Question

Find the speed of sound in a mixture of 1 mole of helium and 2 moles of oxygen at `27^(@)C`. If the temperature is raised by 1K from 300K, find the percentage change in the speed of sound in the gaseous mixture.
Take `R=8.31Jmo l e^(-1)K^(-1)`.

Answer 1

Here, `n_(1)=1, M_(1)=4, n_(2)=2, M_(2)=32`
`:. M_(mix)=(n_(1)M_(1)+n_(2)M_(2))/(n_(1)+n_(2))`
`=(1xx4+2xx32)/(1+2)=(68)/(3)gm//mo l e`
`=(68)/(3)xx10^(-3) kg//mo l e`
As He is monoatomic, `C_(v_(1))=3//2R`
and oxygen is diatomic, `C_(v_(2))=5//2R`
`(C_(v))_(min)=(n_(1)C_(v_(1))+n_(2)C_(v_(1)))/(n_(1)+n_(2))`
`=(1xx(3//2)R+2(5//2)R)/(1+2)=(13)/(6)R`
`(C_(p))=(C_(upsilon))_(mix)+R`
`=(13)/(6)R+R=(19)/(6)R`
`gamma_(mix)=((c_(p))mix)/((C_(upsilon))_(mix))=((19//6)R)/((13//6)R)=(19)/(13)`
`upsilon_(mix)=sqrt((gamma_(mix)RT)/(M_(mix)))=sqrt(((19//3)8.31xx300)/(68//3xx10^(-3)))`
`upsilon_(mix)=400.9m//s`
Again, from `upsilon=sqrt((gammaRT)/(M))`, we find
`(Deltaupsilon)/(upsilon)=(1)/(2)((DeltaT)/(T)), `, as `gamma` , R, M are constants
`=(1)/(2)xx(1)/(300)=(1)/(600)`
`(Deltaupsilon)/(upsilon)xx100=(100)/(600)=0.167%`