Question

The ratio of the speed of sound in hydrogen gas to the speed of sound in oxygen gas at the same temperature is:

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

Hint:

The speed of sound in a gas varies as: \[ v \propto \frac{1}{\sqrt{M}} \] - A lighter gas (lower molar mass) has a higher speed of sound. - The speed of sound in hydrogen is 4 times that in oxygen.

Answer 1

The Correct Option is A

Step 1: Using the formula for the speed of sound in a gas. The speed of sound in a gas is given by: \[ v = \sqrt{\frac{\gamma R T}{M}} \] where: - \( v \) = speed of sound, - \( \gamma \) = adiabatic index (assumed same for both gases), - \( R \) = universal gas constant, - \( T \) = temperature (same for both gases), - \( M \) = molar mass of the gas. Since \( \gamma \), \( R \), and \( T \) are the same for both gases, the speed of sound is inversely proportional to the square root of the molar mass: \[ v \propto \frac{1}{\sqrt{M}} \] 
Step 2: Finding the molar masses. - Molar mass of hydrogen (\( H_2 \)): \( M_H = 2 \) g/mol. - Molar mass of oxygen (\( O_2 \)): \( M_O = 32 \) g/mol. 
Step 3: Taking the ratio of speeds. \[ \frac{v_H}{v_O} = \sqrt{\frac{M_O}{M_H}} \] \[ \frac{v_H}{v_O} = \sqrt{\frac{32}{2}} \] \[ \frac{v_H}{v_O} = \sqrt{16} = 4 \] 
Final Answer: \[ \boxed{4:1} \]