Question

The mass of a hot-air balloon and its cargo (not including the air inside) is 200 kg. The air outside is at 10.0°C and 101 kPa. The volume of the balloon is475 m³. To what temperature must the air in the balloon be heated before the balloon will liftoff? (Air density at 10.0°C is 1.25 kg/m³.)
1 K

Answer 1

Total weight of the balloon + cargo system + filled airis

V*D*g + 200*9.8 = 475*D*9.8 + 1960 N, where D is thedensity of the air filled nside and V is the volume of theballoon.

Buoyant force acting on the balloon is d*V*g,

where d is the density of the air outside, V is volumeof the balloon,

So, for the balloon to lift off,

Buoyant force must be greater than the totalforce.

So, for d = 1.25 kg/m3,

V = 475 m3

1960 + 475*9.8*D = 1.25*475*9.8

So, D = 0.82895 kg/m3

So, using PV/nT = constant,

n/V α P/T

(mn/V) α P/T , where m is the average molarmass.

So, D α P/T

Thus, D/d = t/T,

t = 10 + 273 K = 283 K,

D = 0.82895 kg/m3, d = 1.25 kg/m3

T = 283*(1.25/0.82895) K = 426.746 K