5.2: Thinking About Populations of Molecules

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Within a population of atoms and molecules, the many collisions that occur per second lead to a range of speeds and directions (that is, velocities) of the atoms/molecules. When large numbers of particles are involved in a phenomenon, their individual actions are not important, for example when measuring temperature or pressure (although they are when individual molecules collide, that is, take part in chemical reactions). We treat large numbers of molecules as a population. A population is characterized by the distribution of the number or probability of molecules moving with various velocities.⁸⁹ This makes it possible to use statistical methods to characterize the behavior of the population. Although any particular molecule behaves differently from one moment to the next, depending upon whether it collides with other molecules or not, the behavior of the population is quite predictable.⁹⁰

From this population perspective, it is the distribution of kinetic energies of atoms or molecules that depends upon the temperature of the system. We will not concern ourselves with deriving the equations that describe these relationships, but rather focus on a general description of the behavior of the motions of atoms and molecules in various states of matter.

Let us think about a population of molecules at a particular temperature in the gas phase. Because of their constant collisions with one another, the population of molecules has a distribution of speeds. We can calculate the probability of a particular molecule moving at a particular speed. This relationship is known as the Maxwell–Boltzmann distribution, shown in the graph. Its shape is a function of the temperature of the system; typically it rises fairly steeply from zero (all of the curves begin at zero – why is that do you think?) to a maximum, which then decreases and tails off at higher velocities (which correspond to higher kinetic energies). Because we are plotting probability versus kinetic energy (or rms velocity or speed) we can set the area under the curve to be equal to one (or any other constant). As the temperature changes, the area under the curve stays constant. Why? Because we are completely certain that each particle has some defined amount of kinetic energy (or velocity or speed), even if it is zero and even if we could not possibly know it (remember the uncertainty principle). As the temperature is increased, the relative number of particles that are moving at higher speeds and with more kinetic energy increases. The shape of the curve flattens out and becomes broader. There are still molecules moving very slowly, but there are relatively fewer of them. The most probable speed (the peak of the curve) and the average speed (which is a little higher since the curve is not symmetrical) increase as the temperature increases.

Questions to Answer

1. What happens to the average speed of molecules as temperature increases?

2. When molecules collide, why don’t they stick together?

3. What do you think happens to the average speed as molecular weight increases (assuming the temperature stays the same)?

4. Imagine a system composed of two different types of molecules, one much heavier than the other. At a particular temperature, how do their average kinetic energies compare? Which, on average, is moving faster?

Questions to Ponder

1. How large does a system have to be to have a temperature, 10 molecules or 10,000,000?

2. If one considers the uncertainty principle, what is the slowest velocity at which a molecule can move?

3. If you place a thermometer into a solution, why does it take time for the reading on the thermometer to correspond to the temperature of the solution?

References

⁸⁹ Although this distribution of speeds of atoms was first derived mathematically, it is possible to observe experimentally that atoms in a gas are moving at different speeds.

⁹⁰ Interestingly, this is like our approach to the decay of unstable atoms. We cannot predict when a particular atom will decay, but in a large enough population, we can very accurately predict the number of atoms per second that will decay.