Kinetic Theory

Kinetic Theory The kinetic theory of gases is the study of the microscopic behavior of molecules and the interactions which lead to macroscopic relationships like the ideal gas law.

The Kinetic Theory of Gases The basic postulates of kinetic theory of gases:

1.

Gases are made up of molecules: molecules of the same gas are similar to one another and differ from the molecule of other gases.

2.

We can treat molecules as point masses that are perfect spheres of negligible masses.

3.

Molecules are in constant random motion: There is no general pattern governing either the magnitude or direction of the velocity of the molecules in a gas. At any given time, molecules are moving in many different directions at many different speeds ranging from zero to infinity.

4. Molecules collide with one another and these collisions are instantaneous.

5.

Molecular collisions are perfectly elastic: Molecules do not lose any kinetic energy when they collide with one another.

6. Molecules of a gas do not exert any force of attraction or repulsion upon each other.

7.

There is no change in the number of molecules per cm3 of the gas with time.

Root mean square velocity According to the kinetic theory of gases, molecules of a gas are in random motion. If c1,c2,c3…………cn are the velocities of the molecules in a gas at any instant, then the root mean square velocity of the gas is given by,

c12 + c 22 + c32 + ........ + c n2 c= n Pressure of a gas Pressure of a gas is given by the expression,

P= But,

1M 2 c 3V

(1)

M = ρ , the density of the gas. V

Therefore the equation for pressure of the gas becomes,

1 P = ρc 2 3 From the above equation,

c2 =

3P ρ

(2)

Or,

3P ρ

c=

(3)

Equation (1) can be written as,

P=

2 1 M 2 × × c 3 2 V

1 Mc 2 2 2 P= × 3 V But,

1 Mc 2 is the kinetic energy of the gas. 2

Therefore, we have-

P=

But,

2 Kinetic energy of the gas × 3 V

kinetic energy of the gas = E , energy per unit volume of the gas V ∴P =

2 E 3

(5)

Relation between root mean square velocity and temperature We know, P =

1 1M 2 c or PV = Mc 2 3V 3

According to gas equation,

PV = RT

From equation (6) and (7),

1 Mc 2 = RT 3

Or,

c2 =

(6)

(7)

3RT M

R and M being constants, we get-

c 2 = (a constant) T Or,

c2 ∝ T

or

c∝ T

Root mean square velocity of the molecules of a gas is proportional to the square root of the absolute temperature.

Derivation of gas laws

a) BOYLE’S LAW: it states that, the temperature remaining constant, volume of a given mass of a gas varies inversely as its pressure.

We know,

P=

1M 2 c 3V

or

PV =

1 Mc 2 3

If T is a constant, C is also a constant. Therefore, PV = a constant

a constant V 1 P∝ V

P= Or,

This is Boyle’s Law.

b) CHARLE’S LAW: it states that, the pressure remaining constant, volume of a given mass of a gas varies directly as the absolute temperature of the gas.

We know,

P=

1M 2 c 3V

or

M and P are constants and c 2 We get, Or,

V=

∝T

1M 2 c 3 P

,

V = ( acons tan t )T

V∝T

This is Charle’s Law. IDEAL GAS EQUATION A relation between the pressure, volume and temperature of an ideal gas is called IDEAL GAS EQUATION. Derivation Consider some gas contained in a cylinder fitted with an air tight piston. Let P1, V1, T1 be the parameters of the gas. [fig.(1)]. Heat the gas so that the temperature increases from T1 to T2, keeping the pressure P1 constant. The volume changes from V1 to v. [fig.(2)].

Applying Charle’s law, we getAnd

V1 ∝ T1 v ∝ T2

(1) (2)

Dividing equation (1) by equation (2),

V1 T1 = v T2 V1T2 = vT1 v=

Or

V1T2 T1

(3)

Now, keeping the temperature constant at T2, increase the pressure from P1 to P2. The volume changes from v to V2.[fig.(3)]. Applying Boyle’s law, we get-

1 v 1 P2 ∝ V2 P1 ∝

And

[From fig. (2)]

(4)

[from fig. (3)]

(5)

Dividing equation (4) by equation (5),

1 P1 = v 1 P2 V2

Or

P1 V2 = P2 v

P1 v = V2 P2 Substituting eqn(3) in eqn(6),

Therefore,

PV T

P1V1T2 = P2V2 T1

(6)

or

P1V1 P2V2 = T1 T2

= a constant . This is the IDEAL GAS EQUATION.

The value of the constant depends on the nature and mass of the gas. For one gram mole of gas,

PV T

= R , Where R is the Universal gas constant. Its value is the same for all the

gases. The value of R is 8.3 J mol-1 K-1. For ‘n’ mole of gas, the ideal gas equation can be written as-

PV T

Or

= nR

PV = nRT