Question

An ideal gas is compressed in volume by a factor of 2,while keeping its temperature constant. The speed of sound in it is: 

• A. doubled
• B. unchanged
• C. reduced to half
• D. increased by 4 times
• E. reduced by 4 times

Answer 1

The Correct Option is B

Given:

• An ideal gas is compressed in volume by a factor of 2 (V → V/2).
• The temperature (T) is kept constant during this process.
• We need to determine how the speed of sound in the gas changes.

Step 1: Speed of Sound in an Ideal Gas

The speed of sound (c) in an ideal gas is given by:

\[ c = \sqrt{\frac{\gamma RT}{M}} \]

where:

• γ = adiabatic index (ratio of specific heats)
• R = universal gas constant
• T = absolute temperature
• M = molar mass of the gas

 

Step 2: Analyze the Given Conditions

Key observations:

1. The temperature T is constant (given).
2. γ, R, and M are properties of the gas that remain unchanged.
3. The volume change affects density, but not the speed of sound directly.

 

Step 3: Conclusion

Since none of the terms in the speed of sound equation (γ, R, T, M) are changing, the speed of sound remains unchanged despite the volume compression.

Final Answer: The speed of sound in the gas is unchanged.

Answer: \(\boxed{B}\)

Answer 2

When an ideal gas undergoes isothermal compression (i.e., the temperature remains constant) and its volume is reduced by a factor of 2, we can determine the effect on the speed of sound in the gas.

The speed of sound in an ideal gas is given by the equation:
\[ v = \sqrt{\frac{\gamma P}{\rho}} \]

where:
- \( v \) is the speed of sound,
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( P \) is the pressure,
- \( \rho \) is the density of the gas.

For an ideal gas, we have the relation \( P = \rho RT \), where \( R \) is the specific gas constant and \( T \) is the temperature. The density \( \rho \) can be expressed in terms of pressure and temperature:
\[ \rho = \frac{P}{RT} \]

Substituting this into the speed of sound equation:
\[ v = \sqrt{\frac{\gamma P}{\frac{P}{RT}}} = \sqrt{\gamma RT} \]

Notice that in this form, the speed of sound \( v \) in an ideal gas depends only on the temperature \( T \) and the gas constant \( R \). It does not explicitly depend on the pressure or the volume of the gas.

Given that the temperature \( T \) remains constant during the isothermal compression and \( R \) is a constant for a given gas, the speed of sound in the gas does not change due to the isothermal compression.

Therefore, even though the volume is compressed by a factor of 2, the speed of sound in the gas remains the same as before the compression.

Thus, the speed of sound in the gas after the isothermal compression remains unchanged.
So The correct answer is Option is (B):unchanged