Question

The planet Jupiter has a surface temperature of \(140 \mathrm{~K}\) and a mass 318 times that of Earth. Mercury has a surface temperature between \(600 \mathrm{~K}\) and \(700 \mathrm{~K}\) and a mass \(0.05\) times that of Earth. On which planet is the atmosphere more likely to obey the ideal-gas law? Explain.

Short answer

Based on our comparison of the estimated densities and surface temperatures, Mercury's atmosphere is less dense and has a higher surface temperature than Jupiter's atmosphere. Therefore, the atmosphere on the planet Mercury is more likely to obey the ideal-gas law.

Step-by-step solution

Compute masses of Jupiter and Mercury

First, we need to calculate the masses of Jupiter and Mercury relative to the Earth's mass. Let the Earth's mass be represented by \(M_{earth}\). Then, the masses of Jupiter and Mercury can be calculated as follows: Mass of Jupiter \(= 318 * M_{earth} \) Mass of Mercury \(= 0.05 * M_{earth}\)

Estimate the densities of the atmospheres

Now we will estimate the densities of the atmospheres of both planets. To do this, we will use the equation for density: \(density = \frac{mass}{volume}\) For simplicity, let's assume a constant volume for both planets' atmospheres. Since Jupiter has a much larger mass than Mercury, its atmosphere is denser if we assume the same volume for both planets. Keep in mind that this is just an estimation, and it doesn't consider the effect of gravity on the density of the atmospheres.

Compare the surface temperatures of both planets

Based on the given information, the surface temperature of Jupiter is \(140 K\) and that of Mercury is between \(600 K\) and \(700 K\). The ideal-gas law is more likely to be valid at higher surface temperatures.

Determine which planet's atmosphere is more likely to obey the ideal-gas law

Based on our comparison of the estimated densities and surface temperatures, we can conclude that Mercury's atmosphere is less dense and has a higher surface temperature than Jupiter's atmosphere. Therefore, the atmosphere on the planet Mercury is more likely to obey the ideal-gas law.