Hot air balloons float in the air because of the difference in density between cold and hot air. in this problem, you will estimate the minimum temperature the gas inside the balloon needs to be, for it to take off. to do this, use the following variables and make these assumptions: the combined weight of the pilot basket together with that of the balloon fabric and other equipment is w. the volume of the hot air inside the balloon when it is inflated is v. the absolute temperature of the hot air at the bottom of the balloon is th (where th> tc). the absolute temperature of the cold air outside the balloon is tc and its density is ? c. the balloon is open at the bottom, so that the pressure inside and outside the balloon is the same. as always, treat air as an ideal gas. use g for the magnitude of the acceleration due to gravity. here is the actual what is the density ? h of hot air inside the balloon? assume that this density is uniform throughout the balloon. express the density in terms of th, tc, and ? c.

Explanation:

We have the followin data:

is the combined weight of the pilot basket together with that of the balloon fabric and other equipment

is the combined mass of the pilot basket together with that of the balloon fabric and other equipment

is the acceleration due gravity

is the volume of the hot air inside the balloon when it is inflated

is the absolute temperature of the hot air at the bottom of the balloon, being

is the absolute temperature of the cold air outside

is the density of the cold air outside

is the density of the hot air inside

where is the pressure at the inside and is the pressure at the outside

Well, let's begin by writting the equations for the density:

Density cold air outside:

(1)

Where is the mass of air outside and is the volume of air outside

Isolating we have:

(2)

Density hot air inside:

(3)

Where

Then:

(4)

On the other hand, the The Ideal Gas equation is:  

 (5)

Where:  

is the pressure of the gas  

the number of moles of gas  

is the gas constant  

is the absolute temperature of the gas in Kelvin.

is the volume

This can be rewritten as:

 (6)

Since :

 (7)

Isolating :

 (8)

Substituting (8) in (3):

(9)

Substituting (2) in (9):

(10)

Rearranging:

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