Question

A gas is mixture of two parts by volume of hydrogen and one part by volume of nitrogen at STP. If the velocity of sound in hydrogen at $0^{\circ }C$ is 1300 m/s. Find the velocity of sound in the gaseous mixture at ${27}^{\circ }C$ (in m/s)

Answer 1

Let $V$ be the volume of nitrogen in mixture , therefore volume of oxygen in mixture will be $2V$ .

Total volume of gaseous mixture $=V+2V=3V$ ,

mass of nitrogen in mixture $=V\times 14=14V$ ,

mass of oxygen in mixture $=2V\times 16=32V$ ,

total mass of gaseous mixture $=14V+32V=46V$ ,

density of mixture ,

${\rho }_M=$total mass/total volume$=46V/3V=15.33$ ,

velocity of sound in nitrogen at $0^{o}C$,

$v_N=\sqrt{\gamma P/{\rho }_N}$ , (given , $v_N=1300m/s)$

velocity of sound in mixture at $0^{o}C$,

$v_M=\sqrt{\gamma P/{\rho }_M}$ ,

therefore , $v_M/v_N=\sqrt{{\rho }_N/{\rho }_M}$ ,

or $v_M=v_N\sqrt{{\rho }_N/{\rho }_M}=1300\left(\sqrt{14/15.33}\right)=1241.5m/s$ ,

velocity of sound at $t^{o}C$ in a gas is given by ,

$v_t=v_0+0.61t$ ,

we have , $v_0=v_M=1241.5m/s$ and $t={27}^{o}C$,

therefore , $v_{27}=1241.5+0.61\times 27=1258m/s$