121 21 The mass of a hotair balloon and its cargo not includ

121. 21) The mass of a hot-air balloon and its cargo (not includ- ing the air inside) is 200 kg. The air outside is at 10.0°C and 101 kPa. The volume of the balloon is 400 m3. To what temperature must the air in the balloon be warmed before the balloon will lift off? (Air density at 10.0°C is 1.25 kg/m3.) tbarmometer is constucted as shown in Higure

Solution

Gas law

PV = nRT

n/V = P/RT

Pressure is constant since open hole. So n/V is inversely proportional to Temperature

In FBD of ballon there will be Buoyancy force in upward and Force of mass and air will be downward.

mb*g - Fb = 0

Fb = mb*g

V*rho(air)*g - V*rho(hot air)*g = mb*g

rho(air) - rho(hot air) = mb/V

rho(hot air) = rho(air) - mb/V = 1.25 - 200/400

rho(hot air) = 0.75 kg/m^3

Now,

rho(hot air)/rho(air) = T(air)/T(hot air)

T(hot air) = (1.25*283.15)/0.75 = 471.91 K = 472 K

Let me know if you have any doubt.