# Kinetic Theory of Gases , Postulates , Kinetic Equation

Any law is an empirical generalization which describes the results of several experiments. A law, however, only describes results; it does not explain why they have been obtained.

A theory is a description which explains the results of experiments. The kinetic-molecular theory of gases is a theory of great explanatory power.

We shall see how it explains the ideal gas law, which includes the laws of Boyle and of Charles; Dalton’s law of partial pressures; and the law of combining volumes.

The gas equation PV = nRT was pretty much in its modern form by 1860 with the inclusion of Avogadro’s hypothesis. The remarkable success of this simple equation prompted several scientists to attempt to derive this equation from a set of simple assumptions.

The workers primarily responsible for this were James Clerk Maxwell, Ludwig Boltzmann and Rudolph Clausius.

### The Postulates :

∎ A gas is made up of a large number of particles (atoms or molecules) whose size is negligible compared to the size of the container. These particles are also small compared to the distances between particles.

∎ The molecules are in constant, random motion. The distribution of kinetic energies in the gas are described by the Maxwell-Boltzmann equation.
∎ The attractions and repulsions between the particles are negligible (i.e., small or nonexistant intermolecular forces)

∎ Collisions between molecules can transfer energy, but the total average energy of the collection of molecules remains constant. The collisions are “elastic” meaning that particles bounce off of one another rather than sticking.

∎ The average kinetic energy is proportional to the temperature (in K) of the gas.

Note that the nature or kind of gas does not enter into these equations so the assumptions should be true whether we’re talking about oxygen or sulfur hexa fluoride.

These postulates, which correspond to a physical model of a gas much like a group of billiard balls moving around on a billiard table, describe the behavior of an ideal gas.

At room temperatures and pressures at or below normal atmospheric pressure, real gases seem to be accurately described by these postulates, and the consequences of this model correspond to the empirical gas laws in a quantitative way.

Gas Pressure Explained: In the kinetic molecular theory, pressure is the force exerted against the wall of a container by the continual collision of molecules against it.

Newton’s second law of motion tells us that the force exerted on a wall by a single gas molecule is equal to the mass of the molecule multiplied by the velocity of the molecule.

The molecule rebounds elastically and no kinetic energy is lost in a collision. All the molecules in a gas do not have the same velocity.

The average velocity is used to describe the overall energy in a container of gas.

The pressure-volume product of the ideal gas equation is directly proportional to the average velocity of the gas molecules.

If the velocity of the molecules is a function only of the temperature, then the kinetic molecular theory gives us Boyle’s Law

The statement that the pressure-volume product of an ideal gas is directly proportional to the total kinetic energy of the gas is also a statement of Boyle’s Law, since the total kinetic energy of an ideal gas depends only upon the temperature.

Gas Temperature Explained: If you look at the ideal gas law equation carefully you will see that it shows that the total kinetic energy of a collection of gas molecules is directly proportional to the absolute temeprature of the gas.

The ideal gas law can be rearranged to give an explicit expression for temperature. Temperature is a function only of the mean kinetic energy, the mean velocity and the mean molar mass.

As the absolute temperature decreases, the kinetic energy must decrease and thus the mean velocity of the molecules must decrease also.

At T=0, the absolute zero of temperature, all motion of gas molecules would cease and the pressure would then also be zero No molecules would be moving.

Experimentally, the absolute zero of temperature has never been attained, although modern experiments have made it to temperatures as low as 0.01 K.

As the absolute temperature decreases, the kinetic energy must decrease and thus the mean velocity of the molecules must decrease also.

At T=0, the absolute zero of temperature, all motion of gas molecules would cease and the pressure would then also be zero No molecules would be moving.

Experimentally, the absolute zero of temperature has never been attained, although modern experiments have made it to temperatures as low as 0.01 K.

It has been necessary to use the average velocity of the molecules of a gas because the actual velocities are distributed over a very wide range.

This distribution, can be described by Maxwell’s law of distribution of velocities, or you can think of a superhighway. There is an average speed on the highway.

Some cars are travelling slower, some faster, some at exactly the right speed. Even those on cruise control are never exactly on the right speed because of the discrepancies in the speed control on each individual engine and the ground geometry over which the car must be moved.

It is not necessary to use a Maxwell-Boltzmann distribution of velocities to explain either the nature of temperature or Charles’ law, although it is the correct expression of the distribution.

Charles’ law can be obtained for any distribution in which the velocities of the gas molecules are a function of the nature of the gas and the absolute temperature only.

Partial-Pressure Explained: Dalton’s law of partial pressures follows from the KMT of gases. If the gas molecules in a mixture are in constant and random motion and if there are no forces operating between the molecules except collisions, then on the average, the net effect of collisions with other molecules must be zero. For this reason each gas acts as if it were present alone.

### The Kinetic Equation:

Maxwell derived an equation on the basis of assumptions Of Kinetic Theory Of gases as

$\large P V = \frac{1}{3}m n u^2$

Where P = Pressure of gas

V = Volume of gas

m = mass of one molecule of gas

n = no. of molecules of gas

u = root mean square velocity of molecules

For 1 mole n = N is Avogadro number

m × N = Molecular mass M.

$\large P V = \frac{1}{3}M u^2$

$\large u^2 = \frac{3 P V}{M} = \frac{3 R T}{M}$

$\large u = \sqrt{\frac{3 P V}{M}} = \sqrt{\frac{3 R T}{M}}$