The ratio of the densities of oxygen and nitrogen is 16:14. At what temperature, the velocity of sound in oxygen will be equal to its velocity in nitrogen at 14°C?

Options

A. `45^(@)` C
B. `55^(@)` C
C. `20^(@)` C
D. `35^(@)` C

Answer

B

Explanation

The velocity of sound in a gas is given by:
`v=sqrt((lambda RT)/M)`
where `lambda` is the ratio of specific heats `(C_(p)/C_(V))`
R is the universal gas constant
T is the absolutee temperature
and M is the molecular weight of the gas.
Let the velocity of sound in oxygen at t°C be equal to the velocity of sound in nitrogen at `14^(@)` C
Now, velocity of sound in oxygen at `t^(@) C, v_(0) = sqrt((lambdaR(t+273))/M_(0))`
and velocity of sound in nitrogen at `14^(@)` C,
`v_(N) = sqrt((gammaR(14 + 273))/M_(N))`
The value of `gamma` for both the gases will be same as both are, diatomic.
Now, according to the question, `v_(0) =v_(N)`
or `sqrt((gamma R(14 + 273))/M_(N))` or `(t+273)/287 = M_(O)/M_(N)`
But, ratio of the densities of oxygen and nitrogen is,
`M_(O)/M_(N) = 16/14`
`therefore (t + 273)/287 = 16/14 = 8/7`
or `7t + 273 xx 7 = 8 xx 287` or `t = 55^(@)` C
Therefore, the velocity of sound in oxygen at 55°C will be equal to the velocity of sound in nitrogen at 14°C.