Question

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.)

Answer 1

Let the pressure inside the balloon be P_i and the outside pressure be P_o 

P_i - P_o = 2γ/r

Considering the air to be an ideal gas

P_i V= n_i R T_i where V is the volume of the air inside the balloon, n_i is the number of moles inside and T_i is the temperature inside, and P_o V= n_o R T_o where V is the volume of the air displaced and n_o is the number of moles displaced and T_o is the temperature outside.

n_i = P_iV/RT_i = M_i/M_A where Mi is the mass of air inside and MA is the molar mass of air and n_o = P_oV/RT_o = M_o/M_A where M_o is the mass of air outside that has been displaced. If W is the load it can raise, then

W + M_i g = M_o g

W = M_o g – M_i g

Air is 21% O₂ and 79% N₂