Question

A hot-air balloon stays aloft because hot air at atmospheric pressure is less dense than cool air at the same pressure. The volume of a balloon is V = 1000.0 m3,

and the surrounding air has a temperature T0 which is 10.0 degrees C. The density of air at 10.0 degrees C and atmospheric pressure is p0=1.23 kg m-3.

a. What must the temperature T of the air in the balloon be for it to lift a total load of M plus the mass of the hot air? Express your answer only in terms of the variables defined above and any needed physical constants.

b. If this happens when T = 80.2◦C, what is the mass M in kilograms?

c. What is the density of the air inside the balloon in kgm−3?

Answer 1

The volume V of the hot air ballon is V = 1000 m3

The surrounding temp T = 10.0 deg C , The density of air of 10.0 deg C is = 1.23 kg/m3

the atmospheric pressure is P = 101 Kpa

a) let the density of the air inside the hot air ballon is

where T is the temperature T and The R is the universal gas constant

The mass of the hot air ballon is

hence, the net upward force for lifting the mass M kg is

F = Mg =

putting the values we have

b) putting T = 80 deg C = 353 K , R = 8.314 j /mol/K , V = 1000 m3 , g = 9.8 m/s2 , P = 101300 Pa we get

mass M = 23.5 Kg

c) The density of air at Temp T = 80 deg C = 353 K is

putting the value we have

density is 0.99 kg / m 3