Physics KINETIC THEORY OF GASES

Kinetic Theory of Gases :

Gases are made-up of tiny particles, consisting of molecules, atoms or even ions (sometimes) which retain the chemical properties of the
sample of which they are composed.

The kinetic theory of gases develops a model of the molecular behavior which results in the observed behavior of an ideal gas.

Assumption of Kinetic Theory of Gases :
i. A gas consists of very large number of molecules. These molecules are identical, perfectly elastic and hard sphere. They are so small that the volume of molecules is negligible as compared with the volume of gas.
ii. Molecules move in all direction randomly. Molecules do not exert any appreciable force on one another or on the walls of the container except during collision.
iii. All collisions between the molecules or with the wall of the container are perfectly elastic. It means kinetic energy of molecules IS conserved in each collision.
iv. These molecules travel in straight line. The duration of a collision is negligible in comparison to the time spent by a molecule between collisions.
v. In steady state, the density and the distribution of molecules with different velocities are independent of position, direction and time. This assumption is justified if the number of molecules is very large.
VI. The molecules obey Newton's Laws of motion.

The assumptions of kinetic theory are close to the real situation at low densities.

Gas laws:-

(a) Boyle's law:- It states that the volume of a given amount of gas varies inversely as its pressure, provided its temperature is kept constant.

PV = Constant

(b)Charlers law or Gey Lussac's law:- It states that volume of a given mass of a gas varies directly as its absolute temperature, provided its pressure is kept constant.

`V/T` = constant

(c)Gay Lussac's law of pressure:- It states that pressure of a given mass of a gas varies directly as its absolute temperature provided the volume of the gas is kept constant.

`P/T = P_0/T_0 or P = P_0/T_0 T = 1/273 = γp`

(d) Dalton's law of partial pressures:-

Partial pressure of a gas or of saturated vapors is the pressure which it would exert if contained alone in the entire confined given space.

`P= p_1+p_2+p_3+....`

(e) Avogadro's law:- It states that under similar conditions of pressure and temperature, equal volume of all gases contain equal number of molecules.

For m gram of gas, `PV/T = nR = (m/M) R`

Maxwell-Boltzmann Distribution Law :

A Maxwell-Boltzmann distribution describes the probability that a particle is moving with a particular speed at a given temperature, and we note how the distribution shifts to higher speeds as temperature increases.

The Maxwell-Boltzmann equation is a probability density function for speed `u`.

Maximum number of molecules have speed lying with in a small range centered about the speed corresponding the peak of the curve. This speed is called the 'most probable speed' `V_p`

Curve is not symmetric - there is more of an elongated "tail" at high speeds. Therefore. the average speed `V_(av)` is slightly greater than the most probable speed `V_(mp)`

Average (or Mean speed) speed : `bar v=sqrt((8kT)/(pi m))`

Most probable speed : `v_(mp)=sqrt((2kT)/m)`

Root mean square speed : `v_(n et)=sqrt((3kT)/m)`

Internal Energy of Ideal Gas :

The internal energy of an ideal gas is entirely the kinetic energy of its molecules, as there is no interaction between the molecules, there is no Potential energy of the system.

`:.` Internal energy of `N` molecules, having `'f` degrees of freedom

`U=N xx f xx 1/2 kT=1/2 f n RT` (as `nR=kN`)

Law of Equipartition of Energy :

For a system in equilibrium at absolute temperature `T`, the average energy per molecule associated with each degree of freedom is `1/2 kT` ; Where `k` is Boltzmann's constant. If degree of freedom of a molecule is `f`, then total kinetic energy of that molecule `= 1/2 f kT`