A hot-air balloon stays aloft because hot air at atmospheric...

A hot-air balloon stays aloft because hot air at atmospheric pressure is less dense than cooler air at the same pressure. If the volume of the balloon is 500.0 m$^3$ and the surrounding air is at 15.0$^\circ$C, what must the temperature of the air in the balloon be for it to lift a total load of 290 kg (in addition to the mass of the hot air)? The density of air at 15.0$^\circ$C and atmospheric pressure is 1.23 kg/m$^3$.

Key Concepts

Ideal Gas Law

The ideal gas law (PV = nRT) relates pressure, volume, temperature, and the number of moles of a gas. For a hot-air balloon operating at atmospheric pressure, it explains how heating the air reduces its density since, at constant pressure, density is inversely proportional to temperature (in Kelvin). This relationship is pivotal for understanding how increasing temperature in the balloon can produce the necessary buoyant lift.

Density and Temperature Relationship in Gases

At a constant pressure, the density of a gas is inversely proportional to its temperature. When the air inside a balloon is heated, its molecules move more vigorously, causing the air to expand and its density to decrease. This reduction in density is essential for generating lift, as it allows the weight of the displaced cooler, denser air to exceed the weight of the heated air within the balloon.

Archimedes' Principle

This principle states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. In the context of a hot-air balloon, this means that the buoyant lift is provided by the difference in weight between the cooler denser external air and the lighter heated air inside the balloon.

Transcript

00:01 So her on this question, we are given that this is a hot air balloon and it stays in the air and the volume inside this balloon v.

00:09 We have given us 500 m cube and surrounding air temperature.

00:17 That means that atmospheric pressure temperature is 15°c. and the mass of the total load other than the mass of the hot air.

00:30 We are given to be us to 90 kg.

00:37 And we need to calculate the temperature that must be maintained in the balloon so that it should not fall down.

00:45 And the atmospheric density of air at 15°c is given to be us 1.23 kg, permit a cube.

00:58 Now, if you see the free body diagram of this balloon, so let me make it help.

01:06 The weight force would be acting like this w.

01:10 And the buoyancy force would be acting in the upward direction.

01:14 And if you balance the forces in the y direction, basically in the vertical direction, we can say that the weight force will be called to the buoyancy force.

01:23 Now the weight of the balloon would be calls to the weight of the hot air inside it plus the weight of this mass.

01:33 So the way it would be to 90 plus the mass of air.

01:39 We don't know.

01:39 So let it be m.

01:40 A.

01:41 Times g.

01:42 This would be a total weight of the balloon and it will be equal to the buoyancy force.

01:47 Now buoyancy force we know it is equal to dance three times the volume times g.

01:54 Now here we see that this g and g will cancel out.

02:00 And from here we can calculate though mass of the air inside the hot air balloon.

02:06 So to 90 plus m.

02:09 A.