9.6 Kinetic Molecular Theory
Density fluctuations: why the sky is blue
Distribution of gas molecules in a gravitational field
The properties such as temperature, pressure, and volume, together with others dependent on them (density, thermal conductivity, etc.) are known as macroscopic properties of matter; these are properties that can be observed in bulk matter, without reference to its underlying structure or molecular nature.
By the late 19th century the atomic theory of matter was sufficiently well accepted that scientists began to relate these macroscopic properties to the behavior of the individual molecules, which are described by the microscopic properties of matter. The outcome of this effort was the kinetic molecular theory of gases. This theory applies strictly only to a hypothetical substance known as an ideal gas; we will see, however, that under many conditions it describes the behavior of real gases at ordinary temperatures and pressures quite accurately, and serves as the starting point for dealing with more complicated states of matter.
1 The basic ideas of kinetic-molecular theory
The “kinetic-molecular theory of gases” may sound rather imposing, but it is based on a series of easily-understood assumptions that, taken together, constitute a model that greatly simplifies our understanding of the gaseous state of matter.
The basic tenets of the kinetic-molecular theory are as follows: It is important that you know them!
- A gas is composed of molecules that are separated by average distances that are much greater than the sizes of the molecules themselves. The volume occupied by the molecules of the gas is negligible compared to the volume of the gas itself.
Figure 9.61 Collection of gas molecules
Ref: commons.wikimedia.org/
- The molecules of an ideal gas exert no attractive forces on each other, or on the walls of the container.
- The molecules are in constant random motion, and as material bodies, they obey Newton’s laws of motion. This means that the molecules move in straight lines (see demo illustration at the left) until they collide with each other or with the walls of the container.
Figure 9.62 Brownian Motion of gas molecules
Ref: commons.wikimedia.org/
- Collisions are perfectly elastic; when two molecules collide, they change their directions and kinetic energies, but the total kinetic energy is conserved. Collisions are not “sticky”.
- The average kinetic energy of the gas molecules is directly proportional to the absolute temperature. (Notice that the term “average” is very important here; the velocities and kinetic energies of individual molecules will span a wide range of values, and some will even have zero velocity at a given instant.) This implies that all molecular motion would cease if the temperature were reduced to absolute zero.
Figure 9.63 Collision of gas molecules
Ref: commons.wikimedia.org/
According to this model, most of the volume occupied by a gas is empty space; this is the main feature that distinguishes gases from condensed states of matter (liquids and solids) in which neighboring molecules are constantly in contact. Gas molecules are in rapid and continuous motion; at ordinary temperatures and pressures their velocities are of the order of 0.1-1 km/sec and each molecule experiences approximately 1010 collisions with other molecules every second.
Here is the particulate view of all states of matter including plasma.
Figure 9.64 state of matter
Ref: commons.wikimedia.org/
2 How kinetic-molecular theory explains the gas laws
If gases do in fact consist of widely-separated particles, then the observable properties of gases must be explainable in terms of the simple mechanics that govern the motions of the individual molecules. This nicely illustrates how a single theory (KMT) can explain all of the laws of gas behavior.
Kinetic interpretation of gas pressure
Figure 9.65 Gas Pressure
The kinetic molecular theory makes it easy to see why a gas should exert a pressure on the walls of a container. Any surface in contact with the gas is constantly bombarded by the molecules. At each collision, a molecule moving with momentum mv strikes the surface. Since the collisions are elastic, the molecule bounces back with the same velocity in the opposite direction. This change in velocity ΔV is equivalent to an acceleration a; according to Newton’s second law, a force f = ma is thus exerted on the surface of area A exerting a pressure P = f/A.
Kinetic interpretation of gas temperature
According to the kinetic molecular theory, the average kinetic energy of an ideal gas is directly proportional to the absolute temperature. Kinetic energy is the energy a body has by virtue of its motion:
As the temperature of a gas rises, the average velocity of the molecules will increase; a doubling of the temperature will increase this velocity by a factor of four. Collisions with the walls of the container will transfer more momentum, and thus more kinetic energy, to the walls. If the walls are cooler than the gas, they will get warmer, returning less kinetic energy to the gas, and causing it to cool until thermal equilibrium is reached. Because temperature depends on the average kinetic energy, the concept of temperature only applies to a statistically meaningful sample of molecules. We will have more to say about molecular velocities and kinetic energies farther on.
