A hot-air balloon works by heating the air inside the balloon to a temperature T

Question

A hot-air balloon works by heating the air inside the balloon to a temperature T, higher than the temperature of the surroundings (T_o). Assume that a balloon holds a volume V of air and that the cargo and balloon have mass m_o and negligible volume. If air can be approximated as an ideal gas with molar mass M, find the density of the air inside the balloon at sea level (pressure p_o). Find all the forces (vectors!) acting on the balloon. What is the minimum temperature necessary to lift the balloon?

Explanation / Answer

(a) In the case of an ideal gas -

P0V = nRT = (m0/M)RT

=> m0/V = (P0*M) / (RT) = Density of air inside the balloon.

(b) Force acting in the upward direction = Buoyant force = B = V*(P0*M) / (RT0)*g

Force acting in the downward direction = m0*g + rho*V*g

(c) Now to lift the balloon -

m0*g + rho*V*g = V*(P0*M) / (RT0)*g

=> T = 1 / [1/T0 - R*m0/P0MV]

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