# One mole of ideal monoatomic gas $\dpi{100} (\gamma =5/3)$ is mixed with one mole of diatomic gas $\dpi{100} (\gamma =7/5)$ . What is $\dpi{100} \gamma$ for the mixture ? $\dpi{100} \gamma$ denotes the ratio of specific heat at constant pressure, to that at constant volume. Option 1) $3/2\;$ Option 2) $\; 23/15\;$ Option 3) $\; 35/23$ Option 4) $\; \; 4/3$

As we learnt in

First law in isobaric process -

$\Delta U= n\, C_{v}\Delta T$

$= n\frac{R}{\gamma -1}\Delta T$

- wherein

$\Delta Q= \Delta U+W$

$= n\frac{\gamma \: R}{\gamma -1}\cdot \Delta T$

$\Delta Q= nC_{p}\: \Delta T$

$\gamma ^{mix}= \frac{n _{1}C_{P1}+n _{2}C_{P2}}{n _{1}C_{V1}+n _{2}C_{V2}}$

$n_{1}=1,\ C_{P1}= \frac{\gamma _{1}R}{\gamma _{1}-1}=\frac{7R}{2};\ C_{V1}= \frac{R}{\gamma _{1}-1}=\frac{5R}{2}$

$n_{2} = 1, C_{P2}= \frac{\gamma _{2}R}{\gamma _{2}-1}=\frac{5R}{2}; C_{V2}= \frac{R}{\gamma _{2}-1}=\frac{3R}{2}$

$\therefore \gamma ^{mix}=1.5$

Option 1)

$3/2\;$

This option is correct.

Option 2)

$\; 23/15\;$

This option is incorrect.

Option 3)

$\; 35/23$

This option is incorrect.

Option 4)

$\; \; 4/3$

This option is incorrect.

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