First slide

Specific heat and degrees of freedom

Question

Two moles of an ideal gas with $\frac{C_{P}}{C_{V}} = \frac{5}{3}$ are mixed with 3 moles of another ideal gas with $\frac{C_{P}}{C_{V}} = \frac{4}{3}$ . The values of $\frac{C_{P}}{C_{V}}$ for the mixture is

Moderate

A

1.50
B

1.42
C

1.47
D

1.45

Solution

$\begin{array}{l} γ_{m i x t u r e} = \frac{n_{1} C_{P_{1}} + n_{2} C_{P_{2}}}{n_{1} C_{V_{1}} + n_{2} C_{V_{2}}} = \frac{n_{1} \frac{γ_{1} R}{γ_{1} - 1} + n_{2} \frac{γ_{2} R}{γ_{2} - 1}}{\frac{n_{1} R}{γ_{1} - 1} + \frac{n_{2} R}{γ_{2} - 1}} \\ A f t e r r e a r r a n g i n g w e g e t, \\ \frac{n_{1} + n_{2}}{γ_{m i x} - 1} = \frac{n_{1}}{γ_{1} - 1} + \frac{n_{2}}{γ_{2} - 1} \\ \Rightarrow \frac{5}{γ_{m i x} - 1} = \frac{3}{1 / 3} + \frac{2}{2 / 3} \\ \Rightarrow \frac{5}{γ_{m i x} - 1} = 9 + 3 = 12 \\ \Rightarrow γ_{m i x u r e} = 1 + \frac{5}{12} = \frac{17}{12} \\ \Rightarrow γ_{m i x} = 1.42 \end{array}$

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