First slide

Specific heat and degrees of freedom

Question

Consider a mixture of n moles of helium gas and 2n moles of oxygen gas (molecules taken to be rigid) as an ideal gas. Its $C_{P} / C_{V}$ value will be :

Moderate

A

23/15
B

67/45
C

19/13
D

40/27

Solution

$\begin{array}{l} F o r H e l i u m (m o n o a t o m i c g a s) c_{v_{1}} = \frac{3}{2} R a n d c_{p_{1}} = \frac{5}{2} R H e r e n_{1} = n \\ F o r o x y g e n (d i a t o m i c g a s) c_{v_{2}} = \frac{5}{2} R a n d c_{p_{2}} = \frac{7}{2} R H e r e n_{2} = 2 n \\ γ_{m i x} = \frac{n_{1} c_{p_{1}} + n_{2} c_{p_{2}}}{n_{1} c_{v_{1}} + n_{2} c_{v_{2}}} = \frac{n (\frac{5}{2} R) + 2 n (\frac{7}{2} R)}{n (\frac{3}{2} R) + 2 n (\frac{5}{2} R)} = \frac{5 + 14}{5 + 10} = \frac{19}{13} \end{array}$

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