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

Total mass of balloon:
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