Question

Use the formula \[v = \sqrt {\dfrac{{\gamma P}}{\rho }} \] to explain why the speed of sound in air:
A. Is independent of pressure,
B. Increases with temperature,
C. Increases with humidity.

Answer 1

Hint: \[v = \sqrt {\dfrac{{\gamma P}}{\rho }} \] and \[\rho = \dfrac{{PM}}{{RT}}\]
Rearrange the two equations to find an expression where velocity is a function of time.

Complete step by step answer:
\[v = \sqrt {\dfrac{{\gamma P}}{\rho }} \] …… (1)
Where,
\[\gamma \] indicates atomicity of the gaseous medium.
\[P\] indicates pressure of gas.
\[\rho \] indicates density of gas.

Again, we have a formula of density:
\[\rho = \dfrac{{PM}}{{RT}}\] …… (2)
So, rearranging the equation (2), we get:
$\dfrac{\rho }{P} = \dfrac{M}{{RT}} $
$\dfrac{P}{\rho } = \dfrac{{RT}}{M} $
Since, the fraction \[\dfrac{{RT}}{M}\] is constant, so \[\dfrac{P}{\rho }\] is also constant.
Mathematically, it can be written as:
\[\dfrac{P}{\rho } = \dfrac{{RT}}{M} = \]constant …… (3)

Using the equation (3) in equation (1):
\[v = \sqrt {\dfrac{{\gamma P}}{\rho }} \]

\[v = \sqrt {\dfrac{{\gamma RT}}{M}} \] …… (4)
In the equation (4), all the quantities are constant, except temperature, so it can be concluded that velocity of the sound won’t depend on the pressure. Whether the pressure is increased or decreased, it won’t affect the speed of sound. But when the temperature increases, the velocity of sound also increases.

So, the correct answer is “Option B”.

Additional Information:
The speed of a gas 's molecules is proportional to the temperature, which is inversely proportional to the gas's molar mass. In other words, as the temperature of a gas sample increases, the molecules accelerate and as a result the root mean square molecular velocities rise.

Humidity has no impact on sound speed nor does air pressure alone. In an ideal approximation to petrol, air pressure has no effect at all. This is because pressure and density both contribute equally to the velocity of the sound and the two effects cancel out in an ideal gas, leaving only the temperature effect.

Note:
In this problem, you are asked to check the velocity of the sound if it is dependent on pressure, temperature and humidity. To solve this, first use, the formula of the velocity of sound which also includes atomicity of the gas. It is important to note that all the physical quantities are constant, except the temperature. Velocity is directly proportional to temperature.