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Question
Use the formula `v = sqrt((gamma P)/rho)` to explain why the speed of sound in air is independent of pressure.
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Solution 1
Take the relation:
`v = sqrt((gamma P)/rho)` ....(i)
where
Density, `rho = "Mass"/"Volume" = M/V`
M = Molecular weight of the gas
V = Volume of the gas
Hence, equation (i) reduces to
`v = sqrt((gamma "PV")/"M")` .....(ii)
Now from the ideal gas equation for n = 1:
PV = RT
For constant T, PV = Constant
Since both M and γ are constants, v = Constant
Hence, at a constant temperature, the speed of sound in a gaseous medium is independent of the change in the pressure of the gas.
Solution 2
We are given that `v = sqrt((gamma p)/rho)`
We know PV = nRT (For n moles of ideal gas)
`=> "PV" = "m"/"M" "RT"`
where m is the total mass and M is the molecular mass of the gas
`:. P = "m"/"M" * "RT"/"M"`
`= (rho"RT")/"M"`
`=> "P"/rho = "RT"/"M"`
For a gas at constant temperature `"P"/rho = "constant"`
∴ As P increase, rho also increase and vice versa. This implies that `v = sqrt((gamma P)/rho)` = constant, i.e velocity is independent of pressure of the gas.
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