Lesson 5.6 Kinetic Molecular Theory
Suggested Reading

Zumdahl Chapter 5 Sections 5.6
Essential Questions

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.
Introduction
To better understand the gas laws, we need to understand the Kinetic
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:
1. 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.
2. 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.
3. 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.
4. When molecules collide with one another, the collisions are elastic. In
an elastic collision, the total kinetic energy remains constant, so none is
lost.
5. The average kinetic energy of a collection of gas particles is assumed to
be directly proportional to the Kelvin temperature of the gas.
Qualitative Interpretation of the Gas Laws
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 equation similar 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 Vnb, 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 by P(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