# Factors Affecting Speed of Sound in Gas

## Factors Affecting Speed of Sound in Gas

Factors affecting speed of sound in gas, such as: Pressure, Density, Temperature, Humidity, frequency etc.

Effect of Pressure: The speed of sound in a gas is given by

$$v=\sqrt{\frac{\gamma P}{\rho }}$$,

We know that:

$$PV=nRT=\frac{m}{M}RT$$,

At constant temperature:

$$P\Delta V=\frac{\Delta m}{M}RT$$,

$$P=\frac{\Delta m}{\Delta V}\frac{RT}{M}=\rho \frac{RT}{M}$$,

∴ $$P=\rho \frac{RT}{M}$$,

P/ ρ = Constant.

Therefore, with the change in pressure, the density also changes in such proportion, so that P/ ρ remains constant. Hence, pressure has no effect on the speed of sound in a gas.

Effect of Density: For two gases of densities ρ₁ and ρ₂ at same pressure with ratios of specific heats γ₁ and γ₂.

$$\frac{{{v}_{1}}}{{{v}_{2}}}=\sqrt{\frac{{{\gamma }_{1}}}{{{\gamma }_{2}}}\times \frac{{{\rho }_{1}}}{{{\rho }_{2}}}}$$,

Effect of Temperature:

We have:

$$\frac{P}{\rho }=\frac{RT}{M}$$,

$$\frac{P}{\rho }=\frac{RT}{M}$$,

Clearly, v α √T

Hence, the speed of sound in a gas is proportional to the square root of its absolute temperature. If v₀ and vt are the velocities of sound in gas at 0°C and t°C respectively. Then,

$${{v}_{0}}=\sqrt{\frac{\gamma R(273+0)}{M}}$$,

And

$${{v}_{t}}=\sqrt{\frac{\gamma R(273+t)}{M}}$$,

∴ $$\frac{{{v}_{t}}}{{{v}_{0}}}={{\left( \frac{273+t}{273} \right)}^{1/2}}={{\left( 1+\frac{t}{273} \right)}^{1/2}}$$,

For small value of ,

$${{v}_{t}}={{v}_{0}}\left( 1+\frac{1}{2}\times \frac{t}{273} \right)$$,

$${{v}_{t}}={{v}_{0}}+\frac{{{v}_{0}}t}{546}$$,

∴ $${{v}_{t}}-{{v}_{0}}=\frac{332t}{546}=0.16t$$ (∵ v₀ = 332m/ sec).

vt – v₀ = 0.61 m/ sec. (When t = 1°C)

Hence, the velocity of sound in air increases by 0.61 m/ sec for every 1°C rise in temperature.