Monday, November 10, 2014

Kinetic theory of gases and Expression for Pressure

The purpose of the Kinetic theory of the gases is to link to the macroscopic properties of the gases like pressure volume,temperature with the macroscopic properties of the gas molecules like displacement,velocity, momentum, force and kinetic energy.

To apply the connect theory of the gases to the molecules we shall make some assumptions.

1. We shall assume that the size of the molecule is very small and when compared with the volume of the gas occupied the volume of the gas molecules occupied is negligible.
2. The gas molecules are tiny in size , spherical in shape , neutral with respect to charge and all are identical.
3. The molecules moves  in all directions randomly with all possible speeds.
4. The collision of the gas molecules is an elastic. That means during the collision both momentum and kinetic energy are conserved. There is no wastage of energy in the format of sound light and heat.

Mean free path:

The average distance molecule can travel without colliding the neighboring molecules is called as mean free path.When the molecule starts its journey, its motion is so random under regular that the we cannot predict what is going to be its path is. 

In between any of the two collisions, the molecule travel some distance and by measuring this total distances and by dividing it with the number of the collisions we can calculate the value of the mean free path.

The gas which obeys all gas laws at all temperatures and pressures is called an ideal gas. In reality no gas is actually Ideal and all the existing gases are called real gases.While explaining kinetic theory of the gases, anyway we assume that the gas is Ideal.

Practically real gas obeys all gas laws only at high temperatures and low pressures.

Expression of the pressure of an ideal gas:

Being the gas molecules are having collisions among themselves and with the walls of the container, they are going to  exert some pressure and here we are going to calculate that pressure. Let us  consider a container who is having cube shape. Let the side of the cube is  L and a gas molecule of mass m is moving in parallel to YZ plane along the x-axis.

As the collision is elastic the molecule will come back with the same velocity.Thus we can calculate the change of the momentem with respect to time which leads the calculation of the force. The derivation for the pressure is made are shown below.

This can be further continued by writing all the forces and then further writing equation for the pressure as defined as the force per unit area as shown further.

The expression for the pressure of a gas molecule can be expressed in terms of kinetic energy of the gas molecule also. Here RMS velocity of the gas molecule can be expressed  as the root mean square velocity of the gas molecule. Basing the ideal gas equation even we can measure the impact of the temperature here. The temperature at which the RMS velocity of  gas molecule becomes zero is called a absolute zero temperature and that is taken as a reference to define Kelvin scale.

We can further find a relation between temperature of the gas and the RMS velocity of the gas molecule as shown below.We can also define a particular temparatue at which RMS velocity of gas molecule become zero called absolute temperature.

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