What Temperature Must the Air Inside a Hot-Air Balloon Reach for Adequate Lift?

[notorious big]

I'm confused as to what equation I need to use for the following problem:

A hot-air balloon achieves its buoyant lift by heating the air inside the balloon, which makes it less dense than the air outside. Suppose the volume of a balloon is 1820 m3 and the required lift is 2983 N (weight of equipment and passenger). Calculate the temperature of the air inside the balloon which will produce the desired lift. Assume that the outside air temperature is 0oC and that air is an ideal gas under these conditions. Express your answer in oC. The density of air at STP is 1.29 kg/m3.

Any help would be greatly appreciated.

 

[OlderDan]

I'm confused as to what equation I need to use for the following problem:

A hot-air balloon achieves its buoyant lift by heating the air inside the balloon, which makes it less dense than the air outside. Suppose the volume of a balloon is 1820 m3 and the required lift is 2983 N (weight of equipment and passenger). Calculate the temperature of the air inside the balloon which will produce the desired lift. Assume that the outside air temperature is 0oC and that air is an ideal gas under these conditions. Express your answer in oC. The density of air at STP is 1.29 kg/m3.

Any help would be greatly appreciated.

In most problems involving the ideal gas law, the number of molecules of gas is constant, and P, V and T vary in some way. In this problem, P and V are essentially constant and changing the temperature affects the number of molecules in the balloon, which affects the density of the air.

The buoyant force on the balloon is the weight of air at STP that is displaced by the balloon. The weight of the air inside the balloon is less because of the higher temperature. The difference must equal the weight of the passenger and equipment.

 

[notorious big]

Thanks for the reply Dan. I've been trying for hours to figure out what your explanation means, but I can't figure out how to find the weight of the air inside or outside of the balloon.

 

[OlderDan]

Thanks for the reply Dan. I've been trying for hours to figure out what your explanation means, but I can't figure out how to find the weight of the air inside or outside of the balloon.

PV = nRT

P and V are constant assuming the balloon is fully expanded and it stays at about the same altitude. Let T1 be the temperature of the air in the balloon and T2 be the temperature of the outside air, with n1/V = number density of molecules inside and n2/V = number density of molecules outside.

P = (n1/V)RT1 = (n2/V)RT2

(n1/V)/(n2/V) = T2/T1

The ratio of the number densities is the inverse of the ratio of the temperatures. The ratio of the number densities is the same as the ratio of the mass densities because it is the same air, just heated up. The mass density of the outside air is given, and the volume of the balloon is given. From that you can calculate the weight of the air displaced by the balloon. The weight of the air inside must be less than the outside by the amount of the buoyant force. You can find the weight of the inside air as a function of temperature using the density ratio and the outside temperature (STP means standard temperature and pressure 0C and 1 atmosphere). Since you know how much it has to weigh, you can calculate the temperature it must have

W2 = outside air weight = D2(g)V; D2 = given density
W1 = inside air weight = D1(g)V; D1 = hot air density

D1/D2 = T2/T1
D1 = D2*T2/T1