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JEE Advance - Physics (2015 - Paper 1 Offline - No. 3)

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)
The average energy per mole of the gas mixture is 2RT
The ratio of speed of sound in the gas mixture to that in helium gas is $$\sqrt {{6 \over 5}} $$
The ratio of the rms speed of helium atoms to that of hydrogen molecules is $${1 \over 2}$$
The ratio of the rms speed of helium atoms to that of hydrogen molecules is $${1 \over {\sqrt 2 }}$$

Explanation

The internal energy of one mole of an ideal gas at temperature T is given by $$U = {f \over 2}RT$$, where f is the degrees of freedom of the gas molecule. The degress of freedom of the gas molecule. The degrees of freedom for hydrogen (diatomic) and helium (monatomic) gases are fH2 = 5 and fHe = 3, respectively. Thus, $${U_{{H_2}}} = {5 \over 2}RT$$ and $${U_{He}} = {3 \over 2}RT$$. The total internal energy of the gas mixture is

$${U_{total}} = {U_{{H_2}}} + {U_{He}} = {5 \over 2}RT + {3 \over 2}RT = 4RT$$.

The mixture contains two moles of the gases. The internal energy per mole of the mixture is $${U_{mix}} = {U_{total}}/2 = 2RT$$.

The specific heat at constant volume is given by $${C_v} = dU/dT$$. Thus, the specific heats at constant volume for helium and the mixture are

$${C_{v,He}} = d{U_{He}}/dT = {3 \over 2}R$$, and

$${C_{v,mix}} = d{U_{mix}}/dT = 2R$$

The specific heats at constant pressure, $${C_p} = {C_v} + R$$, for these gases are

$${C_{p,He}} = {C_{v,He}} + R = {5 \over 2}R$$, and

$${C_{p,mix}} = {C_{v,mix}} + R = 3R$$.

The ratio of specific heats, $$\gamma = {C_p}/{C_v}$$, are $${\gamma _{He}} = 5/3$$ and $${\gamma _{mix}} = 3/2$$. The speed of sound, in a gas of molecular mass M, is given by $${v_s} = \sqrt {\gamma RT/M} $$. The molecular mass of the gas mixture is

$${M_{mix}} = {{{n_{{H_2}}}{M_{{H_2}}} + {n_{He}}{M_{He}}} \over {{n_{{H_2}}} + {n_{He}}}}$$

$$ = {{(1)(2) + (1)(4)} \over {1 + 1}} = 3$$ g/mol,

where nH2 = 1 and nHe = 1 are the number of moles of hydrogen and helium in the gas mixture. The ratio of the speeds of sound in the gas mixture and helium is

$${{{v_{s,mix}}} \over {{v_{s,He}}}} = {{\sqrt {{\gamma _{mix}}RT/{M_{mix}}} } \over {\sqrt {{\gamma _{He}}RT/{M_{He}}} }} = \sqrt {{{{\gamma _{mix}}} \over {{\gamma _{He}}}}{{{M_{He}}} \over {{M_{mix}}}}} $$

$$ = \sqrt {{{(3/2)(4)} \over {(5/3)(3)}}} = \sqrt {{6 \over 5}} $$

The rms speed of the atoms/molecules is given by $${v_{rms}} = \sqrt {3RT/M} $$. The ratio of the rms speed of helium atoms to that of hydrogen molecules is

$${{{v_{rms,He}}} \over {{v_{rms,{H_2}}}}} = {{\sqrt {3RT/{M_{He}}} } \over {\sqrt {3RT/{M_{{H_2}}}} }} = \sqrt {{{{M_{{H_2}}}} \over {{M_{He}}}}} = \sqrt {{2 \over 4}} = \sqrt {{1 \over 2}} $$.

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