Two moles of helium are mixed with n moles of hydrogen. The root mean spure (rms) speed of the gas molecules in the mexture is `sqrt2` times the speed of sound in the mixture. Then value of n is

Options

A. 1
B. `3//2`
C. 2
D. 3

Explanation

The root mean square speed of the gas `v" rms"=sqrt((3RT)/(M))` ,
`v" sound"=sqrt((gammaRT)/(M)),v" rms"=sqrt(2)v" sound"`
Solving it , we get `sqrt(3)=sqrt(2gamma)`
`rArr" " gamma=(3)/(2)` for the mixture.
As, `gamma=(C_(p))/(C_(v))=(n_(1)C_(p_(1))+n_(2)Cp_(2))/(n_(1)+n_(2))xx(n_(1)+n_(2))/(n_(1)C_(v_(1)+n_(2))C_(v_(2)))`
`gamma=(n_(1)C_(p_(1))+n_(2)Cp_(2))/(n_(1)C_(v_(1))+n_(2)C_(v_(2)))`
For helium , `Cp_(1)=(5)/(2)R,C_(v_(1))=(3)/(2)R`
For hydrogen, `Cp_(2)=(7)/(2)R,C_(v_(2))=(5)/(2)R`
`therefore" " (3)/(2)=(2((5)/(2)R)+n((7)/(2)R))/(2((3)/(2)R)++n((7)/(2)R))=(10+7n)/(6+5n)`
or 20+14n=18+15n
`rArr` n=2