MasteringPhysics problem 15.86?

Question

Please help me solve the following problem:

"A hot-air balloon stays aloft because hot air at atmospheric pressure is less dense than cooler air at the same pressure.

If the volume of the balloon is 500 m^3 and the surrounding air is at 15.0 C, what must the temperature of the air in the balloon be for it to lift a total load of 290 kg  (in addition to the mass of the hot air)? The density of air at 15.0 C and atmospheric pressure is 1.23 kg/m^3 ."

Thanks!

Anonymous · Accepted Answer

assume hot air as ideal gas.

minimum uplift force occurred when :

∑F = 0

F_archimedes - W_hot air - Total load = 0

F_archimedes = W_hot air + Total load = (ρ_hot) (g) V + 290(g) N

g is earth gravitation.

F_archimedes = ρ' (g) V = (ρ_hot) (g) V + 290(g)

ρ' V = (ρ_hot)  V + 290

(1.23) (500) = (ρ_hot) (500) + 290

ρ_hot = 0.65 kg/m³


ideal gas equation,

PV = nRT

P(m/ρ) = (m/Mr)RT

P/ρ = (1/Mr)RT

1/(ρT) = (1/Mr)(RP)

ρT = constant

at 15°C, ρ air = 1.23 kg/m³, find T if  ρ air = 0.65 kg/m³

(0.65)T = (1.23)(15 + 273) 

T = 544.98 K 

T = 544.98 - 273 = 271.98°C

MasteringPhysics problem 15.86?

1 Answer