Problem 90 - Chemistry The Central Science In Si Units

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.

Step-by-Step Solution

Answer

Mercury's atmosphere is more likely to obey the ideal-gas law due to higher temperatures.

Step 1: Introduction to Ideal Gas Law

The ideal gas law is given by the equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature in Kelvin. An atmosphere is likely to obey the ideal gas law if its temperature is high and pressure and intermolecular forces are low.

Step 2: Consider Jupiter's Conditions

Jupiter has a surface temperature of \( 140 \mathrm{~K} \) which is relatively low. It also has a very high mass, which suggests a high gravity and therefore potentially high atmospheric pressure. The combination of low temperatures and high pressures makes real gas behavior more likely, thus deviating from ideal-gas behavior.

Step 3: Consider Mercury's Conditions

Mercury has a surface temperature between \( 600 \mathrm{~K} \) and \( 700 \mathrm{~K} \), which is significantly higher than Jupiter's. Although it has a very low mass, which implies low gravity and low atmospheric pressure, the higher temperatures increase the likelihood of Mercury's atmosphere behaving like an ideal gas, as gases tend to behave more ideally at higher temperatures.

Step 4: Conclusion Based on Conditions

Considering the temperatures and mass-related pressures, Mercury's atmosphere is more likely to obey the ideal-gas law due to its higher temperatures, which favor ideal-gas conditions even if the atmosphere's density and pressure are lower.

Key Concepts

Planetary Atmospheres

Understanding the atmosphere of any planet involves looking at various factors like temperature, mass, and pressure. Planetary atmospheres are not just simple layers of gas. They are complex systems subject to a range of physical laws. In the study of planetary science, we consider how these factors interact to determine the behavior of the atmosphere.

Atmospheres with different temperatures and masses will behave differently. For example, Jupiter, with its massive size and relatively low surface temperature of \(140 \mathrm{~K}\), has a thick atmosphere with high pressure. This affects how the gases within it interact. In contrast, Mercury's atmosphere is much thinner due to its low mass (0.05 times that of Earth). However, it experiences much higher temperatures between \(600 \mathrm{~K}\) and \(700 \mathrm{~K}\).

The differences in atmosphere are critical in determining how closely the gases on these planets follow the ideal gas law. Generally, hotter and less dense atmospheres, like Mercury's, are more likely to approximate the conditions described by the ideal gas law.

Temperature Influence on Gas Behavior

Temperature plays a crucial role in affecting how gases behave. At higher temperatures:

• Gas molecules move more rapidly.
• The increase in kinetic energy makes collisions more elastic.
• Intermolecular forces have less influence.

These factors contribute to a gas behaving more ideally because the assumptions of the ideal gas law—no intermolecular forces and perfectly elastic collisions—are more closely met.

In the case of Mercury's atmosphere, as it experiences temperatures between \(600 \mathrm{~K}\) and \(700 \mathrm{~K}\), the conditions are conducive to ideal gas behavior. Compared to Jupiter's cooler surface temperature of \(140 \mathrm{~K}\), Mercury's higher temperature means that the atmospheric gases are subject to less intermolecular attraction, allowing them to behave more in line with ideal gas predictions.

Gas Laws in Planetary Science

The Ideal Gas Law, represented by the equation \( PV = nRT \), is a cornerstone for understanding gas behavior in planetary atmospheres. Here:

• \(P\) stands for pressure.
• \(V\) is the volume.
• \(n\) is the number of moles of gas.
• \(R\) is the ideal gas constant.
• \(T\) indicates temperature.

For gases to behave ideally, certain conditions need to be met—higher temperatures and lower pressures are optimal because they minimize interactions between gas particles.

In planetary science, these laws help predict whether a planet's atmosphere behaves as an ideal gas or departs from it. Mercury, with its lower mass and higher temperatures, offers a better approximation of ideal gas conditions compared to Jupiter. Jupiter's massive size and the consequent high pressure result in deviations due to stronger intermolecular forces. Thus, planetary scientists use these gas laws to enhance our understanding of atmospheres in our solar system and beyond.