Laboratory apparatus comparing the temperature of a gas and the RMS speed (root-mean-square speed) of helium and xenon, with a bright central flash showing that their average energetic impact is identical despite different speeds.

Is the Temperature of a Gas Indicative of Its RMS Speed?

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Introduction

Real gases (e.g., air) behave like ideal gases under normal conditions. Consider two ideal gases: one made of very light atoms and another made of very heavy molecules. If the two gases are at the same temperature, they have the same average translational kinetic energy (Kavg). You may ask naturally then, “Do different gases at the same temperature have the same average speed?” As we shall see, the spot-on question is, “Is the temperature of a gas indicative of its RMS speed?”

The Relationship Between Kavg and Average Speed

A gas consists of a chaotic assembly of particles (e.g., atoms, molecules) colliding with one another. Therefore, individual particles can move at vastly different speeds. What is the relationship between Kavg and average speed? The equation is

K avg = 1 2 m v rms 2

where

Kavg = average translational kinetic energy of one particle1

m = mass of a single molecule (in kilograms)

vrms = root-mean-square (RMS) speed of the particles (in meters per second)

What does RMS speed mean? Imagine a large, packed bumper-car arena in which thousands of cars are crashing constantly into each other. At any given second,

  • Some cars are temporarily pinned against the wall, sitting at a dead stop (0 mph).
  • Most cars are cruising around at a medium pace (e.g., 10 mph).
  • A few cars just got rear-ended and are rocketing forward at higher speeds (e.g., 40 mph).

Suppose that we want to calculate the total kinetic energy of the arena in that instant. Adding the speeds of all the cars and then dividing that sum by the number of cars yields the standard average speed. However, using that speed is not the way to go. Why? Because kinetic energy depends on the square of speed. A single car going 40 mph has 16 times the energy of a car going 10 mph. The fast cars contribute considerably more to the total kinetic energy than the slow cars. To capture this dependence of kinetic energy on the square of speed, we use the RMS speed instead. RMS speed, not the ordinary average speed, reflects correctly the total kinetic energy of the arena.

How Is RMS Speed Calculated?

To calculate RMS speed, we work backward through its name (square-mean-root) in three steps:

  1. Square. We square the individual speeds of all the particles. This action aligns the values with how translational kinetic energy behaves.
  2. Mean. We find the mathematical average (mean) of all those squared values.
  3. Root. We take the square root of that average to convert the number back into a familiar speed unit (meters per second).

These three steps produce the RMS speed that appears in the kinetic energy equation.

What Does RMS Speed Mean Pictorially?

We just saw RMS speed mathematically. Now let us see it pictorially. Imagine that the gas particles are moving at many different speeds. Their total kinetic energy has a certain value. Now suppose that we stop the collisions, make every particle travel at the same speed, but do not change the total internal energy of the system (U) or the temperature of the system.2 That uniform speed is the RMS speed. It is the one constant speed that, if every particle moved at it, would leave the system’s total kinetic energy unchanged.

The Temperature of a Gas Measures Its Average Punch, Not Its Average Speed

Now that we understand the relationship between Kavg and vrms, let us consider next a sample of a light gas, say helium, and a sample of a heavy gas, say xenon. Each is in a sealed container having a thermometer, and they both have the same Kavg. Consequently, the helium atoms must move very fast to make up for their tiny mass, and the xenon atoms must move much slower to make up for their large mass. From the kinetic theory of gases,

v rms = 3 k B T m

where kB is the Boltzmann constant (≈ 1.38 x 10-23 J/K), T is the absolute temperature (in kelvin), and m is the mass of one gas particle (in kilograms).

For example, at room temperature (about 300 K), the RMS speed of helium is 1,367 m/s, but the RMS speed of xenon is 239 m/s. Thus, when the helium atoms strike a surface (e.g., the surface of the thermometer), their average energetic impact—their punch—is identical to that of the xenon atoms. The temperature of a gas, therefore, measures the average punch of its particles, not the RMS speed of its particles.

We can now answer our two earlier questions: Do different gases at the same temperature have the same average speed? No, they do not. Is the temperature of a gas indicative of its RMS speed? No, it is not.

Conclusion

In summary, the temperature of a gas is not indicative of the RMS speed of its particles but of the average translational kinetic energy of its particles. That kinetic energy depends on both the mass of the particles and the square of the RMS speed of the particles.

  1. At thermal equilibrium, Kavg = average translational kinetic energy of the entire assembly (Ktotal/N, where Ktotal is the translational kinetic energy of the entire assembly and N is the total number of particles). ↩︎
  2. For an ideal monoatomic gas, U equals Ktotal.  Ktotal = NKavg, where N is the total number of particles. ↩︎

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