A certain balloon maintains an internal gas pressure of $P_{0} = 100$ $k P a$ until the volume reaches $V_{0} = 20 m^{3}$ . Beyond a volume of $20 m^{3}$ , the internal pressure varies as $P = P_{0} + 2 k {(V - V_{0})}^{2}$ where $P$ is in $k P a, V$ is in $m^{3}$ and $k$ is a constant $(k = 1 k P a / m^{3})$ . Initially the balloon contains helium gas at $20^{0} C$ , $100 k P a$ with a $15 m^{3}$ volume. The balloon is then heated until the volume becomes $25 m^{3}$ and the pressure is 150 kPa. Assume ideal gas behaviour for helium. The work done by the balloon for the entire process in $k J$ is

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1256

1414

1512

1083

answer is C.

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Detailed Solution

Work done by the balloon $W = \int_{15 m^{3}}^{20 m^{3}} P_{0} d V + \int_{20 m^{3}}^{25 m^{3}} p d V$

$= \int_{15 m^{3}}^{20 m^{3}} P_{0} d V + \int_{20 m^{3}}^{25 m^{3}} [P_{0} + 2 {(V - V_{0})}^{2}] d V$

$= [100 \times 5 + 100 \times 5 + 2 \times \frac{{(25 - 20)}^{3}}{3}] k J = 1083 k J$

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