If the ideal gas is diatomic and its increase in internal energy is $100 J$ then the work performed by gas is :
(Ignore vibrational degree of freedom)

80 J

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180 J

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100 J

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20 J

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Solution

The correct option is A $80 J$
$U = α \sqrt{V} H e r e U = K i n e t i c e n e r g y o f i d e a l g a s$
$(55) b 5$
$U = n C_{V} T$
$\Rightarrow n C_{V} T = α \sqrt{V} . . . . (i) T = \frac{P V}{n R} . . . . . . (i i)$
sub. (2) in (1)
$(\begin{matrix} \frac{C_{V}}{R} \end{matrix}) . P V = α \sqrt{V}$
$\Rightarrow P = (\begin{matrix} \frac{α . R}{C_{V}} \end{matrix}) \frac{1}{\sqrt{V}}$
$w = - \int P d V = - (\begin{matrix} \frac{α R}{C_{V}} \end{matrix}) \int \frac{d V}{\sqrt{V}}$
$= - {[(α) (γ - 1) . \frac{1}{(1 / 2)} (\sqrt{V})]}_{V_{1}}^{V_{2}}$
$w = - 2 (α) (γ - 1) (\sqrt{V_{2}} - \sqrt{V_{1}})$
Work done by the gas = $- w = 2 (α)$
$(γ - 1) (\sqrt{V_{2}} - \sqrt{V_{1}})$
For diatomic gas with no vibrational degree of freedom
$C_{V} = \frac{3}{2} R + 2 \times \frac{1}{2} R$
$C_{V} = 5 / 2 R$
$Δ U = α (\sqrt{V_{2}} - V_{1}) = 100 J$
$W = 2 (α) (γ - 1) (\sqrt{V_{2}} - V_{1}) = (2) (γ - 1) \times 100$
$∵ γ = \frac{C_{P}}{C_{V}} = \frac{7}{5}$
$(γ - 1) = (\begin{matrix} \frac{7}{5} - 1 \end{matrix}) = (\begin{matrix} \frac{2}{5} \end{matrix})$
$W = (2) (\begin{matrix} \frac{2}{5} \end{matrix}) (100) J = 80 J$

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If the ideal gas is diatomic and its increase in internal energy is 100J then the work performed by gas is : (Ignore vibrational degree of freedom)

If the ideal gas is diatomic and its increase in internal energy is $100 J$ then the work performed by gas is :
(Ignore vibrational degree of freedom)