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

Mercury's atmosphere is more likely to obey the ideal gas law because it has a higher surface temperature (\(600 \mathrm{~K}\) to \(700 \mathrm{~K}\)) and a lower mass (0.05 times that of Earth), leading to lower atmospheric pressure due to its lower gravity. These conditions are favorable for the ideal gas law to hold when compared to Jupiter, which has a lower surface temperature (\(140 \mathrm{~K}\)) and higher mass (318 times that of Earth).

Step-by-step solution

Understand the relation between temperature, pressure, and 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 of gas, R is the ideal gas constant, and T is the temperature. The ideal gas law works better when the pressure is low and the temperature is high. This means that for a planet's atmosphere to follow this law, it should have low pressure and high temperature.

Convert and compare the surface temperature

First, let's compare the surface temperatures of Jupiter and Mercury. The surface temperature of Jupiter is given as \(140 \mathrm{~K}\), while the surface temperature of Mercury ranges between \(600 \mathrm{~K}\) and \(700 \mathrm{~K}\). Since the surface temperature is higher on Mercury, it has a better chance of its atmosphere following the ideal gas law based on this factor alone.

Compare the planet masses

Next, let's compare the planets' masses. Jupiter's mass is given as 318 times that of Earth, while Mercury's mass is only 0.05 times that of Earth. Planets with higher mass generally have higher gravity, which would lead to a more compact atmosphere and higher atmospheric pressure. In this case, Mercury has a lower mass and therefore lower gravity, which would lead to a more dispersed atmosphere and a lower atmospheric pressure on Mercury than on Jupiter. This lower pressure also supports the idea that Mercury's atmosphere is more likely to follow the ideal gas law.

Conclusion based on the comparison of temperature and mass

Since Mercury has a higher surface temperature and lower atmospheric pressure due to its lower mass and gravity, its atmosphere is more likely to obey the ideal gas law compared to Jupiter's atmosphere. Hence, the atmosphere on Mercury is more likely to follow the ideal gas law.

Key concepts

Planetary Atmospheres

Planetary atmospheres are the layers of gases that surround celestial bodies like planets. Each planet's atmosphere varies in composition, thickness, and behavior based on several factors, such as distance from the sun and composition of surface materials.
Understanding these atmospheres involves considering how gases interact under different temperature and pressure conditions.
One key theory used to study these atmospheres is the Ideal Gas Law, which provides insights based on pressure, volume, and temperature of gases. This law helps scientists predict how these gases will behave when conditions change, such as increases in temperature or decreases in pressure.
Atmospheres that are diffuse and at relatively high temperatures might align more closely with the Ideal Gas Law as opposed to those that are dense and cool.

• Planets with significant atmospheres include Jupiter and Earth.
• Thin atmospheres are evident in smaller planets or moons, like Mercury.

Surface Temperature

Surface temperature is a crucial factor in planetary atmospheres and how they might follow physical laws like the Ideal Gas Law. The temperature of a planet’s surface determines how energetic the gas particles are.
Higher surface temperatures mean that the gas particles move faster, potentially behaving more like an ideal gas because ideal gas behavior becomes more accurate at higher energies.
For instance, Mercury, with a surface temperature between \(600 \mathrm{~K}\) and \(700 \mathrm{~K}\), has hotter, possibly more active gas particles compared to a colder planet like Jupiter, which has a surface temperature of only \(140 \mathrm{~K}\).
In general, planets closer to a star, like the sun, tend to have higher surface temperatures, encouraging the atmosphere to display behaviors predicted by the Ideal Gas Law.

• Higher temperatures increase the chance of ideal gas behavior.
• Lower temperatures may result in non-ideal behaviors due to reduced molecular activity.

Planetary Mass

The mass of a planet significantly influences its gravitational pull, which in turn affects its atmosphere's behavior and organization. Heavier planets like Jupiter, which is 318 times more massive than Earth, exert a stronger gravitational force.
This force pulls atmospheric gases into a denser atmosphere, likely increasing atmospheric pressure. Higher pressure conditions do not favor the Ideal Gas Law, which assumes low pressure for ideal behavior.
On the other hand, lighter planets like Mercury have weaker gravity, leading to a less compact, spread-out atmosphere and lower atmospheric pressure. Such conditions are more conducive to the atmosphere following the Ideal Gas Law principles because the interactions between the gas molecules more closely resemble those of an ideal gas.

• Higher planetary mass leads to higher atmospheric pressures.
• Lower planetary mass allows for sparser atmospheres with lower pressures.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the atmosphere as gravity pulls it towards the planet’s surface. It is a crucial factor in determining whether a planet's atmosphere behaves ideally.
In planets with higher gravity (and thus higher mass), like Jupiter, atmospheric pressure is higher. This increased pressure often leads to less ideal gas behaviors due to more frequent interactions between gas molecules, deviating from the assumptions of the Ideal Gas Law.
Conversely, planets like Mercury, with lesser mass and consequently lower gravity, have lower atmospheric pressure. Lower pressure conditions minimize intermolecular forces, allowing gases to act more like an ideal gas.
This explains why Mercury, despite its smaller size, has an atmosphere more likely to obey the Ideal Gas Law when compared to the massive Jupiter.

• Lower atmospheric pressure is associated with higher adherence to the Ideal Gas Law.
• Higher atmospheric pressure results in non-ideal behavior due to stronger molecular interactions.