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If the pressure, temperature and density of an ideal gas are denoted by P, T and ρ, respectively, the velocity of sound in the gas is
A) proportional to$\sqrt P $ , when $T$ is constant
B) proportional to $\sqrt T $
C) proportional to $\sqrt P $ , when $\rho $ is constant
D) proportional to $T$

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Last updated date: 13th Jun 2024
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Answer
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Hint: Ideal gas is a hypothetical gas whose molecules occupy negligible space and have no interactions, and which consequently obeys the gas laws exactly. The speed of sound in a gas can be calculated as the square root of (the coefficient ratio of specific heats × the pressure of the gas / the density of the medium).

Formula used:
$PV = nRT$ , $v = \sqrt {\dfrac{{\gamma RT}}{M}} $ , $v = \sqrt {\dfrac{{\gamma P}}{\rho }} $

Complete step by step solution:
From kinetic theory of gas, we can write
$v = \sqrt {\dfrac{{\gamma RT}}{M}} $
Thus velocity of the sound in gas is directly proportional to square root of temperature
That is, proportional to $\sqrt T $
Also, from the ideal gas law, also called the general gas equation
 $PV = nRT$ (Here taking $n = 1$ , as considering 1 mole of gas)
Therefore $PV = RT$
Substituting the value of $RT$ in $v = \sqrt {\dfrac{{\gamma RT}}{M}} $$\left( M \right)$
We get $v = \sqrt {\dfrac{{\gamma PV}}{M}} $
$v = \sqrt {\dfrac{{\gamma P}}{\rho }} $ (As $\rho $ is the density of the gas as $\rho = \dfrac{M}{V}$ )
Thus velocity of the sound in gas is directly proportional to square root of pressure when density of gas is constant.
From the given formula it is clear that the velocity of the sound in the gas is proportional to $\sqrt T $ and $\sqrt P $ when $\rho $ is constant.

Note: For a given gas, $\left( \gamma \right)$ , gas constant $\left( R \right)$ and molecular mass $\left( M \right)$are constants. Then speed of sound depends only on temperature. It is independent of the pressure. But for a given gas, if adiabatic index $\left( \gamma \right)$ and density $\left( \rho \right)$are constants, the speed of sound depends only on the square root of pressure. It is independent of temperature.