Question

The mass of a hot-air balloon and its cargo (not including the air inside) is 120 kg. The air outside is at 10.0

Answer #1

m = m0 + m_hotair

Required buoyant force:

B = m_net*g

Origin of buoyant force based upon volume and background fluid
density:

B = rho_bg*g*V

Thus:

rho_bg*g*V = m_net*g

Who cares about g?

rho_bg*V = m_net

Mass of hot air:

m_hotair = rho_hot*V

rho_bg*V = m0 + rho_hot*V

Solve for rho_hot:

rho_hot = (rho_bg*V - m0)/V

Find densities from ideal gas law:

rho_hot = P*M/(R*T_hot)

rho_bg = P*M/(R*T_cold)

Thus:

P*M/(R*T_hot) = (P*M/(R*T_cold)*V - m0)/V

Solve for T_hot:

R*T_hot/(P*M) = V/(P*M/(R*T_cold)*V - m0)

Concluding formula:

T_hot = V*P*M/(R*(P*M/(R*T_cold)*V - m0))

Data:

P = 101 kPa

M = 28.97 kg/kmol

R = 8.314 kPa-m^3/kmol-K

T_cold = 283.15 K

m0 = 120 kg

V = 550 m^3

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