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. 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:
1. The combined weight of the pilot basket together with that of the balloon fabric and other equipment is W.
2. The volume of the hot air inside the balloon when it is inflated is V.
3. The absolute temperature of the hot air at the bottom of the balloon is Th (where Th > Tc).
4. The absolute temperature of the cold air outside the balloon is Tc and its density is c.
5. The balloon is open at the bottom, so that the pressure inside and outside the balloon is the same.
6. As always, treat air as an ideal gas.
Use for g the magnitude of the acceleration due to gravity.

Explanation / Answer

We will use the ideal gas equation to find the expression for density

We know that PV = nRT

Let us derive an equation for the density of a gas in terms of volume occupied by the gas V, the number of moles of gas particles n and mass per mole m,

We have = mn /V

Now let us express this in terms of T,P,R and m, the mass of one mole of gas,

= mn/nRT/p

= mP/RT

Now we have an expression for the density of a gas in terms of temperature, mass, and pressure. Use this result to write one equation for the density of the hot air (at the bottom of the balloon) and another for the density of the cold air

h = mP/RTh ---------------------------------(1)

c = mP/RTc----------------------------------(2)

Divide (1)/(2)

The pressure at the bottom of the balloon and mass per mole of the hot air are the sameas those of the cold air.

We use the assumption that the density is constant throughout the balloon, so the density you have found is the density everywhere inside the balloon.

Thus we have,

h/c= Tc/Th

h = cTc/Th

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