Q. 23

# A hot air balloon is a sphere of radius 8 m. The air inside is at a temperature of 60°C. How large a mass can the balloon lift when the outside temperature is 20°C? (Assume air is an ideal gas, R = 8.314 J mole ^{-1} K^{-1}, 1 atm. = 1.013×105 Pa; the membrane tension is 5 N m^{-1}.)

Answer :

Given:

Inside temperature = T_{i} = 60°C = 333K

Outside temperature = T_{o}= 20°C = 293K

Outside pressure = P_{o} = 1.013×10^{5}Nm^{-2}

Let the pressure inside the balloon be P_{i}, T be the surface tension and r be the radius of the balloon, then by formula for excess pressure,

Assume air to be ideal gas, then by formula for an ideal gas,

If there are n_{i} moles of air inside the balloon, then

where M_{i} is the mass of air inside the balloon and M_{A} is the molar mass.

Similarly,

where M_{o} is the mass of air inside the balloon and M_{A} is the molar mass.

Now, in order to calculate P_{i}, lets calculate excess pressure.

Now, since this is negligible, we can assume,

P_{i} = P_{o} = 1.013×10^{5}Nm^{-2}

Assume, 21% of O_{2} and 79% of N_{2} in the atmosphere.

We can calculate M_{A} as,

We know, the mass M lifted by the balloon can be found as,

So, the balloon can carry a mass of 301.46kg.

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