At which temperature, velocity of sound in `H_(2)` gas will be same as that in `O_(2)` at `1200^(@)C` temperature ? Density of O, is 16 times density of `H_(2).`

Explanation

Velocity of sound in a gas with density `rho` and at temperature T is,
`v = sqrt (( gamma P )/( rho )) = sqrt (( gamma P )/( M // V )) = sqrt (( gamma P V )/( M )) = sqrt (( gamma mu RT)/( mu M _(0)))`
`therefore v = sqrt (( gamma RT)/( M _(0)))`
(Where T = absolute temperature of gas, `M _(0)=` molecular weight of gas)
`implies v prop sqrt ((T)/( M _(0))) ( because ` Here `gamma` and R be are constants)
`therefore (v _(1))/(v _(2)) = sqrt (( T _(1))/( T _(2)) xx ((M _(0) ) _(2))/( ( M _(0)) _(1)))`
Taking `H _(2) and O _(2)` gas as first and second gas respectively,
`1 = sqrt (( T _(1))/( (1200 + 273) ) xx (16)/(1)) (because ` Here ` v _(1) = v _(2)) `
`therefore 1 = (15 T _(1))/( 1473) (because rho _(2) = 16 rho _(1) implies (M _(0)) _(2) = 16 (M _(0)) _(1))`
`therefore T _(1) = (14 73)/(16) = 92.06 K`
`OR T _(1) = 92.06 -273 =- 180 . 94 ^(@)C`