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A balloon is 1 m in diameter and contains air at 25° C and 1 bar pressure. It is filled with air isothermally and reversibly until the pressure reaches 5 bar. Assume pressure is proportional to diameter of balloon. Calculate work done by air (atm $m^{3}$ ):

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$78 π$

$l 56 π$

$624 π$

$625 π$

answer is A.

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Detailed Solution

We have

$P \propto D$

$P = k D$

$\Rightarrow k = 1; D_{f} = 5$

Calculation of the value

$\begin{aligned} W = - \int P d V \\ V = \frac{4}{3} π \frac{D^{3}}{8} \end{aligned}$

$d V = \frac{π}{6} \times 3 D^{2} \cdot d D$

$w = - \frac{π}{2} \int_{1}^{5} D^{3} d D = - {\frac{π}{2} \frac{D^{4}}{4}|}_{1}^{5}$

$= - \frac{π}{8} \times 624$ $= - 78 π$

$W . D by air = + 78 π$

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