**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 v_{t}
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).

v_{t} – 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.