Q. 23

A hot air balloon is a sphere of radius 8 m. The air inside is at a temperature of 60°C. How large a mass can the balloon lift when the outside temperature is 20°C? (Assume air is an ideal gas, R = 8.314 J mole -1 K-1, 1 atm. = 1.013×105 Pa; the membrane tension is 5 N m-1.)

Answer :

Given:


Inside temperature = Ti = 60°C = 333K


Outside temperature = To= 20°C = 293K


Outside pressure = Po = 1.013×105Nm-2



Let the pressure inside the balloon be Pi, T be the surface tension and r be the radius of the balloon, then by formula for excess pressure,



Assume air to be ideal gas, then by formula for an ideal gas,




If there are ni moles of air inside the balloon, then



where Mi is the mass of air inside the balloon and MA is the molar mass.


Similarly,



where Mo is the mass of air inside the balloon and MA is the molar mass.


Now, in order to calculate Pi, lets calculate excess pressure.



Now, since this is negligible, we can assume,


Pi = Po = 1.013×105Nm-2


Assume, 21% of O2 and 79% of N2 in the atmosphere.


We can calculate MA as,




We know, the mass M lifted by the balloon can be found as,





So, the balloon can carry a mass of 301.46kg.


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