Question

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

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

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.

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.

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.

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.

Key concepts

Surface Temperature

Surface temperature plays a significant role in determining whether a planet's atmosphere behaves closer to the predictions of the ideal-gas law. The reason behind this is that higher temperatures increase the kinetic energy of gas molecules, making them behave more like an ideal gas. This means that at higher temperatures, the interactions between gas molecules are minimized, and they move more freely. When reviewing the exercise, the surface temperature of Jupiter is noted as 140 K, whereas Mercury's temperature ranges from 600 K to 700 K. With temperatures significantly higher on Mercury, its gases are more likely to mimic ideal behavior, satisfying the conditions of high temperature outlined in the ideal-gas law.

Planetary Mass

Planetary mass, while not directly a component of the ideal-gas law, influences the density of a planet's atmosphere. A larger planetary mass generally implies a stronger gravitational pull, which can result in a denser atmosphere. Let's compare Jupiter and Mercury:

• Jupiter's mass is 318 times that of Earth.
• Mercury's mass is only 0.05 times that of Earth.

The stronger gravitational force of Jupiter can pull the gas molecules closer together, leading to a denser atmosphere. On the other hand, Mercury's smaller mass suggests a less dense atmosphere. Since the ideal-gas approximation works best at low densities, Mercury's atmosphere is better positioned to adhere to the ideal gas law.

Gas Density

Gas density is another critical factor when considering the applicability of the ideal-gas law. The lower the gas density, the fewer interactions occur between gas molecules, which aligns with the conditions under which the ideal-gas law is most accurate. With Jupiter hosting a substantially higher mass, it consequentailly can contribute to a denser atmosphere due to its stronger gravitational influence. Conversely, with its significantly low mass, Mercury is likely to have a less dense atmosphere. While the actual density cannot be directly derived without specific volumes, the large mass differences imply that Mercury, with its presumably lighter atmosphere, will better satisfy the low-density condition of the ideal-gas law.

Equation of State

The equation of state for an ideal gas is represented as: \[ PV = nRT \]- \(P\) denotes the pressure of the gas,- \(V\) is the volume,- \(n\) represents the number of moles, - \(R\) is the universal gas constant,- \(T\) is the temperature.This mathematical expression helps to understand how gases behave under different conditions, notably low pressure and high temperature, as well as low densities. In the context of planetary atmospheres, it provides a framework to assess whether an atmosphere tends to display ideal gas traits.For the planets in question, applying this equation shows that higher temperature and lower density conditions on Mercury suggest that its atmosphere will more likely conform to this ideal behavior, compared to the denser, cooler atmosphere of Jupiter. Understanding these conditions is key in utilizing the state equation to predict atmospheric behaviors on different planets.