The planet Jupiter has a surface temperature of \(140 \mathrm{~K}\) and a mass 318 times that of Earth. Mercury (the planet) 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

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In conclusion, Mercury's atmosphere is more likely to obey the ideal-gas law than Jupiter's atmosphere, as Mercury has a higher temperature (\(T_M = 600-700 \mathrm{~K}\)) and lower mass (mass ratio to Earth, \(M_M = 0.05\)), which implies a lower density compared to Jupiter's temperature (\(T_J = 140 \mathrm{~K}\)) and mass ratio to Earth (\(M_J = 318\)). The ideal-gas law is more accurate for gases with high temperatures and low densities.

Step by step solution

01

Understand the ideal-gas law

The ideal-gas law is an equation of state for a gas, given by the formula: \(PV = nRT\) where, P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature. The approximation is accurate for gases at high temperatures and low densities. As we are not given any specific information regarding pressure and volume for both planets, we should focus on relative temperatures and density.
02

Compare the temperatures of Jupiter and Mercury

We are given the surface temperatures of Jupiter and Mercury: - Jupiter: \(T_J = 140 \mathrm{~K}\) - Mercury: \(T_M = 600-700 \mathrm{~K}\) As Mercury has a higher temperature compared to Jupiter, Mercury's atmosphere is more inclined towards obeying the ideal-gas law based on the temperature.
03

Compare the densities of Jupiter and Mercury

To estimate the densities, let's first consider the mass ratios of Jupiter and Mercury compared to Earth: - Jupiter: Mass ratio to Earth, \(M_J = 318\) - Mercury: Mass ratio to Earth, \(M_M = 0.05\) Jupiter has a significantly higher mass ratio compared to Mercury. The ideal-gas law is more accurate for low-density gases. Since Jupiter's mass is much larger than Mercury's mass, Jupiter's density might be higher, making it less compatible with the ideal-gas law.
04

Conclusion

In conclusion, based on the given information, Mercury has a higher temperature and lower mass (thus lower density) compared to Jupiter. Therefore, Mercury's atmosphere is more likely to obey the ideal-gas law than Jupiter's atmosphere.

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