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The exact relationship derived mathematically shows that the temperature T is actually proportional to the average kinetic energy of the gas particles, since the particles don’t all have the same speed. The complete result gives this relationship as
KE = \frac{3}{2} RT
where KE is the average kinetic energy of the gas particles and R is the same constant, which appears in the Ideal Gas Law.
This is a wonderful result for many reasons. The relationship of temperature to molecular kinetic energy is used constantly by Chemists in interpreting experimental observations. Even though we have worked it out for a gas, it turns out that this proportionality is also valid in liquids and solids.
We can use the results of the previous section to understand many of the observations we have made about gases. For example, Boyle’s Law tells us that the pressure of a gas is inversely proportional to the volume of the gas, if we have a fixed number of molecules and a fixed temperature. Our results above show us that decreasing the volume for a fixed number of molecules increases the frequency with which the particles hit the walls of the container. This produces a greater force and thus a higher pressure.
We also know that the pressure of a gas increases with the number of particles, if the volume and temperature are fixed. Our results show us again that, for a fixed volume, more particles will create more collisions with the walls, producing a greater force and a higher pressure.
Finally, we know that the pressure of a gas increases with the temperature. Our results above show that increasing the temperature increases the speed of the particles. This increases the frequency of collisions and increases the force of each collision. Therefore, the increase in pressure is proportional to v ^{2} .
We can also interpret the deviations from the Ideal Gas Law observed in Figures 1, 2, and 3. Remember that a gas at high density may have a greater pressure or a lower pressure than predicted by the Ideal Gas Law. But the postulates of the Kinetic Molecular Theory lead us to predictions that match the Ideal Gas Law. This means that, if the pressure of a gas under some special conditions does not match the prediction of the Ideal Gas Law, then one or more of the postulates of the Kinetic Molecular Theory must not be correct for those conditions.
We only see deviations from the Ideal Gas Law at high particle density, and in this case, the particles are much closer together on average than at lower density. Looking back at the postulates above, this means that we can no longer assume that the gas particles do not interact with each other. If the particles do interact, they exert forces on each other, which will consequently change their speeds.
If the speeds are reduced by these forces, there will be fewer impacts with the wall, and each impact will exert a weaker force on the wall. Therefore the pressure will be lower than if the particles don’t exert these forces on each other. What force between the particles would cause the speeds to be lower? If the particles attract each other, then this attraction will cause each particle to slow down as it moves towards the wall, since it will be attracted to particles behind it. We can conclude that attractions between molecules will lower the pressure, which we called a negative deviation from the Ideal Gas Law. Therefore, when we see a large negative deviation from the Ideal Gas Law, we can conclude the molecules have strong intermolecular attractions.
If the speeds are increased by the intermolecular forces, there will be more impacts with the wall and each impact will exert a great force on the wall, causing a higher pressure. Following our reasoning above, a positive deviation from the Ideal Gas Law must be due to molecules with strong intermolecular repulsions.
Our experiments tell us that, as we increase the density of the gas particles to a high value, we first see negative deviations from the Ideal Gas Law. Therefore, as we increase the density, the molecules are on average closer together and the first intermolecular force they experience is attraction. This means that the attraction of particles is important when the particles are still rather far apart from each other. Only when the particle density gets very high and the particles are on average much closer together will repulsions become important resulting in positive deviations from the Ideal Gas Law.
This analysis of experimental observations from the Kinetic Molecular Theory can be applied to understanding properties of liquids and solids. In particular, it is very helpful in understanding why substances have high or low melting or boiling points. We begin our study of these different phases of matter in the next Concept Development Study.
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