First slide

Heat ;work and internal energy

Question

If one mole of a mono atomic gas $γ= \frac{5}{3}$ , is mixed with one mole of diatomic gas $γ= \frac{7}{5}$ , what is the value of $γ$ for the mixture.

Moderate

A

$\frac{3}{2}$
B

$\frac{5}{2}$
C

$\frac{6}{5}$
D

$\frac{8}{5}$

Solution

$\begin{array}{rcl} s p e c i f i c h e a t o f m i x t u r e o f g a s e s a t c o n s \tan t v o l u m e i s C_{v_{m i x t u r e}} = \frac{n_{1} C_{V_{1}} + n_{2} C_{v_{2}}}{n_{1} + n_{2}} \\ C_{v_{m i x t u r e}} & = & \frac{(1) \frac{3}{2} R+(1) \frac{5}{2} R}{2} =2R \\ A v e r a g e C_{P_{m i x t u r e}} & = & C_{v_{m i x t u r e}} + R \\ \Rightarrow & C_{P_{m i x t u r e}} =2R+R \\ \Rightarrow & C_{P_{m i x t u r e}} =3R \\ ∴ f o r m i x t u r e o f g a s e s γ= \frac{C_{P_{m i x t u r e}}}{C_{v_{m i x t u r e}}} \\ {⇒γ}_{m i x} = \frac{3R}{2R} \\ \Rightarrow & γ_{m i x} = \frac{3}{2} \end{array}$

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