# Temperature, Kinetic Energy and Gases

Now here is an unexpected fact: the average kinetic energies of molecules of any gas at the same temperature are equal (since KE = 3/2kT, the identity of the gas does not matter). Let us think about how that could be true and what it implies about gases. Under most circumstances the molecules in a gas do not significantly interact with each other; all they do is collide with one another like billiard balls. So when two gases are at the same temperature, their molecules have the same average kinetic energy. However, an even more unexpected fact is that the mass of the molecules of one gas is different from the mass of the molecules of the other gas. Therefore, given that the average kinetic energies are the same, but the molecular masses are different, the average velocities of molecules in the two gases must be different. For example, let us compare molecular hydrogen (H2) gas (molecular weight = 2 g/mol) with molecular oxygen (O2) gas (molecular weight = 32 g/mol), at the same temperature. Since they are at the same temperature the average kinetic energy of H2 must be equal to the average kinetic energy of O2, then the H2 molecules must be moving, on average, faster than the O2 molecules.91

So the average speed at which an atom or molecule moves depends on its mass. Heavier particles move more slowly, on average, which makes perfect sense. Consider a plot of the behavior of the noble (monoatomic) gases, all at the same temperature. On average helium atoms move much faster than xenon atoms, which are over 30 times heavier. As a side note, gas molecules tend to move very fast. At 0 °C the average H2 molecule is moving at about 2000 m/s, which is more than a mile per second and the average O2 molecule is moving at approximately 500 m/s. This explains why smells travel relatively fast: if someone spills perfume on one side of a room, you can smell it almost instantaneously. It also explains why you can’t smell something unless it is a gas. We will return to this idea later.

1. Why don’t all gas particles move with the same speed at a given temperature?

2. Where would krypton appear on the plot above? Why?

3. Consider air, a gas composed primarily of N2, O2, and CO2. At a particular temperature, how do the average kinetic energies of these molecules compare to one another?

4. What would a plot of kinetic energy versus probability look like for the same gas at different temperatures?

5. What would a plot of kinetic energy (rather than speed) versus probability look like for different gases (e.g., the noble gases) at the same temperature?

## Questions to Ponder

1. If gas molecules are moving so fast (around 500 m/s), why do most smells travel at significantly less than that?

2. Why does it not matter much if we use speed, velocity, or kinetic energy to present the distribution of motion of particles in a system (assuming the particles are all the same)?

## References

91 We use average speed and velocity to describe the motion of the particles in a gas, but it is more accurate to use the root mean square (rms) of the velocity, that is, the square root of the average velocity. However, for our purposes average speed (or velocity) is good enough.