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: Understanding the Concept:
The velocity of sound in a gas depends on the properties of the gas, specifically its elasticity (bulk modulus) and its density. For an ideal gas, these properties can be related to the temperature.
Step 2: Key Formula or Approach:
The formula for the velocity of sound (\( v \)) in an ideal gas is given by the Laplace-Newton formula:
\[ v = \sqrt{\frac{\gamma P}{\rho}} \] where \( \gamma \) is the adiabatic index (ratio of specific heats), P is the pressure, and \( \rho \) is the density.
We also use the ideal gas law, which relates pressure, volume, and temperature: \( PV = nRT \). This can be rewritten in terms of density.
Step 3: Detailed Explanation:
From the ideal gas law, \( PV = nRT \), where n is the number of moles.
The number of moles \( n \) can be written as \( \frac{m}{M} \), where \( m \) is the mass of the gas and \( M \) is the molar mass.
So, \( PV = \frac{m}{M}RT \).
Rearranging, we get \( P = \frac{m}{V} \frac{RT}{M} \).
The density \( \rho \) is defined as mass per unit volume, \( \rho = \frac{m}{V} \).
Substituting this, we get \( P = \rho \frac{RT}{M} \), which gives us the ratio \( \frac{P}{\rho} = \frac{RT}{M} \).
Now, substitute this ratio back into the formula for the velocity of sound:
\[ v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\gamma \left(\frac{RT}{M}\right)} \] In this final expression, \( \gamma \), R (the universal gas constant), and M (the molar mass of the gas) are constants for a given gas. Therefore, the velocity of sound \( v \) is directly proportional to the square root of the absolute temperature T.
\[ v \propto \sqrt{T} \] Step 4: Final Answer:
The velocity of sound in a gas is directly proportional to \( \sqrt{T} \). This corresponds to option (A).