The ratio of densities of nitrogen and oxygen | vedclass

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Question

(D) The speed of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$.Since $v_N = v_O$,we have $\sqrt{\frac{\gamma_N R T_N}{M_N}} = \sq

Vedclass · Accepted Answer

$14$

Vedclass · Answer

(D) The speed of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$.Since $v_N = v_O$,we have $\sqrt{\frac{\gamma_N R T_N}{M_N}} = \sqrt{\frac{\gamma_O R T_O}{M_O}}$.Assuming $\gamma_N = \gamma_O$ for diatomic gases,we get $\frac{T_N}{M_N} = \frac{T_O}{M_O}$,which implies $\frac{T_N}{T_O} = \frac{M_N}{M_O}$.Given the ratio of densities $ho_N : ho_O = 14:16$,and since density $ho \propto M$ at constant pressure and temperature,we have $\frac{M_N}{M_O} = \frac{14}{16} = \frac{7}{8}$.The temperature of oxygen is $T_O = 55^oC = 273 + 55 = 328 \, K$.Thus,$T_N = T_O 	imes \frac{7}{8} = 328 	imes \frac{7}{8} = 41 	imes 7 = 287 \, K$.Converting back to Celsius: $T_N(^oC) = 287 - 273 = 14 \, ^oC$.

The ratio of densities of nitrogen and oxygen is $14:16$. The temperature at which the speed of sound in nitrogen will be the same as that in oxygen at $55^oC$ is ..... $^oC$.

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