JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Heat & Thermodynamics

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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)

(A)The average energy per mole of the gas mixture is 2RT.✓ Correct
(B)The ratio speed of sound in the gas mixture to that in helium gas is $6/5$ .✓ Correct
(C)The ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/2.
(D)The ratio of the rms speed of helium atoms to that of hydrogen molecules is $1/ 2$ .✓ Correct

Answer: (A), (B), (D)

Solution

Total energy =

\frac{5}{2} R T

(H₂, diatomic) +

\frac{3}{2} R T

(He, monoatomic) =

4 R T

. Average per mole =

4 R T /2 = 2 R T

. ✓ (A)

For speed of sound:

v_{so u n d} = γ R T / M

For the mixture:

γ_{mi x} = \frac{n _{1} C _{p 1} + n _{2} C _{p 2}}{n _{1} C _{v 1} + n _{2} C _{v 2}} = \frac{\frac{7}{2} + \frac{5}{2}}{\frac{5}{2} + \frac{3}{2}} = \frac{6}{4} = \frac{3}{2}

M_{mi x} = \frac{1 \times 2 + 1 \times 4}{2} = 3

g/mol

\frac{v _{so u n d, mi x}}{v _{so u n d, H e}} = \frac{γ _{mi x} \times M _{H e}}{γ _{H e} \times M _{mi x}} = \frac{3/2 \times 4}{5/3 \times 3} = \frac{6}{5}

✓ (B)

For rms speeds:

v_{r m s} = 3 R T / M

\frac{v _{r m s, H e}}{v _{r m s, H_{2}}} = \frac{M _{H_{2}}}{M _{H e}} = \frac{2}{4} = \frac{1}{2}

✓ (D)

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JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Heat & Thermodynamics

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