Kinetic Molecular Theory of Gases

Kinetic molecular theory of gases:

Kinetic theory of gas was formulated to explain the macroscopic properties of a gas such as its temperature, pressure, volume etc. by considering their molecular composition and motion.

The postulates for the kinetic theory of gases are:

• The gas particles are very small with negligible mass.
  
• The total volume of all the molecules is negligible as compared to the volume of the container , or in other words the volume of the molecule  is negligible as compared to the volume of the gas.
• The number of the molecules is large such that statistical treatment is possible.
• The interaction between the molecules is negligible except during collisions.Thus they move independently without any force field.
• The molecules are in constant, random motion. They keep on colliding with the walls of the container.
• The molecules are perfectly elastic and the collisions they undergo do not result in a change in their Kinetic energy.
• The average K.E of the system depends only on the temperature of the system and is directly proportional to it.
• All the molecules of the gas move in different velocities and as they collide their speed changes but the distribution of speed remains constant at any particular temperature.This distribution is called Maxwell-Boltzman distribution, and is shown in graph below:

• Thus, it can be seen that a very small fraction of molecules have extreme velocities (i.e, very high or very low velocities), and most of them have a common velocity corresponding to the point of the maxima in the curve.This velocity is called the Most probable velocity.
• As temperature increases, the value of most probable velocity increases though the number of molecules at the most probable velocity decreases.

The equation obtained on the basis of above postulates is known as kinetic gas equation and is expressed as :

                                                                pV = (1/3)mNCrms²           

where, m is the mass of the molecules , N is the number of the molecules and Crms is known as root mean square speed as defined in the next section.

 Root mean square speed:

 If the molecules are moving with individual speeds of C₁,C₂,C₃, .......................................C_N., then the root mean square speed is defined as :

       C_rms = ( C_1²+ C_2²+ C_3²+  .......................................+C_N² / N )^1/2

 The term root mean square is self - explanatory , it is the square root of the mean of the square of the individual speeds.It has the dimensions of speed.

Also,  C_rms = (3RT/M)^1/2  ( M is in Kg)

From the ideal gas equation , pV = nRT

?RT = pV/n

?C_rms = (3pV/nM)^1/2 = (3pV/ρ)^1/2 

Where, m is the mass of the given quantity of  the gas and ρ is the density of  the gas.It should be kept in mind that  rms speed depends only on the temperature of the gas and it doesnsinglequotet depend on the pressure and volume variations if temperature is constant as the product pV and the factor p/ρ are constant at constant temperature.

Average Speed:

The average speed is defined as:

                                                C_av = (C₁ + C₂ + C₃ ........................C_N / N)

?                                             C_av = (8RT/M)^1/2

 Most probable speed: (C_MP):

The term most probable sped was coined by Maxwell on the basis of probability distribution of molecular speeds.

Its expression is:

                                C_mp = (2RT/M)^_1/2

Ratio between the three speeds:

                                       C_mp : C_av : C_rms :: (1/2)_^1/2 : (8/ π )^1/2 : (3)^1/2 = 1.2248 : 1.1284 : 1