19.13
Escape Velocities of Gases
To escape the Earth's gravity, an object near the top of the atmosphere at an altitude of 100 km must travel away from Earth at 11.1 km/s. This speed…
The escape velocity is the minimum velocity required for a non-propelled object to escape from the influence of the gravitational force of the primary object. The magnitude of the escape velocity of the Earth is 11.1 km/s.
What is the temperature at which hydrogen molecules attain the rms speed equal to the escape velocity of the Earth?
According to the kinetic theory of gases, the average kinetic energy of any gas is related to its temperature. The mass of hydrogen molecules can be calculated by taking the ratio of its molar mass to Avogadro's number.
Now, rearranging the equation for kinetic energy and substituting the known quantities, the temperature required to attain the escape velocity is calculated.
The estimated temperature is a few orders of magnitude higher than the temperature at high altitudes of Earth's atmosphere. This means the hydrogen molecules cannot escape from the Earth's atmosphere.
However, during multiple collisions, significant hydrogen molecules attain the escape velocity allowing the loss of hydrogen from Earth's atmosphere.
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Q1: What is escape velocity and why does it matter for atmospheric gases?
Escape velocity is the minimum speed an object must travel to break free from a planet's gravitational pull. For Earth, this speed is 11.1 km/s at an altitude of 100 km. Understanding escape velocity helps explain why lighter gases like hydrogen gradually leave Earth's atmosphere while heavier gases remain.
Q2: At what temperature do hydrogen molecules reach Earth's escape velocity?
According to kinetic theory, hydrogen molecules attain an rms speed equal to Earth's escape velocity at approximately 99,700 K. This temperature is several orders of magnitude higher than the 250 K found at 100 km altitude. Since actual atmospheric temperatures are far lower, hydrogen molecules cannot escape through thermal motion alone.
Q3: How does the distribution of molecular speeds explain hydrogen loss from Earth?
Although average hydrogen molecules lack escape velocity, the distribution of molecular speeds means some molecules move faster than others at any instant. During collisions, individual hydrogen molecules occasionally reach speeds exceeding escape velocity, allowing them to escape. Over billions of years, this process has removed nearly all hydrogen from Earth's atmosphere.
Q4: Why do heavier gases like nitrogen and oxygen remain in Earth's atmosphere?
Heavier molecules have lower rms speeds than hydrogen at the same temperature. The probability that oxygen or nitrogen molecules reach escape velocity is extremely small, requiring billions of years to lose significant amounts. Their greater mass makes atmospheric escape far less likely than for lightweight hydrogen.
Q5: How is the relationship between temperature and molecular kinetic energy used to calculate escape velocity?
The kinetic theory of an ideal gas relates average kinetic energy directly to temperature. By equating the rms kinetic energy formula to the escape velocity energy and solving for temperature, scientists determine the thermal conditions required for gas molecules to escape. This calculation uses the gas molecule's mass and Avogadro's number.
Q6: Why does the Moon lack an atmosphere while Earth retains one?
The Moon's gravitational pull is much weaker than Earth's, resulting in a lower escape velocity. Consequently, gas molecules on the Moon reach escape velocity more easily, and the Moon has lost nearly its entire atmosphere over time. The same escape velocity principles apply to both bodies, but the Moon's weaker gravity accelerates atmospheric loss.
Q7: What role do molecular collisions play in determining whether gases escape Earth's atmosphere?
Molecular collisions continuously redistribute kinetic energy among gas molecules, causing individual speeds to fluctuate. At each collision, a molecule's speed changes, creating a finite probability that it temporarily exceeds escape velocity. Over geological timescales, these rare high-speed events allow lighter gases to gradually escape despite average temperatures being far too low.