Kinetic explanation of Boyle’s law
Boyle’s law is easily explained by the kinetic molecular theory. The pressure of a gas depends on the number of times per second that the molecules strike the surface of the container. If we compress the gas to a smaller volume, the same number of molecules are now acting against a smaller surface area, so the number striking per unit of area, and thus the pressure, is now greater.
Kinetic explanation of Charles’ law
Kinetic molecular theory states that an increase in temperature raises the average kinetic energy of the molecules. If the molecules are moving more rapidly but the pressure remains the same, then the molecules must stay farther apart, so that the increase in the rate at which molecules collide with the surface of the container is compensated for by a corresponding increase in the area of this surface as the gas expands.
Kinetic explanation of Avogadro’s law
If we increase the number of gas molecules in a closed container, more of them will collide with the walls per unit time. If the pressure is to remain constant, the volume must increase in proportion, so that the molecules strike the walls less frequently, and over a larger surface area.
Kinetic explanation of Dalton’s law
“Every gas is a vacuum to every other gas”. This is the way Dalton stated what we now know as his law of partial pressures. It simply means that each gas present in a mixture of gases acts independently of the others. This makes sense because of one of the fundamental tenets of KMT theory that gas molecules have negligible volumes. So Gas A in mixture of A and B acts as if Gas B were not there at all. Each contributes its own pressure to the total pressure within the container, in proportion to the fraction of the molecules it represents.
3 Some important practical applications of KMT
The molecules of a gas are in a state of perpetual motion in which the velocity (that is, the speed and direction) of each molecule is completely random and independent of that of the other molecules. This fundamental assumption of the kinetic-molecular model helps us understand a wide range of commonly-observed phenomena.
Kinetic Molecular Theory and its Postulates
All the gas laws Boyles’s law, Charles’s law, Dalton’s law of partial pressure can be explained in terms of kinetic molecular theory. The most important relation is the the relationship between kinetic energy and temperature:
The kinetic energy of an object is the energy associated with its motion. It is related to object’s mass and speed according to the formula:
This equation shows that a heavy object and a light object have the same kinetic energy, heavy object must be moving more slowly. Different gases at the same temperature have same kinetic energy. For this to be true, molecules with higher mass have lower speed.
We can use the kinetic molecular theory to derive the expression for Root Mean Square Velocity (rms) for the gas particles. This velocity of any gas particle is inversely proportional to the molar mass of the gas. Therefore, at particular temp, smaller the gas particles, higher the speed.
Figure 9.86 Energy Distribution with Temperature
Ref: commons.wikimedia.org/
Distribution of Molecular speed: Refinement of the basic kinetic molecular theory provide us with a view into the world of gas molecules. In a sample of gas, several different molecules are moving at different speed.
Figure 9.87 Molecular speed Distribution
Ref: commons.wikimedia.org/
Q: What is the root-mean-square speed for a sample of oxygen gas at 298 K?
Mean free path (distance travelled between two collisions) of gas particles and the rate of effusion and diffusion.
What you should be able to do
Make sure you thoroughly understand the following essential ideas which have been presented above. It is especially important that you know the principal assumptions of the kinetic-molecular theory. These can be divided into those that refer to the nature of the molecules themselves, and those that describe the nature of their motions:
The molecules – Negligible volume, absence of inermolecular attractions (think of them as very hard, “non-sticky” objects.)
Their motions – Completely random in direction, in straight lines only (this is a consequence of their lack of attractions), average velocities proportional to the absolute temperature..
The idea that random motions of individual molecules can result in non-random (directed) movement of the gas as a whole is one of the most important concepts of chemistry, exemplified here as the principle of diffusion.
In most courses you will be expected to know and be able to use Graham’s law.