All gas particles have kinetic energy due to their motions as they bounce around. Temperature is a measure of this microscopic kinetic energy. But how exactly is the temperature scale related to the velocities of the particles? The answer was worked out around 1860 by physicists James Clerk Maxwell from Scotland and Ludwig Boltzmann from Austria. Their simple law gives us insights that will be useful throughout our discussion of planets.
James Clerk Maxwell. Click here for original source URL
The kinetic energy of any moving particle is ½mv2, where m is the particle mass and v is its velocity. Maxwell and Boltzmann deduced that the mean kinetic energy is proportional to T. This statement is usually written as:
1/2 mv2 = 3/2 kT
In this equation, k is a fundamental constant called the Boltzmann constant, which has the tiny value of 1.38 × 10-23 Joules per Kelvin. We can see the main features of this equation easily. The temperature is proportional to the square of the average velocity and it is proportional to the mass of the particle.
Ludwig Boltzmann. Click here for original source URL
As an example, consider the air that surrounds you. What is the typical velocity of a molecule? Let's assume we are dealing with nitrogen, since most air is composed of nitrogen. A single nitrogen molecule has an atomic weight of 28 and so a mass 28 times that of a hydrogen atom or 4.68 × 10-26 kilograms. Assume the air is 20° C or 293 K. We can rearrange the equation above to solve for velocity
v = √(3 kT / m)
Now we plug in the numbers. If we use the correct units the answer will come out in meters per second. The result is that v = √(3 × 1.38 × 10-23 x 293) / 4.68 × 10-26 = 509 meters per second! It sounds like an amazingly high speed. But of course the kinetic energy of each molecule is only 3/2 kT = 3/2 × 1.38 × 10-23 × 293 = 6.1 × 10-21 Joules, so each of the hits on our skin is very puny. Nevertheless, the cumulative effect of all these tiny collisions is the pressure of ordinary air.
The velocity that is calculated by equating temperature with kinetic energy is a typical or close to an average velocity for the particles in a gas. However, in practice there is a broad distribution of velocities. Some gas particles travel much faster than the typical velocity and some travel much slower. There is a bell curve distribution centered on the average velocity.
We can use the equation above to show how the velocity of a gas particle depends on the type of particle and the temperature. Particle velocity is proportional to the inverse square root of the mass. Gas of low atomic or molecular weight moves faster than gas of high atomic or molecular weight. How fast would hydrogen move at room temperature? In this example, the temperature is the same and the masschanges. Since a nitrogen molecule is 28 times more massive than a hydrogen atom, the hydrogen atom would move √28 = 5.3 times faster, or a speed of 509 × 5.3 = 2700 meters per second.
What about nitrogen at a lower temperature? Particle velocity is proportional to the square root of temperature. Cold gas particles move slower than hot gas particles. In this second example, the gas is the same and the temperature changes. The coldest temperature ever recorded on Earth is around -90° C or 183 K. The nitrogen molecules in that miserable place move √(183 / 293) = 0.79 times slower than the nitrogen molecules in the air you are breathing. Their velocity would be 0.79 × 509 = 402 meters per second.
There is an interesting consequence of the spread of particle velocities in a gas. As a rule of thumb, there are significant numbers of particles up to about six times the typical velocity. The escape velocity of the Earth is 11.2 kilometers per second. This applies to any moving object — from a rocket to a single atom. The fastest nitrogen molecules will travel 509 × 6 = 3050 meters per second or about 3.1 kilometers per second. This is well under the escape velocity.
However, hydrogen in the Earth's atmosphere will move as fast as 2700 × 6 = 16,200 meters per second or 16.2 kilometers per second. This is well above the Earth's escape velocity. So the fastest hydrogen atoms, those in the tail of the distribution, are energetic enough to overcome the grip of gravity. Hydrogen will therefore seep into space. Earth can retain heavy gases but will lose light gases. This same idea applies to other planetary atmospheres.