Kinetic Theory of Gases

Assumption of Kinetic Theory of Gases

(i) All gases are made of molecules moving randomly in all directions .

(ii) The size of molecules is much smaller than the average separation between the molecules .

(iii) The molecules exert no force on each other or on the walls of container except during collision .

(iv) All collisions between two molecules or between a molecule and a wall are perfectly elastic . Also the time spent during a collision is negligibly small .

(v) The molecules obey Newton’s Laws of motion .

(vi) When a gas is left for sufficient time , it comes to a steady state . The density and distribution of molecules with different velocities are independent of position , direction and time .

Pressure exerted by an Ideal gas

Let m = mass of a molecule

N = Total number of molecules

M (= m N) = Mass of the gas

M0 (= m NA) = Molecular mass of gas ; Where NA = Avogadro’s Number

n = M/M0 = N/NA = Number of moles of gas

ρ = M/V = Density of gas ; Where V = volume of gas

Pressure exerted by gas

$\large P = \frac{1}{3}\rho v_{rms}^2 $

Where vrms = Root Mean Square Speed of molecules .

R.M.S Speed :

$\large v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + …+ v_n^2}{N}}$

$\large v_{rms} = \sqrt{\frac{3P}{\rho}}$

$\large v_{rms} = \sqrt{\frac{3P}{M/V}} = \sqrt{\frac{3PV}{M}}$

$\large v_{rms} = \sqrt{\frac{3 n R T}{M}} = \sqrt{\frac{3 n R T}{n M_0}} $

$\large v_{rms} = \sqrt{\frac{3 R T}{M_0}} $

$\large v_{rms} = \sqrt{\frac{3 R T}{ m N_A}} $

$\large v_{rms} = \sqrt{\frac{3 k T}{ m}} $ ; Where k = R/NA = Boltzmann constant

Mean Speed or Average Speed :

$\large v_{ms} = \frac{v_1 + v_2 + v_3 + …+ v_n}{N}$

$\large v_{ms} = \sqrt{\frac{8 R T}{\pi M_0}} = \sqrt{\frac{8 k T}{\pi m }} $

Most Probable Speed :

It is the speed which maximum number of molecules in a gas have at constant temperature .

$\large v_{ms} = \sqrt{\frac{2 R T}{M_0}} = \sqrt{\frac{2 k T}{ m }} $

Translational Kinetic Energy of a gas

The total translational Kinetic energy of all the molecules of the gas is

$\large K = \Sigma \frac{1}{2} m v^2 = \frac{1}{2} (m N )\frac{\Sigma v^2}{N} = \frac{1}{2} M v_{rms}^2 $

Average K.E of a molecule is

$\large \frac{K}{N} = \frac{1}{2} \frac{M}{N} v_{rms}^2 = \frac{1}{2} m v_{rms}^2 $

$\large = \frac{1}{2} m (\frac{3 k T}{m}) = \frac{3}{2} k T $

NOTE : Hence molecules of different gases at same temperature will have same Translational Kinetic Energy .

Degree of Freedom ( f ):

Number of independent ways in which molecule can posses energy is known as Degree of Freedom .

or , Number of independent motion of the particle is equal to its degree of freedom .

For monoatomic gases ; f = 3

For diatomic gases ; f = 5

For triatomic non linear gases : f = 6

Law of Equipartition of Energy :

For an ideal gas , Average energy associated with per molecule per degree of freedom is $\large \frac{1}{2} k T $

If f be the degree of freedom for a gas . Average energy associated with its any molecule $\large = f (\frac{1}{2} k T ) $

Kinetic Energy of one mole of gas $\large = f (\frac{1}{2} R T ) $

Internal Energy of a gas

Internal Energy of a gas should be Sum of K.E & P.E of its constituent molecules .

But for an Ideal gas P.E = 0 (Since force of interaction between molecules is Zero )

Thus , Internal Energy of a gas = K.E of molecules

If f be the degree of freedom for a gas molecules then Total K.E (Translation + Rotation of each molecule ) for one mole $\large U = f(\frac{1}{2}R T ) $

For n mole of gas & having f degree of freedom are at temperature T1 kelvin and heated to temperature T2 then

At T1 , $\large U_1 = n f(\frac{1}{2}R T_1 ) $

At T2 , $\large U_2 = n f(\frac{1}{2}R T_2 ) $

Change in Internal Energy $\large \Delta U = \frac{n f R}{2} (T_2 – T_1 )$

Change in internal Energy of ideal gas is a function of temperature only . As (fR/2) is represented by Cv
$\large C_v = \frac{f R}{2} $

$\large \Delta U = n C_v \Delta T $

Mean Free Path:

The average distance traveled by by a gas molecule between two successive collisions is called mean free path .

Mean Free Path $\large \lambda = \frac{1}{\sqrt{2} \pi d^2 n} $ ;

Where , n= number of molecules per unit volume

d = diameter of the molecule

As , $\large P V = N k T $

$\large P = \frac{N}{V} k T = n k T$ ; Where N/V = n = number of molecules per unit volume

$\large \lambda = \frac{k T}{\sqrt{2} \pi d^2 P } $ ;

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