How to Estimate the Minimum Temperature Required for a Hot Air Balloon to Take Off

What is the density of hot air inside the balloon?

Assume that this density is uniform throughout the balloon.

Final answer:

To estimate the minimum temperature needed for a hot air balloon to take off, the density of the hot air inside the balloon can be calculated using the ideal gas law. Hot air must be less dense than the surrounding colder air due to its higher temperature, which allows the balloon to become buoyant and rise.

The question pertains to the physical principles that allow a hot air balloon to float, specifically by understanding how the density of hot air inside the balloon compares to the cooler external atmosphere. To calculate the density of the hot air inside the balloon which contributes to its ability to rise, one must apply the ideal gas law under the assumption that the pressure inside the balloon remains the same as the external pressure due to it being open at the bottom.

The ideal gas law is PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in kelvins. Since n/V is the molarity, which can be related to density (ρ) through ρ = molar mass (M) × molarity (n/V), we can rearrange the equation to get ρ = P × M / RT. The same molar mass of dry air is used for hot air as the composition of the gases is assumed to be the same.

As per the laws of thermodynamics, the density of a gas is inversely proportional to its temperature when pressure is held constant. Therefore, for the hot air balloon to lift off, the temperature of the air enclosed within the balloon must be higher than the temperature of the surrounding atmosphere, making the internal air less dense. This decrease in density leads to a buoyant force which enables the balloon to rise. The balloon's lifting capacity is then a result of the difference in the mass of cooler air displaced by the balloon and the mass of the warmer air inside it.