Question

In class we showed that hot air at atmospheric pressure is less dense than cooler air at the same pressure.

Because of this difference hot-air balloons can stay aloft. Assume that you have an air-balloon that has

the volume of 500 m3 and the surrounding air is at 100C. You want to lift 250 kg object (in addition to

the mass of the hot air that is inside the balloon). What must be the temperature of the air in the balloon?

03
The density of air at 10 C and atmospheric pressure is 1.23 kg/m . The heated air inside the balloon is at

roughly the same pressure as the outside air.

Answer 1

For the balloon to lift off, the buoyant force B must be greater than its weight.

The buoyant force is equal to the weight of the displaced air at 10 oC = 283 K.

which is equal to weight of the balloon and weight of the hot air

Then mass of the hot air = 3577/9.8 = 365 kg

then density of the hot air = d = mass/volume = 365/500 = 0.73 kg/m3

At constant pressure, the volume of a gas is proportional to the absolute temperature. (Law of Gay-Lussac) The pressures on the inside and outside of the inflated balloon are nearly equal. The pressure on the outside is the constant atmospheric pressure. The Law of Gay-Lussac therefore applies.
Since the volume of a gas at constant pressure is proportional to its temperature, its density d = m/V is proportional to 1/T.

(1.25/0.73)*283 = T2 = 484.58 K = 211.58 degree Celcius

The air in the balloon must be heated to more than 484.58 K = 211.58 oC