Heat and Thermodynamics 4 Question 32
34. A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature $T$. Assuming the gases are ideal, the correct statements is/are
(2015 Adv.)
(a) The average energy per mole of the gas mixture is $2 R T$
(b) The ratio of speed of sound in the gas mixture to that in helium gas is $\sqrt{\frac{6}{5}}$
(c) The ratio of the rms speed of helium atoms to that of hydrogen molecules is $\frac{1}{2}$
(d) The ratio of the rms speed of helium atoms to that of hydrogen molecules is $\frac{1}{\sqrt{2}}$
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Solution:
- (a) Total internal energy $U=\frac{f _1}{2} n R T+\frac{f _2}{2} n R T$
$$ \begin{gathered} \left(U _{\text {ave }}\right) _{\text {per mole }}=\frac{U}{2 n}=\frac{1}{4}[5 R T+3 R T]=2 R T \\ \text { (b) } \gamma _{\text {mix }}=\frac{n _1 C _{p _1}+n _2 C _{p _2}}{n _2 C _{V _1}+n _2 C _{V _2}}=\frac{(1) \frac{7 R}{2}+(1) \frac{5 R}{2}}{(1) \frac{5 R}{2}+(1) \frac{3 R}{2}}=\frac{3}{2} \\ M _{\text {mix }}=\frac{n _1 M _1+n _2 M _2}{n _1+n _2}=\frac{M _1+M _2}{2}=\frac{2+4}{2}=3 \\ \text { Speed of sound } v=\sqrt{\frac{\gamma R T}{M}} \Rightarrow v \propto \sqrt{\frac{\gamma}{M}} \\ \frac{V _{\text {mix }}}{V _{He}}=\sqrt{\frac{\gamma _{\text {mix }}}{\gamma _{He}} \times \frac{M _{He}}{M _{\text {mix }}}}=\sqrt{\frac{3 / 2}{5 / 3} \times \frac{4}{3}}=\sqrt{\frac{6}{5}} \end{gathered} $$
(d) $V _{rms}=\sqrt{\frac{3 R T}{M}} \Rightarrow V _{rms} \propto \frac{1}{\sqrt{M}}$,
$$ \frac{V _{He}}{V _H}=\sqrt{\frac{M _H}{M _{He}}}=\sqrt{\frac{2}{4}}=\frac{1}{\sqrt{2}} $$
