Consider a hot-air balloon. The deflated balloon,gondola, and two passengers hav

Question

Consider a hot-air balloon. The deflated balloon,gondola, and two passengers have a combined mass of 315 kg. When inflated, the balloon contains 994 m^3 of hot air. Find the temperature (C) of the hot air required to lift the balloon o the ground. The air outside the balloon has a temperature of 20.2 C and a pressure of 0.916*10^5 Pa. The pressure of the hot air inside the balloon is the same as the pressure of the air outside the balloon. The molar mass of air is 29 g/mol. Hint: First nd the density of the air outside the balloon. Then use Archimedes' principle to nd the density required for the air inside the balloon. Then find the temperature of the air inside the balloon.

Explanation / Answer

P1 V1 / T1 = P2 V2 / T2 considering air as an ideal gas
V2 / V1 = T2 / T1 since P1 = P2 or the balloon would change size
Let p = density of air
F = (V2 - V1) p g where F = bouyant force (weight of air displaced)
F = (T2 V1 / T1 - V1) p g = V1 (T2/T1 - 1) p g
T2/T1 = F / (V1 p g) + 1 = m / (V1 p) + 1
I don't think you need to compare densities of air outside and inside balloon.
The volume of the outside air displaced is V2 - V1 and the
weight of the air displaced is (V2 - V1) p g which is the buoyant force.
Also, the weight of the air in the balloon does not change.
The density of air is around .01 kg/m^3.
The problem implies that you use Avagadro's number, but that applies
to Standard Conditions of 0 deg C and density of air varies with temperature.

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