Imagine a giant dry-cleaner's bag full of air at a temperature of -31 floating like a balloon with a string hanging from it 15 above the ground.

Estimate what its temperature would be if you were able to yank it suddenly back to Earth's surface.

To estimate the temperature of the giant dry-cleaner's bag of air if it were suddenly yanked back to Earth's surface, we need to understand some basic principles of thermodynamics.

First, let's consider the ideal gas law, which relates the pressure (P), volume (V), and temperature (T) of a gas. The ideal gas law is given by the equation: PV = nRT, where n is the number of moles of gas and R is the ideal gas constant.

In this case, we have a fixed volume (the giant dry-cleaner's bag) and a fixed amount of air inside it. So, if we assume that the number of moles of air remains constant, the ideal gas law can be simplified to: P/T = constant.

Now, let's consider what happens when the bag is suddenly yanked back to Earth's surface. As the bag descends, the pressure on the air inside increases due to the increased weight of the air column above it. This increase in pressure will also cause an increase in temperature.

We can estimate the final temperature of the air inside the bag using the equation: P1/T1 = P2/T2, where P1 and T1 are the initial pressure and temperature, and P2 and T2 are the final pressure and temperature.

Given that the initial temperature is -31°C, we need to convert it to Kelvin since we are dealing with absolute temperatures. To convert from Celsius to Kelvin, we add 273.15. So, T1 = -31°C + 273.15 = 242.15K.

The air pressure at Earth's surface is approximately 101325 Pa (assuming standard atmospheric pressure). Let's assume that the initial pressure inside the bag is also 101325 Pa, so P1 = P2 = 101325 Pa.

Now, we can rearrange the equation and solve for T2:

P1/T1 = P2/T2
101325/242.15 = 101325/T2

Solving for T2:

T2 ≈ (101325 * 242.15) / 101325 ≈ 242.15K

Therefore, if the giant dry-cleaner's bag of air were suddenly yanked back to Earth's surface, the estimated temperature of the air inside would be approximately 242.15 Kelvin (equivalent to -31°C).