The mass of a hot-air balloon and its occupants is 373 kg (excluding the hot air

Question

The mass of a hot-air balloon and its occupants is 373 kg (excluding the hot air inside the balloon). The air outside the ballocn has a pressure of 1.01 x 105 Pa and a density of 1.29 kg/m3. To lift off, the air inside the balloon is heated. The volume of the heated balloon is 466 m3. The pressure of the heated air remains the same as that of the outside air. To what temperature in kelvins must the air be heated so that the balloon just lifts off? The molecular mass of air is 29 u. Buoyant force Weight

Explanation / Answer

The condition for the balloon to just lifts off -

Mass of balloon = Mass of the displaced air by the balloon

Now, you can work out the mass that the air in the balloon needs to be in order for it to float -

Density = m/V

So, m = densityxV = 1.29 x 466 = 601.14 kg

So, the mass of the air = total mass-mass of balloon = 601.14 - 373 = 228.14 kg

Then you can use the perfect gas equation pV=nRT to relate the mass to temperature, since p and V are constant.

n = mass/molar mass = 228.14 / (6.02x10^23 x 29 x 1.66x10^-27) = 7872 mol

Use the expression -

T = pV/nR

= (1.01x10^5 x 466) / (8.31 x 7872) = 719.5 K

So, the required temperature, air must be heated = 719.5 K

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