Kinetic Theory of Gases (Basics)

What is Kinetic Theory of gases?

Take a system of gas molecules in a container, then the pressure applied by the gas and the temperature of gas can be solely expressed in terms of the velocities, position and number of the molecules.

Following approximations are taken:
1. Only Translational Motion (2 dimensional)
2. No intermolecular forces.
3. Elastic collision between molecules so that both kinetic energy and momentum is conserved.
4. Large Inter-molecular forces and Low density.

Some important concepts:

Equipartition Theorem:

Equipartition Theorem states that for a system which is at a temperature T, each degree of freedom is associated with internal energy given by:

Thus for a system in 2-D which has only 2 degrees of freedom, total internal energy of a single particle is given by

Now since, the only energy that is available to molecules is kinetic energy (we have neglected intermolecular potential). This kinetic energy is random and thus constitute no net motion at macroscopic level, i.e., if you take a container of gas, the molecules are moving randomly but the container isn’t moving itself. This random motion is associated with the internal energy of the gas molecules. So, if a molecule has a mass “m”, then the kinetic energy associated with it is:

Therefore total energy of a system with N molecules can be written as:

Assuming all particles has same mass,

The above equation related Temperature with the velocities of the molecules.

And gives,

We will come back to this equation after going through the velocity distribution.

Boltzmann-Maxwell velocity Distribution:

If you leave a system undisturbed, the velocities of the gas molecule settle down in a given velocity probability distribution given by Boltzmann-Maxwell velocity distribution. The calculation of the distribution is the beauty of statistical mechanics (a branch of physics that deals with all the stuff we have been talking about).

This all starts with Boltzmann distribution. It is a distribution of probability of a system being in a certain state which is described by energy and temperature. In other words, what is the probability that a particle will have given energy at the given temperature.

The proportionality can be removed by normalizing the probability distribution. So how do we calculate the velocity distribution?

Thus the probability of having velocity v is given by,

Till now, “v” was representing a single velocity component. In other words, the above probability distribution represents velocity distribution in 1-D. Now since in 2-D, the two components of velocity represents two independent states, we will have to consider the probabilities for both. Thus giving,

[The above transformation involves going from Cartesian to Polar while reducing one variable, direction (Θ), which is of no importance in kinetic theory because of randomness.]

Thus, we get the probability density of speed being between v and dv as,

On normalization (note limit of v is from 0 to infinity, since v is speed and thus always positive),

gives,

And the probability distribution of speed in 2-D is given by,

Now, by definition,

which as same as we got using Equipartition theorem, so all our hard work finally paid off.

Here we got an important conversion relation,

And so, velocity distribution we will use is,

Entropy

Entropy is normally considered as a measure of disorder. Physically, it’s the measure of possible micro-states available to a system. For example, a gas in small container is more ordered as compared to the same amount of gas in big container, OR in small container gas has less number of possible micro-states as compared to in big container.

A screenshot from “Concept of Thermal Physics by Blundell” to help you understand the difference between micro-states and macro-states,

Entropy in statistical mechanics if defined as

where, Ω is the number of possible micro-states. However, these relation is valid for on when the system is in equilibrium. For example, if there is heat transfer going between two systems, then we can calculate the Entropy before the heat transfer started and only when the heat transfer is complete and the systems have reached thermal equilibrium as we can’t define the micro-states in between. My aim with the simulation is to study this entropy change between two equilibrium states.

Summary

Things that we will use:

Enjoy !!