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CBSE NOTES CLASS 11 CHAPTER 13

KINETIC THEORY

Kinetic Theory for Ideal Gases

1. Every gas consists of extremely small particles known as molecules. The molecules of a given gas are all identical but are different from those of another gas.

2. The molecules of a gas are identical, spherical, rigid and perfectly elastic point masses.

3. Their molecular size is negligible in comparison to intermolecular distance.

4. The speed of gas molecules lies between zero and infinity (very high speed).

5. The distance covered by the molecules between two successive collisions is known as free path and mean of all free paths is known as mean free path.

6. The number of collision per unit volume in a gas remains constant.

7. No attractive or repulsive force acts between gas molecules.

8. Gravitational attraction force among the molecules is ineffective due to small masses and very high speed of molecules.

Gas Laws

Ideal gases irrespective of their nature obey the following laws,

Boyle’s Law

At constant temperature the volume (V) of given mass of a gas is inversely proportional to its pressure (p), i.e.,

For a given gas,

Where kB is called Boltzmann constant and N is the number of molecules in the given sample of the gas.

Its value in SI units is 1.38 × 10–23 J K–1

Charles’ Law

At constant pressure the volume (V) of a given mass of gas is directly proportional to its absolute temperature (T), i.e.,

For a given gas,

$\frac{{\mathrm{V}}_{1}}{{\mathrm{T}}_{1}}=\frac{{\mathrm{V}}_{2}}{{\mathrm{T}}_{2}}$

At constant pressure the volume (V) of a given mass of a gas increases or decreases by $\frac{1}{273.15}$ of its volume at 0°C for each 1°C rise or fall in temperature.

Volume of the gas at t °C is,

where V0 is the volume of gas at T0 = 0 °C = 273.15 K

Gay Lussac’s Law

At constant volume the pressure p of a given mass of gas is directly proportional to its absolute temperature T, i.e. ,

For a given gas,

$\frac{{\mathrm{p}}_{1}}{{\mathrm{T}}_{1}}=\frac{{\mathrm{p}}_{2}}{{\mathrm{T}}_{2}}$

At constant volume (V) the pressure p of a given mass of a gas increases or decreases by $\frac{1}{273.15}$ of its pressure at 0°C for each 1°C rise or fall in temperature.

Pressure of the gas at t °C,

where p0 is the pressure of gas at 0°C.

Equal volume of all the gases under similar conditions of temperature and pressure contain equal number molecules. This statement is called Avogadro’s hypothesis.

(i) Avogadro’s number - The number of molecules present in 1 gram-mole of a gas is defined as Avogadro’s number.

NA = 6.022 × 1023 per gram mole

(ii) At STP or NTP (T = 273 K and p = 1 bar = 105 Pa), 1 mole of gas occupies volume of 22.4 L and contains 6.022 × 1023 molecules.

Ideal Gas Equation

Gases which obey all gas laws in all conditions of pressure and temperature are called ideal gases.

Ideal gas equation

pV = n R T

or pV = NkBT = nRT

where p = pressure,

V = volume,

T = absolute temperature,

R = universal gas constant

n = number of moles of a gas.

N = number of particles in the sample of gas.

Universal gas constant R = 8.31 J mol-1 K-1.

${\mathrm{k}}_{\mathrm{B}}=\frac{\mathrm{R}}{{\mathrm{N}}_{\mathrm{A}}}$

Real or Van der Waal’s Gas Equation

Real gases deviate slightly from ideal gas laws because

• Real gas molecules interact with each other.

• Real gas molecules occupy a finite volume.

At high pressures molecules of gases are very close to each other. Molecular interactions start operating. At high pressure, molecules do not strike the walls of the container with full impact because these are dragged back by other molecules due to molecular attractive forces. This affects the pressure exerted by the molecules on the walls of the container. Thus, the pressure exerted by the gas is lower than the pressure exerted by the ideal gas.

${\mathrm{p}}_{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}}={\mathrm{p}}_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}+\frac{\mathrm{a}{\mathrm{n}}^{2}}{{\mathrm{V}}^{2}}$

Repulsive forces also become significant.

At high pressures or low volumes, the repulsive forces also start operating and the molecules behave as small but impenetrable spheres. The volume occupied by the molecules also becomes significant because instead of moving in volume V, these are now restricted to volume (V – nb) where nb is approximately the total volume occupied by the molecules themselves. Here, b is a constant.

So the gas equation for real gases becomes,

where a and b are called van der Waals’ constants.

Daltons Equation of Partial Pressures

In a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases.

Partial Pressure

The pressure that would be exerted by the component gas in a mixture if it occupied the same volume on its own, is called partial pressure of the component gas.

Proof of Daltons Equation of Partial Pressures

From ideal gas equation, we have,

PV = n RT

Mole fraction are defined as,

Graham’s Law

Diffusion

Diffusion of a gas is the process in which particles of one gas are spread throughout another gas by molecular motion.

Rate of diffusion ∝ $\frac{1}{\sqrt{\mathrm{molar mass}}}$

Effusion

Effusion is the process in which gas molecules escape from an evacuated container though a small hole.

The kinetic energy of any gas, with mass m and velocity v, is given by

Because temperature is a measure of the average kinetic energy of a gas, two gases at the same temperature will also have the same kinetic energy. Thus,

Kinetic Interpretation of Pressure

Consider a gas enclosed in a cube of side l. Take the axes to be parallel to the sides of the cube, as shown.

A molecule with velocity vx, vy, vz ) hits the planar wall parallel to yz plane of area A (= l2).

Since the collision is elastic, the molecule rebounds with the same velocity; its y and z components of velocity do not change in the collision but the x-component reverses sign. That is, the velocity after collision is (-vx, vy, vz ) .

