Any consideration of the properties of a collection of particles such as molecules requires knowledge of their energy. Part of this energy is kinetic energy, the energy of motion. The kinetic energy (KE) of an object is determined by the equation

KE

=

1 2

mv2

where m = mass, v = velocity

This equation states that the kinetic energy of an object is dependent on both its mass and its velocity. A semitrailer truck and a subcompact car traveling at the same velocity have different kinetic energies. You would be aware of this difference if they crashed, for the truck would demolish the subcompact. For the two vehicles to have the same kinetic energy, the subcompact would have to be traveling at a much higher velocity than the truck.

A. The Distribution of Kinetic Energy
In a collection of molecules, each molecule has a kinetic energy that can be calculated by the equation given above. Even if the molecules have a constant mass, they differ in velocity, so that a collection of molecules will have a wide range of kinetic energies, from very low to very high. Each molecule may change its kinetic energy often, but the overall distribution will remain the same.

Figure 9.3 shows a typical distribution of kinetic energies in a collection of molecules. In the graph, kinetic energy is plotted along the horizontal axis, and the percent of molecules having a particular kinetic energy is shown by the height of the curve at that point. Several observations can be made by studying the graph:

FIGURE 9.3 Distribution of kinetic energy in a collection of molecules.

1. The area under the curve represents the total number of molecules in the sample. Between any two points on the horizontal axis, the area under the curve represents the number of molecules that have kinetic energies in that range. For example, the shaded area between A and B represents the number of molecules that have kinetic energies between A and B.

   

2. The peak of the curve shows the most probable kinetic energy. More molecules have this energy than any other.

3. The average kinetic energy is slightly greater than the most probable kinetic energy.

4. Some molecules have a kinetic energy much higher than the average value.

5. Some molecules have a kinetic energy much lower than the average value.

Notice that the distribution of energies is much like the distribution of grades on a test. Figure 9.4 shows the distribution of grades on a standardized test. There is a most probable score. Most of the grades fall close to the most probable score; some grades are higher and others are lower.

FIGURE 9.4 Graph of test grades. Each bar represents the number of students who received a particular grade.

B. Kinetic Energy and Temperature
The average kinetic energy of a collection of molecules is directly proportional to its temperature. At absolute zero (-273°C), the molecules have a minimum kinetic energy. As the temperature of the sample increases, so does its average kinetic energy. As the temperature rises, the distribution of kinetic energies among the molecules in the sample also changes. Figure 9.5 shows the distribution of kinetic energies in a sample at two different temperatures. Curve A is at the lower temperature; curve B is at the higher temperature. Notice the following differences between the two curves:

1. The peak of curve B is lower and broader than the peak of curve A. This difference in the curves means that, at the higher temperature, fewer molecules have the average kinetic energy and the distribution of energies is more spread out.

2. The peak of curve B is at a higher kinetic energy than the peak of curve A. This difference means that, at the higher temperature, the average kinetic energy of the molecules is higher.

We can conclude that, as the temperature of a sample increases, not only does the average kinetic energy increase but also fewer molecules have the average energy and the distribution of energies among the molecules is more uniform.

FIGURE 9.5 Distribution of kinetic energy in the same collection of molecules at two different temperatures.