While the hot air balloon is still in the air. The mass of the hot air and the average temperature in the balloon needs to be determined.

Assumptions:

The air is ideal gas with gas constant R = 287 J/(kg-K).

(1) The mass of air in the balloon

Since the air in the balloon is ideal gas, it obeys the ideal-gas equation of state.

      PV = m_airRT

where
       R = gas constant
       V = volume of the balloon
       P = pressure in the balloon
       m_air = mass of the air in the balloon
       T = temperature of the air in the balloon

Force Balance on The Balloon

Since the balloon is open to the air, the pressure P in the balloon is the same as the ambient pressure.

According to the force diagram shown on the left, the sum of all forces should be zero when the balloon is still in the air.

      F_B - (G_b + G_p + G_air) = 0

      F_B = ρ_{cool air}gV_balloon = ρ_{cool air}gπ (4/3)(D/2)³

      ρ_{cool air} = P/(RT)
                 = 90(10³)/(287(15+273.15))
                 = 1.089 kg/m³

Rearranging the equation to give the expression of mass of the hot air.

      m_air = F_B/g - m_b - 3(m_p)

 With all the data known

      m_air =  (1.089)(9.8)(4/3)π(10)³ /9.8 -  80 - 3(65)
             = 4287 kg

(2) The average temperature in the balloon

After the mass of hot air is determined, the average temperature of the hot air can be determined using the ideal-gas equation of state.

      PV = m_airRT
      T = PV/ (m_airR)
         = 90(10³)(4/3)(3.14)(10)³/ ((4287)(287))
         = 306.5 K = 33.34 ^oC

(3) The movement of the balloon if the local air temperature is 30 ^oC

The density of the air decreases with the temperature increases. Hence the buoyancy force will decrease and the balloon will move downward. Repeating the solution above, the temperature of the hot air equals 50.38 ^oC if the balloon keeps still again.