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Question
If c is r.m.s. speed of molecules in a gas and v is the speed of sound waves in the gas, show that c/v is constant and independent of temperature for all diatomic gases.
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Solution
We know that for molecules, c = `sqrt((3P)/P)`.
We know that `p/ρ = (PT)/M`. Therefore, we get
⇒ `p/ρ = ((RT)/V)/(M/V)`
⇒ `p/ρ = (RT)/M`
Thus, we can write the equation for the molecules as,
⇒ c = `sqrt((3RT)/M)`
Where M is the molar mass of gas.
For sound waves, we have
⇒ v = `sqrt((ϒP)/ρ)`
We know that PV = nRT. For n = 1, we have
⇒ P = `(RT)/V`
Thus, we get ⇒ v = `sqrt((ϒRT)/M)`
⇒ `c/v = sqrt((3RT)/M)/(sqrt((ϒRT)/M)`
⇒ `c/v = sqrt(3/ϒ)`
We know that ϒ = `C_P/C_v = 7/5` is an adiabatic constant for diatomic gas. Therefore, we get
⇒ `c/v = sqrt(3/(7/5)`
⇒ `c/v = sqrt((3 xx 5)/7`
⇒ `c/v = sqrt(15/7)` = constant.
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