Lesson 5.6 Kinetic Molecular Theory
Suggested Reading- Zumdahl Chapter 5 Sections 5.6
- What does kinetic molecular theory predict about the behavior of gases?
Learning Objectives
- Explain the kinetic molecular theory of gases.
- Related KMT to ideal gas behavior.
To better understand the gas laws, we need to understand theKinetic Molecular Theory of Gases(KMT). This model is used to describe the behavior of gases. More specifically, it is used to explain macroscopic properties of a gas, such as pressure and temperature, in terms of its microscopic components, such as atoms. Like the ideal gas law, this theory was developed in reference to ideal gases, although it can be applied reasonably well to real gases.
Kinetic Molecular Theory of an Ideal Gas
The word kinetic describes something in motion. Kinetic energy, Ek,is the energy associated with the motion of an object of mass. Recall that Ek =1/2mv2. We will use this concept of kinetic energy in describing the kinetic theory. Our present explanation of gas pressure is that it results from the continual bombardment of the container walls by constantly moving molecules. Although this kinetic interpretation of gas pressure was first put forth by Robert Hook in 1676, it took about 200 years for this idea to gain acceptance by scientists. The present kinetic molecular theory of gases was developed by a number of scientists throughout the last half of the 19th century. This theory is now a cornerstone of our present view of molecular substances.
Postulates of Kinetic Theory
In order to apply the kinetic model of gases, five assumptions are made:
- Gases are composes of molecules whose size is negligible compared with the average distance between them. Thus, most of volume occupied by a gas is empty space. This means that you can usually ignore the volume occupied by the molecules.
- The particles are in constant motion. The collisions of the particles with the walls of the containers are the cause of the pressure exerted by the gas.
- The forces of attraction or repulsion between two molecule (intermolecular forces) in a gas are very weak or negligible. This means that a molecule will continue moving with undiminished speed until it collides with another gas molecule or with the walls of the container.
- When molecules collide with one another, the collisions are elastic. In an elastic collision, the total kinetic energy remains constant, so none is lost.
- The average kinetic energy of a collection of gas particles is assumed to be directly proportional to the Kelvin temperature of the gas.
According to KMT, the pressure of a gas results from the bombardment of container walls by molecules. Both the concentration and the average speed of the molecules are factors in determining this pressure. Molecular concentration and average speed determine the frequency of collision with the wall. Average molecular speed determines the average force of collisions.
You can make sense of Boyle's law in these terms. Constant temperature means the that the average kinetic energy of a molecule is constant. Therefore, the average molecular speed and force of collision remain constant. Suppose you increase the volume of a gas. This decreases the concentration and frequency of collisions. Thus, when you increase the volume of a gas while keeping the temperature constant, the pressure decreases.
Now consider Charles' law. If you raise the temperature, you increase the average molecular speed. The average force of a collision increases. If all other factors remain constant, the pressure increases. For the pressure to remain constant as it does in Charles' law, it is necessary for the volume to increase so that the concentration and frequency of collisions decreases. Thus, when you raise the temperature of a gas while keeping the pressure constant, the volume increases.
The Ideal Gas Law from Kinetic Theory
One of the most important features of KMT is its explanation for the ideal gas law. According to KMT, the pressure of a gas is proportional to the frequency and force of molecular collisions with a surface.
P ∝ frequency of collisions x average force
The average force exerted by a molecule during a collision depends on its mass, m, and its average speed, u, which is the average momentum of a molecule, mu. In other words, the greater the mass of the molecule and the faster it is moving, the greater the force exerted during a collision. The frequency of collision is also proportional to the average speed, u. because the faster the molecule is moving the more often it strikes the container wall. Frequency of collisions is inversely proportional to the gas volume, V, because the larger the volume, the the less often a given molecule strikes the container walls. Finally, the frequency of collisions is proportional to the number of molecules, N, in a given volume. Putting these factors together gives
Bring the volume to the left side gives
Because the average kinetic energy of a molecule of mass, m, and average speed, u is 1/2mu2, PV is proportional to the average kinetic energy of the molecule. Since the average kinetic energy is proportional to temperature (assumption 5) and the number of molecules, N, is proportional to moles, n, you have
Finally, inserting the constant of proportionality gives PV = nRT.
Real Gases
Experiments show that the ideal gas law describes the behavior of real gases quite wall at low pressures and moderate temperatures, but not at high pressures and low temperatures. When working with the ideal gas law, we must remember that ideal gases are hypothetical constructs. The simplifying assumptions of KMT do not hold at high pressures and temperatures, and the law breaks down as a result.
The figure to the right shows the behavior of real gases at various pressures. The gases deviate noticeably from ideal behavior at high pressures. Also, the deviations differ for each kind of gas. We can explain why this by examining the postulates of KMT, from which the ideal gas law is derived.
Assumption 1 of KMT says that the volume of space occupied by molecules is negligible compared with the total volume of gas. At low pressures, where the volume of individual molecules is negligible, the ideal gas law is a good approximation. At higher pressures, the volume of individual molecules becomes important, since the total volume is smaller.
Assumption 3 says that the forces of attraction between gas molecules
(intermolecular forces) are very weak or negligible. This is also a good approximation at low pressure, where the molecules are far apart, because these forces decrease rapidly as the distance between molecules increases. However, intermolecular forces become significant at higher pressures, because the molecules are closer together. Because of these intermolecular forces, the actual pressure of a gas is less than that predicted by ideal gas behavior. As a molecule begins to collide with a surface, neighboring molecules pull this colliding molecule slightly away from the wall, giving a reduced pressure.
The Dutch physicist J.D, van der Waals was the first to account for these deviations of a real gas from ideal behavior. The van der Waals equation is an equationsimilar to the ideal gas law, but includes two constants, a & b, to account for deviations from ideal behavior.
The constants a and b differ depending on the gas you are working with. You will not have to solve numerical problems using this equation, but if you are curious, Table 5.3 of your textbook on page 210 lists some van der Waals constants for some common gases. However, you will be asked to apply this equation in a qualitative way.
You can obtain the van der Walls equation by substituting P = P+an2/V2 and V = V-nb into the ideal gas law PV = nRT. To obtain these substitutions, van der Waals reasoned as follows. The volume available for a molecule to move in equals the gas volume, V, minus the volume occupied by the molecules. So he replaced V in the ideal gas law with V-nb, where nb represents the volume occupied by n moles of molecules. Then he noted that the total force of attraction on any molecule about to hit a wall is proportional to the concentration of neighboring molecules, n/V as well as the total concentration, n/V. The net result is that the pressure is reduced from that assumed in the ideal gas law by a factor proportional to n2/V2. Letting a be the proportionality constant, we can write
P(actual) = P(ideal) - n2a/V2
or
P(ideal) = P(actual) +n2a/V2
In other words, you replaced P in the ideal gas law byP(actual)+n2a/V2. The van der Waals equation is used just like the ideal gas equation, except the conditions are not quite ideal. Therefore, corrections for deviation from ideal behavior must be made using the substitutions given in the van der Waals equation.
HOMEWORK: Go over power point notes examples relating to the van der Waals equation. Complete practice exercises 10.14 – 10.15