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.