Question

A spherical balloon of 1 m diameter contains a gas at 150 kpa. the gas inside the balloon is heated until the pressure reaches 450 kpa. during the pressure of heating, the pressure of the gas inside the balloon is proportional to the diameter of the balloon (p = c*d where c is a constant). determine the work done by the gas inside the balloon.

Answer 1

Pressure inside the balloon is given by the equation

$P = cd$

when pressure is 150 kPa the diameter is 1 m

$150 = c*1$

Now when pressure will reach at 450 kPa its diameter will be 3 meter

so radius of the balloon will increase from r = 0.5 m to r = 1.5 m

also the pressure in balloon depends on radius as per following relation

$P = 300 r$

now work done to increase the pressure in balloon is given by

$W = \int PdV = \int P 4\pi r^2 dr$

now we have

$W = \int 300* 4\pi r^3dr$

$W = 1200\pi*\frac{r^4}{4}$

$W = 300\pi(1.5^4 - 0.5^4)$

$W = 4.7 * 10^6 J$

So above is the work done to increase the radius of balloon