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 lifi

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

of the air outside the balloon, with the difference in density compensu.

$($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?