91Ó°ÊÓ

(a) Show that a projectile with mass \(m\) can "escape" from the surface of a planet if it is launched vertically upward with a kinetic energy greater than \(m g R_{p},\) where \(g\) is the acceleration due to gravity at the planet's surface and \(R_{\mathrm{p}}\) is the planet's radius. Ignore air resistance. (See Problem \(18.70 .\) ) (b) If the planet in question is the earth, at what temperature does the average translational kinetic energy of a nitrogen molecule (molar mass \(28.0 \mathrm{~g} / \mathrm{mol}\) ) equal that required to escape? What about a hydrogen molecule (molar mass \(2.02 \mathrm{~g} / \mathrm{mol}\) )? (c) Repeat part (b) for the moon, for which \(g=1.63 \mathrm{~m} / \mathrm{s}^{2}\) and \(R_{\mathrm{p}}=1740 \mathrm{~km}\) (d) While the earth and the moon have similar average surface temperatures, the moon has essentially no atmosphere. Use your results from parts (b) and (c) to explain why.

Step by step solution

01

Understanding escape velocity

We know that in order for an object to escape the gravitational field of a planet, the kinetic energy must be equal or greater than the gravitational potential energy. The condition for escape then becomes \( K.E. >= G.P.E \n). This equates to \( {1}/{2} m \cdot v_{escape}^2 >= m \cdot g \cdot R_p \) where \(v_{escape}\) is the escape velocity.

02

Simplification and relationship with the temperature of the gas

The equation derived in step 1 can be written as \( v_{escape} >= \sqrt{2gR_p} \) \n . Now, considering the average translational kinetic energy of a gas molecule, which is \( K.E.avg = {3}/{2} k \cdot T \), the escape velocity of a gas molecule is then linked with the temperature as \( \sqrt{(3kT)/m} = \sqrt{2gR_p} \), where \( k \) is the Boltzmann constant and \( m \) is the mass of a gas molecule. Solving for temperature we get \(T_{escape} = \frac{2 g m R_{p}}{3k}\)

03

Calculate the temperature for nitrogen and hydrogen molecules on Earth

We can insert the values for Earth (g = 9.8 m/sec^2, R = 6.4 x10^6 m), k (Boltzmann constant, 1.38 x 10^-23 J/K) and m (mass of a molecule), into the formula found in step 2. For nitrogen, m = \(28.0 \times 1.67 \times 10^{-27} kg \), and we will get the \(T_{Nitrogen escape} \). Similarly, for hydrogen, m = \(2.02 \times 1.67 \times 10^{-27} kg \), and we will get the \(T_{Hydrogen escape} \).

04

Repeat the calculation for the Moon

Step 3 is repeated using the values for the moon (g=1.63m/sec^2, Rp=1740km), to get the temperatures for nitrogen and hydrogen molecules escape on the Moon.

05

Clarify the discrepancy between Earth and Moon

Comparing the results of steps 3 and 4, we see that the temperature required for atmospheric molecules to escape from the Moon's gravity is much lower than that from the Earth. Therefore, at a temperature similar to Earth's, the Moon's gravity is unable to retain the gas molecules, resulting in the lack of atmosphere on the Moon.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Potential Energy

Gravitational Potential Energy (GPE) is a concept used to describe the energy that an object possesses because of its position in a gravitational field. On a planet's surface, the GPE is largely determined by the planet's mass and radius. The formula for gravitational potential energy is expressed as \( GPE = m \, g \, R_p \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity at the planet's surface, and \( R_p \) is the radius of the planet.

In the context of escape velocity, GPE signifies the minimum energy needed for an object to overcome the gravitational pull of the planet and safely reach into space without returning. To achieve this, the object must have sufficient kinetic energy to balance or exceed the GPE. This concept is crucial in understanding why objects launched with insufficient energy cannot break free from planetary bounds.

Kinetic Energy

Kinetic Energy (KE) represents the energy that an object holds due to its motion. The basic formula for kinetic energy is \( KE = \frac{1}{2} m \cdot v^2 \), where \( m \) is mass and \( v \) is velocity. When discussing escape velocity, kinetic energy must reach or exceed gravitational potential energy for a mass to escape a planet's gravitational field.

The relationship between kinetic and gravitational potential energy in reaching escape velocity is expressed through \( KE \geq GPE \). When an object achieves exactly this balance, it reaches what's known as the escape velocity. For any moving object, especially those aiming to leave Earth or another celestial body's influence, understanding kinetic energy and how it converts or compares to potential energy is key.

Translational Kinetic Energy

Translational Kinetic Energy refers to the kinetic energy due to the linear motion of an object or molecule. This means it's one aspect of the energy associated with the motion along a path. For gases, this is particularly important as it contributes to the total kinetic energy needed for molecules to escape from a gravitational field, like Earth's or the Moon's.

For molecules in a gas, such as nitrogen or hydrogen, their average translational kinetic energy can be given by \( KE_{avg} = \frac{3}{2} k \cdot T \), where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature. This relationship shows how the temperature of a gas is linked to its molecules' ability to overcome gravitational forces. Therefore, the higher the temperature, the greater the velocity and thus the more likely molecules can escape a planet's pull.

Boltzmann Constant

The Boltzmann Constant (\( k \)) is a fundamental constant that links the average kinetic energy of particles in a gas with the temperature of the gas. Its value is approximately \( 1.38 \times 10^{-23} \) J/K. This constant provides a bridge between macroscopic and microscopic physics, connecting the energy at an atomic or molecular level to temperature.

In the context of escape velocity and atmospheric retention, the Boltzmann constant is crucial for calculating how much energy, on average, is carried by the molecules in a gas. It helps us determine the temperature needed for these molecules to have sufficient translational kinetic energy to overcome gravitational forces and escape into space. Thus, differences in the gravitational force of celestial bodies, like Earth and the Moon, heavily influence whether a substantial atmosphere can be retained or if molecules will escape due to thermal motion.