One Fall day morning $($the temperature is $10C),$ a family of three $($and a dog$),$ decided to take a hot

air balloon flight. The hot air balloon is powered by burning propane gas. The passengers, dog, basket,

ropes, burners, gas cylinder and the envelope of the balloon weigh $678kg$ all together.

The oxidation reaction of propane is given as

$C_3{H}_8(g)+5O_2(g)3C{O}_2(g)+4H_{2}O(l)$

Molar enthalpies are

$C_3{H}_8(g)$:$-103\mathrm{.}85kJmo{l}^{-1}$

$C{O}_2(g)$:$-393\mathrm{.}509kJmo{l}^{-1}$

$H_{2}O(l)$:$-285\mathrm{.}83kJmo{l}^{-1}$

Molecular weight of propane is $44\mathrm{.}1gmo{l}^{-1}$

Average molecular weight of air is $28\mathrm{.}97gmo{l}^{-1}$

$C_P$ of water is $4\mathrm{.}187kJk{g}^{-1}{K}^{-1}$

$C_P$ of $($hot$)$ air is $1\mathrm{.}01kJk{g}^{-1}{K}^{-1}$

Hint: A hot air balloon is based on the principle of buoyancy. For lij

off the density of air inside the balloon should be less than the densi.

of the air outside the balloon, with the difference in density compensating for the weight of the load.

$($a$)$ Given that the inflated balloon is a nearly perfect sphere of $9m$ radius, what should be the average

temperature of the air inside the balloon for a lift$-$off?

$($b$)$ Shortly after the take$-$off, a sudden rainstorm strikes, $($temperature of rain water $\{$:$=6C)$ and the

envelope gets soaked with cold rain uniformly and absorbs $12kg$ of rain water. Assuming that the heat

capacity of the dry envelope material itself is negligible, and the envelope temperature is equal to the

temperature of the air inside the balloon, calculate the heat lost by the air inside the balloon to the rain

water on the envelope. Assume that heat transfer is instantaneous, and the temperature of the envelope

never drops below the inside temperature.

$($c$)$ If all this heat is replenished by burning propane gas calculate the mass of propane gas to be burnt.