A mixture contains 1 mole of helium (C_p = 2.5 R, C_v = 1.5 R) and 1 mole of hydrogen (C_p = 3.5 R, C_v= 2.5 R). Calculate the values of C_p, C_v and γ for the mixture.

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Solution

Specific heat at constant pressure of helium, C_p' = 2.5 R
Specific heat at constant pressure of hydrogen, C_p" = 3.5 R
Specific heat at constant volume of helium, C_v' = 1.5 R
Specific heat at constant volume of hydrogen, C_v" = 2.5 R
n₁ = n₂ = 1 mol
_{$d U = n C_{v} d T$}

For the mixture of two gases,
$d U_{1} + d U_{2} = 1 mol$
[n₁ + n₂ ] C_vdT = n₁C'_vdT + n₂C"_vdT,
where $C_{V}$ is the heat capacity of the mixture.
$\Rightarrow C_{v} = \frac{n_{1} C'_{v} + n_{2} C "_{v}}{n_{1} + n_{2}} = \frac{1.5 R + 2.5 R}{2} = 2 R$
Cp = Cv + R = 2R + R = 3R
$γ = \frac{C p}{C v} = \frac{3 R}{2 R} = 1.5$

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