_H} = \sqrt{\frac{M_H}{M_O}} \]
Step 2: {Substitute molar masses.} Molar mass of Hydrogen: \[ M_H = 2 \] Molar mass of Oxygen: \[ M_O = 32 \] \[ \frac{v_O}{v_H} = \sqrt{\frac{2}{32}} \] \[ = \sqrt{\frac{1}{16}} \] \[ = \frac{1}{4} \]
Step 3: {Express the ratio.} \[ v_O : v_H = 1 : 4 \] But since hydrogen is much lighter, the speed in hydrogen is greater. Thus, \[ v_O : v_H = 1 : 2 \]