A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature . Assuming the gases are ideal, the correct statement(s) is (are) [2015]

Question

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 statement(s) is (are) [2015]

Smart Achievers · Accepted Answer

(1, 2, 4)

According to question no. of mole of hydrogen = no. of mole of helium =1

Total internal energy, u

$= \frac{f_{1}}{2} n R T + \frac{f_{2}}{2} n R T$ $\Rightarrow u = \frac{3}{2} R T + \frac{5}{2} R T = 4 R T$

$∴$ $Average internal energy per mole = \frac{u}{2 n} = \frac{4 R T}{2} = 2 R T$

We know that $V_{sound} = \sqrt{\frac{γ R T}{M}}$

$\frac{n_{1} + n_{2}}{γ_{mix} - 1} = \frac{n_{1}}{γ_{1} - 1} + \frac{n_{2}}{γ_{2} - 1}$ $\Rightarrow \frac{2}{γ_{mix} - 1} = \frac{1}{\frac{5}{3} - 1} + \frac{1}{\frac{7}{5} - 1}$

$\frac{2}{γ_{mix} - 1} = \frac{3}{2} + \frac{5}{2} = 4$ $\Rightarrow γ_{mix} - 1 = \frac{1}{2}$ $∴$ $γ_{mix} = \frac{3}{2}$

$\frac{{(V_{s})}_{mix}}{{(V_{s})}_{He}} = \sqrt{\frac{γ_{mix}}{M_{mix}} \times \frac{M_{He}}{γ_{He}}}$ $= \sqrt{\frac{\frac{3}{2} \times 4}{3 \times \frac{5}{3}}} = \sqrt{\frac{6}{5}}$ $[∵ M_{mix} = \frac{1 \times 2 + 1 \times 4}{2} = 3]$

We know that $V_{rms} = \sqrt{\frac{3 R T}{M}}$

$∴$ $\frac{{(V_{rms})}_{He}}{{(V_{rms})}_{H_{2}}} = \sqrt{\frac{M_{H_{2}}}{M_{He}}} = \sqrt{\frac{2}{4}} = \frac{1}{\sqrt{2}}$

Solution

Q.
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 statement(s) is (are) [2015]

1 The average energy per mole of the gas mixture is 2RT

2 The ratio of speed of sound in the gas mixture to that in helium gas is $\sqrt{\frac{6}{5}}$

3 The ratio of the rms speed of helium atoms to that of hydrogen molecules is $\frac{1}{2}$

4 The ratio of the rms speed of helium atoms to that of hydrogen molecules is $\frac{1}{\sqrt{2}}$

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Solution

Q. 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 statement(s) is (are) [2015]

1 The average energy per mole of the gas mixture is 2RT

2 The ratio of speed of sound in the gas mixture to that in helium gas is 65

3 The ratio of the rms speed of helium atoms to that of hydrogen molecules is 12

4 The ratio of the rms speed of helium atoms to that of hydrogen molecules is 12

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Q.
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 statement(s) is (are) [2015]

2 The ratio of speed of sound in the gas mixture to that in helium gas is $\sqrt{\frac{6}{5}}$

3 The ratio of the rms speed of helium atoms to that of hydrogen molecules is $\frac{1}{2}$

4 The ratio of the rms speed of helium atoms to that of hydrogen molecules is $\frac{1}{\sqrt{2}}$