Question

If \(T\) and \(\rho\) represent the temperature and density of a gas, then the velocity of sound in the gas is directly proportional to

• A. \(\sqrt{T}\)
• B. \(\sqrt{\rho}\)
• C. \(T\)
• D. \(\rho^2\)
• E. \(T^2\)

Hint:

Velocity of sound in a gas depends only on temperature, not density, when expressed using the ideal gas law.

Answer 1

The Correct Option is A

Step 1: Recall the formula for velocity of sound in a gas.
The velocity of sound in a gas is given by: \[ v=\sqrt{\frac{\gamma P}{\rho}} \] where \(\gamma\) is the ratio of specific heats, \(P\) is pressure, and \(\rho\) is density.

Step 2: Use the ideal gas equation.
From the ideal gas law: \[ P=\rho RT \] Substitute this into the expression for velocity: \[ v=\sqrt{\frac{\gamma (\rho RT)}{\rho}} \]

Step 3: Simplify the expression.
\[ v=\sqrt{\gamma RT} \] Thus, density \(\rho\) cancels out.

Step 4: Identify the dependency.
From the simplified expression: \[ v\propto \sqrt{T} \]

Step 5: Analyze the options.
Velocity is proportional to the square root of temperature, not directly to \(T\), \(\rho\), or their powers.

Step 6: Interpret physically.
Higher temperature means molecules move faster, so sound waves travel faster.

Step 7: State the final answer.
\[ \boxed{\sqrt{T}} \] which matches option \((1)\).