# A hot air balloon is a sphere of radius 8 m. The air inside is at a temperature of 60°C.

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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–1K-1, 1 atm. = 1.013×105 Pa; the membrane tension is 5 N m–1.)

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Let the pressure inside the balloon be Pi and the outside pressure be Po

Pi - Po = 2γ/r

Considering the air to be an ideal gas

Pi V= ni R Ti where V is the volume of the air inside the balloon, ni is the number of moles inside and Ti is the temperature inside, and Po V= no R To where V is the volume of the air displaced and no is the number of moles displaced and To is the temperature outside.

ni = PiV/RTi = Mi/MA where Mi is the mass of air inside and MA is the molar mass of air and no = PoV/RTo = Mo/MA where Mo is the mass of air outside that has been displaced. If W is the load it can raise, then

W + Mi g = Mo g

W = Mo g – Mi g

Air is 21% O2 and 79% N2 