Question 94: Thermodynamics & KTG

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)

Accepted Answer

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 f
H
2
= 5 and f
He
= 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 n
H
2
= 1 and n
He
= 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
$${{{v_{rms,He}}} \over {{v_{rms,{H_2}}}}} = {{\sqrt {3RT/{M_{He}}} } \over {\sqrt {3RT/{M_{{H_2}}}} }} = \sqrt {{{{M_{{H_2}}}} \over {{M_{He}}}}}  = \sqrt {{2 \over 4}}  = \sqrt {{1 \over 2}} $$.

Answer

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

Answer

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

Answer

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

Answer

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

Thermodynamics & KTG

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)

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