Here we would like to figure out why we do not seem to have much molecular hydrogen or helium in Earth’s atmosphere. Although this can get very complicated quickly IRL, we can simplify things by asking a couple of questions, one at a time. We will examine just helium as a monatomic ideal gas and extend our reasoning to molecular hydrogen. The molar mass of helium is about 4 x 10^-3 kg/mol. First, we would like to see if Earth’s gravity has what it takes to hold helium. Calculate the mean escape velocity at the surface of the Earth. (Answer: 11.2 X 10^3 m/s) Then, we would like to see if Earth is too hot to hold helium. Calculate the rms speed of helium at a temperature of 300 K. Is helium expected to escape at this speed? (Answer: 1.37 X 10^3 m/s) Perhaps if we wanted to be really sure we did not miss anything, we can examine another scenario. The Earth’s upper atmosphere (thermosphere and up) is much hotter (but it may FEEL cold because it is so rarified). Let us repeat our rms speed calculation for helium at 2000 K instead, which can be typical near the “edge of outer space.” Can the helium escape Earth’s gravity then? (Answer: 3.53 X 10^3 m/s (Even though the rms speed is lesser than the escape velocity, some of the helium atoms may have a velocity greater than the escape velocity and hence would escape Earth’s gravity) New Questions: So far, we really only used contact forces of collisions and the force of gravity. Are there any other forces (not necessarily fundamental ones) that can get helium into the upper atmosphere for us? What fundamental quantity/quantities do we need to calculate its/their value? This is not a calculation question, this is a conceptual one. The rms speed is an energy-scaled average of an ensemble of particles, but particles can move slower and even much faster than the rms average. If we need more speed, is there hope that helium can escape Earth’s gravity still? (Special note: the scenario you have been guided through is actually not the main mechanism for terrestrial helium loss, but it is one less significant one.)
Here we would like to figure out why we do not seem to have much molecular hydrogen or helium in Earth’s atmosphere. Although this can get very complicated quickly IRL, we can simplify things by asking a couple of questions, one at a time. We will examine just helium as a monatomic ideal gas and extend our reasoning to molecular hydrogen. The molar mass of helium is about 4 x 10^-3 kg/mol.
First, we would like to see if Earth’s gravity has what it takes to hold helium. Calculate the mean escape velocity at the surface of the Earth. (Answer: 11.2 X 10^3 m/s)
Then, we would like to see if Earth is too hot to hold helium. Calculate the rms speed of helium at a temperature of 300 K. Is helium expected to escape at this speed? (Answer: 1.37 X 10^3 m/s)
Perhaps if we wanted to be really sure we did not miss anything, we can examine another scenario. The Earth’s upper atmosphere (thermosphere and up) is much hotter (but it may FEEL cold because it is so rarified). Let us repeat our rms speed calculation for helium at 2000 K instead, which can be typical near the “edge of outer space.” Can the helium escape Earth’s gravity then? (Answer: 3.53 X 10^3 m/s
(Even though the rms speed is lesser than the escape velocity, some of the helium atoms may have a velocity greater than the escape velocity and hence would escape Earth’s gravity)
New Questions:
So far, we really only used contact forces of collisions and the force of gravity. Are there any other forces (not necessarily fundamental ones) that can get helium into the upper atmosphere for us? What fundamental quantity/quantities do we need to calculate its/their value? This is not a calculation question, this is a conceptual one.
The rms speed is an energy-scaled average of an ensemble of particles, but particles can move slower and even much faster than the rms average. If we need more speed, is there hope that helium can escape Earth’s gravity still? (Special note: the scenario you have been guided through is actually not the main mechanism for terrestrial helium loss, but it is one less significant one.)
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