Question

The mass of one mole of a gas is $22.4 \times 10^{-3}$ kg and its specific heat ratio is 1.6. The speed of sound in the gas at STP is nearly

• A. $402 \text{ ms}^{-1}$
• B. $292 \text{ ms}^{-1}$
• C. $302 \text{ ms}^{-1}$
• D. $312 \text{ ms}^{-1}$

Hint:

Speed of Sound: \[ v = \sqrt\frac\gamma R TM, \quad M~\textin kg/mol \] STP: $T = 273~K$, $R = 8.314$, ideal gas assumed.

Answer 1

The Correct Option is A

The speed of sound in an ideal gas is given by: \[ v = \sqrt{\frac{\gamma R T}{M}} \] Given: $\gamma = 1.6$, $R = 8.314$, $M = 22.4 \times 10^{-3}$ kg/mol, $T = 273$ K. Substituting: \[ v = \sqrt{\frac{1.6 \cdot 8.314 \cdot 273}{0.0224}} \approx \sqrt{162228} \approx 402.77 \text{ ms}^{-1} \] Hence, the answer is approximately $402 \text{ ms}^{-1}$, matching option (1).

Answer 2

Step 1: Understand the given data
- Molar mass of the gas, M = 22.4 × 10⁻³ kg/mol = 0.0224 kg/mol
- Specific heat ratio (γ) = 1.6
- Standard Temperature and Pressure (STP): T = 273 K, P = 1 atm (but pressure is not directly needed for speed of sound calculation in ideal gas)

Step 2: Formula for speed of sound in a gas
The speed of sound (v) in an ideal gas is given by:
v = √(γ × R × T / M)
where,
R = universal gas constant = 8.314 J/mol·K
T = temperature in Kelvin
M = molar mass in kg/mol

Step 3: Substitute values into the formula
v = √(1.6 × 8.314 × 273 / 0.0224)
Calculate inside the square root:
= √(1.6 × 8.314 × 273 / 0.0224)
= √(1.6 × 8.314 × 273 / 0.0224) ≈ √(162342)
= 402.92 m/s

Step 4: Conclusion
The speed of sound in the gas at STP is approximately 402 m/s.
This result shows how molar mass and specific heat ratio influence the propagation speed of sound in gases.