Hot air balloons float in the air because of the difference in density between c

Question

Hot air balloons float in the air because of the difference in density between cold and hot air. Consider a balloon in which the mass of the pilot basket together with the mass of the balloon fabric and other equipment is m b  . The volume of the hot air inside the balloon is V 1   and the volume of the basket, fabric, and other equipment is V 2  . The absolute temperature of the cold air outside the balloon is T c   and its density is ? c  . The absolute temperature of the hot air at the bottom of the balloon is T h   (where T h >T c  ). The balloon is open at the bottom, so that the pressure inside and outside the balloon is the same here. Assume that we can treat air as an ideal gas. Use g  for the magnitude of the acceleration due to gravity. What is the magnitude of the buoyant force F B   on the balloon? For the balloon to float, what is the minimum temperature T min   of the hot air inside it?

Explanation / Answer

buoyant force F = rho*g*V

for the ballon to float,

buoyant force > weight of the ballon

and then u cna use the ideal gas equation to calculate minimu temperature

rho*V*g>W

=>V>W / g*rho

=>n*R*T / P >W / g*rho

=>T>W*P / g*rho*n*R

now, n=rho*V / 28.8(molar mass of air)

so, Tmin = W*P*28.8 / (R*g*rho^2*V)

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