Question

Which of the following represents the quantity which is independent of the velocity of sound in gas?
${\text{A}}{\text{.}}$ Pressure
${\text{B}}{\text{.}}$ Temperature
${\text{C}}{\text{.}}$ Density
${\text{D}}{\text{.}}$ Humidity

Answer 1

Hint: Here, we will proceed by writing down the formula for the velocity of sound in gas. Then, we will use the ideal gas equation for one mole of the gas such that the mass is replaced by molecular mass of the gas.

Formula used:
${\text{v}} = \sqrt {\dfrac{{\gamma {\text{P}}}}{\rho }} $ and PV = RT.

Complete answer:
As we know that the velocity of sound in any gas is given by the formula as under
i.e., velocity of sound in gas ${\text{v}} = \sqrt {\dfrac{{\gamma {\text{P}}}}{\rho }} $ where P denotes the pressure of gas, $\gamma $ denotes the ratio of the specific heats at constant pressure and at constant volume and $\rho $ denotes the density of gas.
Also, we know that the ideal gas equation can be written as mentioned under
PV = nRT where P denotes the pressure of gas, V denotes the volume of gas, n denotes the number of moles, R denotes the gas constant and T denotes the temperature of gas
Considering one mole of the gas i.e., n = 1
PV = RT $ \to {\text{(1)}}$
Also, the density of any gas can be written as the ratio of the mass of gas to the volume of gas
i.e., Density of gas $\rho = \dfrac{{\text{m}}}{{\text{V}}}$ where m is the mass of gas and V is the volume of gas
$ \Rightarrow {\text{V}} = \dfrac{{\text{m}}}{\rho }\; \to {\text{(2)}}$
By substituting the equation (2) in equation (1), we get
$
\Rightarrow {\text{P}}\left[ {\dfrac{{\text{m}}}{\rho }} \right] = {\text{RT}} \\
\Rightarrow \dfrac{{\text{P}}}{\rho } = \dfrac{{{\text{RT}}}}{{\text{m}}} \\
$
In the RHS of the above equation, universal gas constant R and mass m of the gas corresponding to one mole of the gas (which is the molecular mass of the gas) are always constant for a particular gas. Assuming the temperature T to be constant, $\dfrac{{{\text{RT}}}}{{\text{m}}}$ will be a constant for a particular gas
i.e., $\dfrac{{\text{P}}}{\rho }$ = constant at constant temperature T
Velocity of sound in gas ${\text{v}} = \sqrt {\dfrac{{\gamma {\text{P}}}}{\rho }} $ where $\gamma $ denotes the ratio of the specific heats at constant pressure and at constant volume (always constant for a particular gas)
At constant temperature T, $\dfrac{{\text{P}}}{\rho }$ will also be constant. This means that velocity of sound in gas is constant if temperature is constant and is independent of the pressure of gas.

So, the correct answer is “Option A”.

Note:
For sound to travel, medium is required and is essential for the propagation of energy through the medium. The speed of sound is different in different media. The speed of sound in any media basically depends on the rapidness of the propagation of vibrational energy through the medium.