Two narrow cylindrical pipes `A and B` have the same length. Pipe `A` is open at both ends and is filled with a monoatomic gas of molar mass `M_(A)`. Pipe `B` is open at one end and closed at the other end, and is filled with a diatomic gas of molar mass `M_(B)`. Both gases are at the same temperature.
(a) If the frequency of the second harmonic of the fundamental mode in pipe `A` is equal to the frequency of the third harmonic of the fundamental mode in pipe `B`, determine the value of `M_(B)//M_(B)`.
(b) Now the open end of pipe `B` is also closed (so that the pipe is closed at both ends). Find the ratio of the fundamental frequency in pipe `A` to that in pipe `B`.

Explanation

(a)
`f_(A) = 2 . (v_(A))/(2L), f_(B) = 3 . (v_(B))/(4L)`
`f_(A) = f_(B)`
`2 . (v_(A))/(2L) = 3 . (v_(B))/(4L)`
`sqrt((gamma _(A) RT)/(M_(A))) = (3)/(4) sqrt((gamma_(B) RT)/(M_(B)))`
`sqrt((M_(A))/(M_(B))) = (4)/(3) sqrt((gamma _(A))/(gamma _(B)))`
`(M_(A))/(M_(B)) = (16)/(9) .(gamma_(A))/(gamma_(B)) = (16)/(9) . (5//3)/(7//5) = (16)/(9) . (25)/(21) = (400)/(189)`
(b)
`(f_(A))/(f_(B)) = (v_(A)//2L)/(v_(B)//2L) = (v_(A))/(v_(B)) = (3)/(4)`