An ideal gas- having ratio of specific heat - undergoes a process in which its internal energy relates to the volume as U - -V where - is a constant- If the gas is expanded from volume V1 to V2-The work performed by gas is -

The correct option is B (A) 2α(γ 1)[ √(V_2) √(V_1)] U = α √(V) Here U = Kinetic energy of ideal gas (5 5) b 5 ...

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Byju's Answer

Standard XII

Physics

RMS and KE Expressions

An ideal gas,...

Question

An ideal gas, having ratio of specific heat $γ$ undergoes a process in which its internal energy relates to the volume as $U = α \sqrt{V}$ where $α$ is a constant. If the gas is expanded from volume $V_{1} t o V_{2}$ .

The work performed by gas is :

(A)

2 α (γ - 1) [\sqrt{V_{2}} - \sqrt{V_{1}}]

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(B)

α (γ - 1) [\sqrt{V_{2}} - \sqrt{V_{1}}]

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(C)

2 α (γ - 1) [V_{2} - V_{1}]

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(D)

α (γ - 1) [V_{2} - V_{1}]

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Solution

The correct option is B (A) $2 α (γ - 1) [\sqrt{V_{2}} - \sqrt{V_{1}}]$
$U = α \sqrt{V} H e r e U = K i n e t i c e n e r g y o f i d e a l g a s$
$(55) b 5$
$U = n C_{V} T$
$\Rightarrow n C_{V} T = α \sqrt{V} . . . . (i) T = \frac{P V}{n R} . . . . . . (i i)$
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 = $- w = 2 (α)$
$(γ - 1) (\sqrt{V_{2}} - \sqrt{V_{1}})$

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An ideal gas, having ratio of specific heat γ undergoes a process in which its internal energy relates to the volume as U=α√V where α is a constant. If the gas is expanded from volume V1toV2.The work performed by gas is :

An ideal gas, having ratio of specific heat $γ$ undergoes a process in which its internal energy relates to the volume as $U = α \sqrt{V}$ where $α$ is a constant. If the gas is expanded from volume $V_{1} t o V_{2}$ .

The work performed by gas is :