Two monoatomic gases of equal masses are in two different containers at S.T.P. if the ratio of velocities of sound in them is `1 :2` , then find the ratio of their volumes .

Explanation

Ratio of the velocities of sound in them = `1:2` we know that velocity of sound in a gas is inversely proportional to square root of molecular weight .
`therefore V porp (1)/(sqrtM)`
`M prop (1)/(V^(2))`
`(M_(1))/(M_(2)) = ((V_(2))/(V_(1)))^(2) = ((2)/(1))^(2) = 4 :1`
But PV = n.RT
`V = nR((T)/(p))`
`V prop (m)/(M) [ because T, p and R` are constant and n = `(m)/(M)`]
`therefore V prop (1)/(M) [ therefore` m is also constant ]
`therefore (V_(1))/(V_(2)) = (M_(2))/(M_(1)) = 1 :4`
Alternate method :
`V prop (1)/(sqrtd) ` or `V prop (1)/(sqrt((m)/(V)))` or `(V_(1))/(V_(2)) = sqrt((V_(1))/(V_(2)))`
`therefore (V_(1)^(1))/(V_(2)^(1)) = (V_(1)^(2))/(V_(2)^(2)) = (1)/(2^(2)) = (1)/(4)`