If the speed of sound in a mixture of 2 moles | vedclass

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(A) Given: $n_{He} = 2$ moles, $n_{H_2} = 2$ moles, $T = \frac{972}{5} \,K$, $R = \frac{25}{3} \,J \,mol^{-1} \,K^{-1}$.$1$. Calculate the molar mass

Vedclass · Accepted Answer

$9$

Vedclass · Answer

(A) Given: $n_{He} = 2$ moles, $n_{H_2} = 2$ moles, $T = \frac{972}{5} \,K$, $R = \frac{25}{3} \,J \,mol^{-1} \,K^{-1}$.$1$. Calculate the molar mass of the mixture $(M_{mix})$:$M_{mix} = \frac{n_1 M_1 + n_2 M_2}{n_1 + n_2} = \frac{2 	imes 4 + 2 	imes 2}{2 + 2} = \frac{8 + 4}{4} = 3 \,g/mol = 3 	imes 10^{-3} \,kg/mol$.$2$. Calculate the adiabatic index $(\gamma_{mix})$ of the mixture:For Helium (monatomic), $f_1 = 3$. For Hydrogen (diatomic), $f_2 = 5$.$f_{mix} = \frac{n_1 f_1 + n_2 f_2}{n_1 + n_2} = \frac{2 	imes 3 + 2 	imes 5}{2 + 2} = \frac{16}{4} = 4$.$\gamma_{mix} = 1 + \frac{2}{f_{mix}} = 1 + \frac{2}{4} = 1.5$.$3$. Calculate the speed of sound $(v)$:$v = \sqrt{\frac{\gamma_{mix} R T}{M_{mix}}} = \sqrt{\frac{1.5 	imes \frac{25}{3} 	imes \frac{972}{5}}{3 	imes 10^{-3}}} = \sqrt{\frac{1.5 	imes 5 	imes 972}{3 	imes 10^{-3}}} = \sqrt{\frac{7.5 	imes 972}{3 	imes 10^{-3}}} = \sqrt{2.5 	imes 324000} = \sqrt{810000} = 900 \,m/s$.Given $v = n 	imes 100 \,m/s$, therefore $n = 9$.

If the speed of sound in a mixture of $2$ moles of Helium and $2$ moles of Hydrogen at temperature $\frac{972}{5} \,K$ is $n \times 100 \,ms^{-1}$, then the value of $n$ is (Take, $R=\frac{25}{3} \,J \,mol^{-1} \,K^{-1}$)

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