PV = nRT question?

Question

A hot-air balloon stays aloft because hot air is less dense than cooler air at the same pressure. Suppose the volume of such a balloon is 500 m^3 and its mass (in addition to the mass of the hot air) is 290 kg. The surrounding air is at 14.0 C. At this temp the density of air is 1.23 kg./m^3. The pressure inside and outside the balloon is the same.

What is the density of hot air in the balloon if it hovers in equilibrium?

What is the Celsius temp of hot air in the balloon?

Please explain and show your work, so that I may understand.

Answer 1

ρ = ((500 m^3)(1.23 kg./m^3) - 290 kg)/(500 m^3)
ρ = 0.65 kg/m^3
T = 1.23*287.12/0.65 - 273.12
T = 270.20 °C