First slide

Heat ;work and internal energy

Question

One mole of mono atomic gas $(γ = \frac{5}{3})$ is mixed with one mole of diatomic gas $(γ = \frac{7}{5})$ . What will be the value of $γ$ for the mixture.

Moderate

A

1.45
B

1.4
C

1.54
D

1.5

Solution

$\begin{array}{rcl} f o r m u l a t o f i n d γ_{mix} i s \frac{n_{1} {+n}_{2}}{γ_{mix} -1} = \frac{n_{1}}{γ_{1} -1} + \frac{n_{2}}{γ_{2} -1} ---(1) \\ h e r e n_{1} {&n}_{2} {=number of moles; γ}_{1} {&γ}_{2} = specific heat ratio \\ g i v e n n_{1} {=n}_{2} & = & 1; γ_{1} = \frac{5}{3} {;γ}_{2} = \frac{7}{5} \\ s u b s t i t u t e g i v e n v a l u e s i n (1) \\ \frac{1 + 1}{γ_{mix} -1} & = & \frac{1}{\frac{5}{3} - 1} + \frac{1}{\frac{7}{5} - 1} \\ \frac{2}{γ_{mix} -1} & = & \frac{1}{\frac{2}{3}} + \frac{1}{\frac{2}{5}} \\ \frac{2}{γ_{mix} -1} & = & \frac{3}{2} + \frac{5}{2} \\ \Rightarrow & \frac{2}{γ_{mix} -1} = 4 \\ γ_{mix} -1 & = & \frac{1}{2} \\ \Rightarrow & γ_{mix} = 1.5 \end{array}$

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