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

The atmosphere of Mercury is more likely to obey the ideal gas law due to its higher surface temperature (600 K - 700 K) and lower pressure compared to Jupiter (140 K), which has lower surface temperature and higher pressure. The ideal gas law holds better at higher temperatures and lower pressures.

Step-by-step solution

Estimate Pressures

Using the given masses of Jupiter and Mercury, we can estimate the gravitational force on the gases at the planets' surfaces. A higher mass will create a stronger gravitational force, resulting in a higher pressure. Jupiter: mass = 318 * Earth_mass Mercury: mass = 0.05 * Earth_mass Since Jupiter has a much larger mass than Mercury, the pressure at the surface of Jupiter is likely to be much higher than at the surface of Mercury.

Analyze Surface Temperatures

We are given the surface temperatures for Jupiter and Mercury: Jupiter: T = 140 K Mercury: T = 600 K - 700 K As we can see, the temperature at the surface of Mercury is much higher than the temperature at the surface of Jupiter.

Compare Jupiter and Mercury

Now, we need to compare the atmospheres of Jupiter and Mercury in terms of the ideal gas law. The ideal gas law states that: PV = nRT where P is pressure, V is volume, n is the number of moles of the gas, R is the gas constant, and T is temperature. An atmosphere is more likely to obey the ideal gas law at higher temperatures and lower pressures. From our estimations in Steps 1 and 2, we see that: - Jupiter has a lower temperature and higher pressure. - Mercury has a higher temperature and lower pressure.

Conclusion

Based on the given information, the atmosphere of Mercury is more likely to obey the ideal gas law than the atmosphere of Jupiter. The higher surface temperature and lower pressure on Mercury create conditions that are more favorable for the ideal gas law to hold.

Key concepts

Understanding Gas Laws

Gas laws play a crucial role in understanding the behavior of gases under various conditions. The ideal gas law is a fundamental equation derived from these gas laws, encapsulating the relationship between pressure (P), volume (V), temperature (T), and moles of gas (n). The equation is expressed as \( PV = nRT \), where R represents the ideal gas constant.

The ideal gas law assumes that real gases behave as 'ideal' or perfect gases and thus, can be used to predict the behavior of gases at the macroscopic level. This law is particularly accurate when the gas molecules do not interact significantly and are at a high temperature and low pressure. Under these conditions, the volume occupied by the individual molecules of the gas is relatively negligible compared to the overall volume of the gas, and the impact of intermolecular forces is minimal.

When answering problems such as the behavior of planetary atmospheres, applying the ideal gas law requires us to consider both the physical size and mass of the planet affecting gravitational force and surface temperature that can affect the temperature component of the equation. This allows us to predict if the atmospheric conditions are close to ideal or not.

The Role of Gravitational Force

Gravitational force is a fundamental force acting between all masses in the universe, and it significantly influences the behaviors of planetary atmospheres. The force of gravity exerts pressure on the gases in a planet's atmosphere, compressing them and affecting their ability to behave as an ideal gas.

The larger a planet's mass, the stronger its gravitational pull, which leads to a higher atmospheric pressure. When the atmosphere is compressed by gravity, the gas molecules are closer together, and intermolecular forces become more influential, causing the gas to deviate from ideal behavior. In the context of our problem, Jupiter's massive size creates a high-pressure environment, potentially pushing the behavior of the gases away from the ideal model.

A primary consideration in the exercise improvement advice is to estimate how the different masses of Jupiter and Mercury affect the potential for their atmospheres to adhere to the ideal gas law. By understanding the correlation between mass, gravitational force, and atmospheric pressure, students can gain insight into why Mercury's lower mass and, thus, lower gravity, might favor ideal gas law behavior.

Surface Temperature and its Effect on Gases

Surface temperature is another vital factor influencing the behavior of gases and their adherence to the ideal gas law. High temperatures provide the thermal energy that allows gas molecules to overcome intermolecular forces and move more freely, closely resembling the conditions of an ideal gas.

In contrast, low temperatures tend to increase the effects of intermolecular forces, causing gases to deviate from the expected behavior of an ideal gas. This is because, at lower temperatures, gas molecules have less kinetic energy and are more likely to interact and clump together.

Applying this knowledge to Jupiter and Mercury, we understand that Mercury's significantly higher surface temperature contributes to an environment where gas molecules have more kinetic energy and are thus less influenced by intermolecular forces, aligning better with the conditions predicted by the ideal gas law compared to the colder Jupiter.