A "rule of thumb" in planetary science is that a planet can...

A "rule of thumb" in planetary science is that a planet can hold on to a gas for the age of the Solar System if the average speed of molecules in the gas is less than one-sixth the escape velocity of the planet (this takes into account the fact that molecules high in an atmosphere are influenced by a smaller pull from gravity ) . The average speed of a molecule depends on the gas temperature and the mass of the molecule according to the following equation: $V_{g a s}=157 \frac{\mathrm{m}}{\mathrm{sec}} \sqrt{\frac{\text { temperature }}{\text { molecule's mass }}}$ The molecular mass used in this formula is determined by adding up the atomic masses of its constituents, so hydrogen molecules $\left(\mathrm{H}_{2}\right)$ have mass $2,$ nitrogen molecules $\left(\mathrm{N}_{2}\right)$ have mass 28 a. What is the average speed of a hydrogen molecule in Earth's atmosphere if the gas temperature is 300 K? The Earth's escape velocity is $11,200 \mathrm{m} / \mathrm{sec}$. Can the Earth hold on to a hydrogen atmosphere? b. What is the average speed of a nitrogen molecule in Earth's atmosphere? Can the Earth maintain a nitrogen atmosphere? c. What is the average speed of a hydrogen molecule on Jupiter, where the temperature is $160 \mathrm{K}$ ? Jupiter's escape velocity is $59,500 \mathrm{m} /$ sec. Can Jupiter hold on to a hydrogen atmosphere? d. If Jupiter were placed at Mercury's distance from the Sun, where the temperature would rise to about $450 \mathrm{K}$, would it be able to hold on to its hydrogen atmosphere?

A "rule of thumb" in planetary science is that a planet can hold on to a gas for the age of the Solar System if the average speed of molecules in the gas is less than one-sixth the escape velocity of the planet (this takes into account the fact that molecules high in an atmosphere are influenced by a smaller pull from gravity ) . The average speed of a molecule depends on the gas temperature and the mass of the molecule according to the following equation: $V_{g a s}=157 \frac{\mathrm{m}}{\mathrm{sec}} \sqrt{\frac{\text { temperature }}{\text { molecule's mass }}}$ The molecular mass used in this formula is determined by adding up the atomic masses of its constituents, so hydrogen molecules $\left(\mathrm{H}_{2}\right)$ have mass $2,$ nitrogen molecules $\left(\mathrm{N}_{2}\right)$ have mass 28
a. What is the average speed of a hydrogen molecule in Earth's atmosphere if the gas temperature is 300 K? The Earth's escape velocity is $11,200 \mathrm{m} / \mathrm{sec}$. Can the Earth hold on to a hydrogen atmosphere?
b. What is the average speed of a nitrogen molecule in Earth's atmosphere? Can the Earth maintain a nitrogen atmosphere?
c. What is the average speed of a hydrogen molecule on Jupiter, where the temperature is $160 \mathrm{K}$ ? Jupiter's escape velocity is $59,500 \mathrm{m} /$ sec. Can Jupiter hold on to a hydrogen atmosphere?
d. If Jupiter were placed at Mercury's distance from the Sun, where the temperature would rise to about $450 \mathrm{K}$, would it be able to hold on to its hydrogen atmosphere?

Key Concepts

Atmospheric Retention

Atmospheric retention is the ability of a planet to keep its gaseous envelope over billions of years. This retention is determined by comparing the average molecular speeds in the atmosphere to a threshold fraction of the planet’s escape velocity. If the molecular speeds are significantly lower, the atmosphere is more likely to remain bound to the planet.

Kinetic Theory of Gases

The kinetic theory of gases explains the motion of particles in a gas and relates macroscopic properties such as temperature and pressure to the velocities of these particles. It provides the foundation for understanding how temperature and molecular mass affect the average speed of gas molecules, which is crucial for assessing their potential to escape a planet’s gravity.

Temperature Effects on Gas Behavior

Temperature is a key factor that influences the kinetic energy and hence the speed of gas molecules. Higher temperatures provide more energy to the molecules, increasing their speeds and the probability that they might exceed the threshold velocity required for escape from a planet's gravitational field.

Escape Velocity

Escape velocity is the minimum speed that an object must reach to break free from a planet's gravitational field without further propulsion. It depends on the mass and radius of the planet, serving as a critical threshold in planetary science for determining if a planet can hold onto its atmosphere, as molecules must have speeds below a certain fraction of this value to be retained over geological timescales.

Average Molecular Speed

Average molecular speed describes how fast gas molecules are moving on average and is derived from the kinetic theory of gases. This speed increases with higher temperature and decreases with greater molecular mass, influencing whether the molecules can overcome the gravitational pull of a planet if they approach the escape velocity.

Transcript

00:01 Hello, so for the first part of this question, we want to calculate the root mean square speed of the hydrogen molecules on jupiter and the earth.

00:14 We're going to compare these answers to their respective sk velocities.

00:22 So i have two equations.

00:24 The first one is the root mean square speed.

00:28 The second one is the sk velocity.

00:32 So for the first part of this question, i'm going to go.

00:35 Go ahead and calculate the root mean square speed of the hydrogen molecules on jupiter.

00:46 So the vrms on jupiter is going to be 3 times the molar.

01:00 That will be the molar gas constant, which is 8 .315.

01:10 The temperature on jupiter is 140, divided by the molar mass of the hydrogen molecules which is 2 .02 times 10 to minus 3.

01:31 This is in kilogram per more.

01:35 So this should give me 1 .31 times 10 to the 3rd, meters per second.

01:50 Now find b -s skip on jupiter.

02:00 So that's the formula right there.

02:03 That's 2 times the gravitational constant, which is 6 .67, 10 to the negative 11.

02:15 The mass of jupiter have on the right, that's 1 .9, that is 10 to the 27, divided by the radius of jupiter.

02:34 To extend this, radius of jupiter is 6 .91 times 10 to the 7.

02:42 So this will be equal to 6 .06 times 10 to the 4 minutes per second.

03:08 Alright, so if you compare the 2, so that's the vrms, that's going to be 3 .1 to the 3rd, 6 .064.

03:36 That is so in other ways we can say that vrms for the molecules is 0 .022 for the v escape then we do that for the earth vrms for the earth so you start in formula again 3 times the mullah gas constant 815 times the temperature which is 2 to divided by the molar mass 10 to 3.

05:01 That gives me 1 .65.

05:10 It is the second.

05:15 You find a v escape for the earth.

05:25 The formula is 2gm over r so that's 2.

05:31 The constants is 0 .67 minus 11 times the mass of the earth is 5 .97.

05:50 Point nine seven times 10 to the 24 divided by the radius of the earth that 6 .38 times 1036 so that gives me 1 .12 and 10 to the 4th give 2 all right so if you compare that's like we if you compare vrms for the earth by the escape velocity, the earth is going to be 1 .65 to the 3rd over 1 .14, so this 1 .12.

07:11 And that should give me 0 .15.

07:20 So you can see that the vrms, the hydrogen molecules on the earth, because it's 0 .15...

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