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 on Mercury is more likely to obey the ideal-gas law due to its higher temperature (\(600 \mathrm{~K}\) to \(700 \mathrm{~K}\)) and lower pressure as compared to Jupiter with a temperature of \(140 \mathrm{~K}\). Mercury's higher average kinetic energy and lower mass relative to Earth also contribute to this conclusion.

Step-by-step solution

Find the Kinetic Energy of the gas particles on each planet

To compare the behavior of gases on both planets, we need the average kinetic energy of gas particles, which is given by the equation: \(K_{avg} = \dfrac32 k_B T\) Here, \(K_{avg}\) is the average kinetic energy, \(k_B\) is Boltzmann's constant (\(1.38 × 10^{-23} \mathrm{~J/K}\)), and \(T\) is the temperature in Kelvin.

Calculate the kinetic energies for Jupiter and Mercury

For Jupiter, we are given \(T = 140 \mathrm{~K}\). Therefore, we can calculate \(K_{avg}\) for Jupiter as follows: \(K_{avg_{Jupiter}} = \dfrac32 (1.38 × 10^{-23} \mathrm{~J/K}) (140 \mathrm{~K})\) For Mercury, we are given a temperature range from \( 600 \mathrm{~K}\) to \( 700 \mathrm{~K}\). We can calculate the average kinetic energy at the lower and higher ends of the temperature range: \(K_{avg_{Mercury_{min}}} = \dfrac32 (1.38 × 10^{-23} \mathrm{~J/K}) (600 \mathrm{~K})\) \(K_{avg_{Mercury_{max}}} = \dfrac32 (1.38 × 10^{-23} \mathrm{~J/K}) (700 \mathrm{~K})\)

Compare the kinetic energies and temperatures

For the ideal gas behavior, we would expect high temperature and low pressure. Given that Mercury has a higher surface temperature and lower mass relative to Earth, the pressure on Mercury would be lower than that on Jupiter, which has a higher mass relative to Earth. Also, comparing the average kinetic energies calculated in Step 2, we can observe that Mercury has a higher average kinetic energy than Jupiter, as Mercury's average kinetic energy falls in between \(K_{avg_{Mercury_{min}}}\) and \(K_{avg_{Mercury_{max}}}\).

Conclude the results

Based on the comparison of the average kinetic energies and the temperatures of both planets, we can conclude that the atmosphere on Mercury is more likely to obey the ideal-gas law due to its higher temperature and lower pressure as compared to Jupiter.

Key concepts

Kinetic Energy

Understanding kinetic energy is crucial when discussing the behavior of gas particles. Kinetic energy is the energy objects possess due to their motion. For gas particles, it determines how they move and interact within a planetary atmosphere. The average kinetic energy of gas particles can be computed using the equation:\[ K_{avg} = \dfrac{3}{2} k_B T \]where:

• \(K_{avg}\) is the average kinetic energy,
• \(k_B\) is Boltzmann's constant, and
• \(T\) is the temperature in Kelvin.

A higher temperature results in greater kinetic energy since the particles will move more energetically. Conversely, a lower temperature means a slower movement.

In comparing Jupiter and Mercury, Mercury shows higher kinetic energy due to its greater surface temperature. This higher energy allows gas particles to move more freely, promoting conditions consistent with the ideal gas law.

Boltzmann's Constant

Boltzmann's constant \(k_B\) is a fundamental constant in thermodynamics that bridges the macroscopic and microscopic worlds. It's used to relate the temperature of a gas to the average kinetic energy of its particles.

The constant has a value of \(1.38 \times 10^{-23} \text{ J/K}\), and it plays a pivotal role in the expression for kinetic energy:\[ K_{avg} = \dfrac{3}{2} k_B T \]This element explains how individual molecule behavior connects to observable physical properties, like temperature.

With a planetary perspective, Boltzmann's constant helps us understand how planets like Mercury and Jupiter compare in terms of their atmospheric conditions. Applying this constant allows us to determine potential gas behavior under given temperatures, aiding in predicting where the ideal gas law might apply more readily.

Planetary Atmospheres

In terms of atmospheric conditions, planets can exhibit a variety of behaviors based on their temperature, mass, and other properties. The ideal gas law suggests that gas behavior becomes more predictable under high temperatures and low pressures.

For Mercury and Jupiter:

• Mercury has a high surface temperature and low mass relative to Earth.
• Jupiter, in contrast, has a lower surface temperature and much greater mass.

These conditions influence how closely their atmospheres behave like ideal gases.

Mercury, with its higher temperatures, provides an environment where gas particles are more energized. Despite its low mass, these characteristics lead to reduced atmospheric pressure, reinforcing compliance with the ideal gas law.

Jupiter's higher mass leads to increased pressure, which can hinder this ideal behavior, even with different temperature conditions than Mercury. Thus, understanding these dynamics can determine which planet is more likely to present atmospheric behaviors aligning with the ideal gas law.