Two moles of oxygen is mixed with one moles of helium. The effective specific heat of the mixture at constant volume is:

CONCEPT:

• The molar specific heat capacity of a gas at constant volume is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.

\(\Rightarrow {C_v} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;volume}}\)

• The molar specific heat of a gas at constant pressure is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant pressure.

\(\Rightarrow {C_p} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;pressure}}\)

• The relation between the ratio of Cp and Cv with a degree of freedom is given by

\(\Rightarrow \gamma = \frac{{{C_p}}}{{{C_v}}} = 1 + \frac{2}{f}\)

Where f = degree of freedom

• The molar specific heat of gaseous mixture at constant volume is given by the formula:

\(\Rightarrow {\left( {{{\rm{C}}_{\rm{V}}}} \right)_{mix}} = \frac{{{{\rm{n}}_1}{{\rm{C}}_{{{\rm{V}}_1}}} + {{\rm{n}}_2}{{\rm{C}}_{{{\rm{V}}_2}}}}}{{{{\rm{n}}_1} + {{\rm{n}}_2}}}\)

Where n1 and n2 are moles of the gases.

CALCULATION:

Given: n₁ = 2 and n₂ = 1

• Oxygen is a diatomic gas, therefore the molar specific heat capacity of a gas at constant volume is 

\(\Rightarrow C_{v_1}= \frac{5}{2}R\)

• Helium is a monoatomic gas, therefore the molar specific heat capacity of a gas at constant volume is 

\(\Rightarrow C_{v_2}= \frac{3}{2}R\)

• The effective specific heat of the mixture at constant volume is

\(\Rightarrow (C_v)_{mix}=\frac{2\times \frac{5}{2}R+1\times \frac{3}{2}R}{3}=\frac{13R}{6}=2.17\,R\)