The change in momentum of the molecule is

– m vx – (mvx) = – 2mvx

By the principle of conservation of momentum, the momentum imparted to the wall in the collision = 2mvx

In a small time interval Δt, a molecule with x-component of velocity vx will hit the wall if it is within the distance vx Δt from the wall. That is, all molecules within the volume Avx Δt only can hit the wall in time Δt.

But, on the average, half of these are moving towards the wall and the other half away from the wall. Thus the number of molecules with velocity (vx, vy, vz ) hitting the wall in time Δt is , where n is the number of molecules per unit volume.

The total momentum transferred to the wall by these molecules in time Δt is:

The force on the wall is the rate of momentum transfer = $\frac{\mathrm{Q}}{\mathrm{\Delta }\mathrm{t}}$.

Now, the pressure is force per unit area,

Actually, all molecules in a gas do not have the same velocity; there is a distribution in velocities. The above equation therefore, stands for pressure due to the group of molecules with speed vx in the x-direction and n stands for the number density of that group of molecules. The total pressure is obtained by summing over the contribution due to all groups:

where is the average of .

Since the gas is isotropic, there is no preferred direction of velocity of the molecules in the vessel.

Therefore, by symmetry,

where v is the mean speed and denotes the mean of the squared speed. Thus

Kinetic Interpretation of Temperature

From equation above, we,

where N (= nV) is the number of molecules in the sample.

Since the internal energy E of an ideal gas is purely kinetic,

Combining this equation with the ideal gas Equation

= Average translational energy of a molecule.

Therefore, we can summarise,

• Average translation energy of 1 molecule of gas

• Average translation energy of 1 mole of gas

$\mathrm{E}=\frac{3}{2}\mathrm{R}\mathrm{T}$

• For a given gas kinetic energy

• Root mean square (rms) velocity of the gas molecules is proportional to the square root of absolute temperature and inversely proportional to the molecular mass of the gas.

, and,

For a gas at two different temperatures, T1 & T2,

$\frac{{\left({\mathrm{v}}_{\mathrm{r}\mathrm{m}\mathrm{s}}\right)}_{1}}{{\left({\mathrm{v}}_{\mathrm{r}\mathrm{m}\mathrm{s}}\right)}_{2}}=\sqrt{\frac{{\mathrm{T}}_{1}}{{\mathrm{T}}_{2}}}$

For different gases,

$\frac{{\left({\mathrm{v}}_{\mathrm{r}\mathrm{m}\mathrm{s}}\right)}_{1}}{{\left({\mathrm{v}}_{\mathrm{r}\mathrm{m}\mathrm{s}}\right)}_{2}}=\sqrt{\frac{{\mathrm{M}}_{2}}{{\mathrm{M}}_{1}}}$

Degrees of Freedom

The degrees of freedom for a dynamic system is the number of directions in which it can move freely or the number of coordinates required to describe completely the position and configuration of the system.

Number of Degree of Freedom for Various Types of Gases

1. For monoatomic gas = 3 (only translational, in x y and z directions)

2. For diatomic gas = 5 (3 translational + 2 rotational, explain)

3. For non-linear triatomic gas or diatomic gas with oscillations = 6

4. For linear triatomic gas = 7

Law of Equipartition of Energy

In equilibrium, the total energy is equally distributed in all possible energy modes, with each mode having an average energy equal to $\frac{1}{2}$ kBT. This is known as the law of equipartition of energy.

• Each translational and rotational degree of freedom of a molecule contributes $\frac{1}{2}$ kBT to the energy

• Each vibrational frequency contributes 2 ×$\frac{1}{2}$ kBT = kBT , since a vibrational mode has both kinetic and potential energy modes.

Specific Heat Capacity of Monatomic Gases

The molecule of a monatomic gas has only three translational degrees of freedom. Thus, the average energy of a molecule at temperature T is $\frac{1}{2}$ kBT . The total internal energy of a mole of such a gas is

Molar Specific heat capacity at constant volume

Therefore, γ = $\frac{{\mathrm{C}}_{\mathrm{P}}}{{\mathrm{C}}_{\mathrm{V}}}$=

Specific Heat Capacity of Diatomic gases

Molar Specific heat capacity at constant volume

Therefore, γ = $\frac{{\mathrm{C}}_{\mathrm{P}}}{{\mathrm{C}}_{\mathrm{V}}}$= $\frac{7}{5}$

If there is also a vibrational degree of freedom, then

Therefore, γ = $\frac{{\mathrm{C}}_{\mathrm{P}}}{{\mathrm{C}}_{\mathrm{V}}}$= $\frac{9}{7}$

Specific Heat Capacity of Polyatomic Gases

• 3 translational, 3 rotational and f vibrational degrees of freedom

• At constant volume,

• At constant pressure, Cp = CV + R

Specific Heat Capacity of Solids

There is no translational or rotational degree of freedom, but there are 3 vibrational degrees of freedom with each atom.

An oscillation in one dimension has average energy of 2 × $\frac{1}{2}$ kBT = kBT .

In three dimensions, the average energy is 3 kBT.

The molar energy = U = 3 kBT × NA = 3 RT

Now at constant pressure ΔQ = ΔU + PΔV = ΔU,

[Since for a solid ΔV is negligible]

Hence, Molar heat capacity of a solid = 3R.

Specific Heat Capacity of Water

Water is treated like a solid. For each atom average energy is 3kBT. Water molecule has three atoms, two hydrogen atoms and one oxygen atom. So it has

and

Temperature Dependence of Specific Heat Capacities

The predicted specific heats are independent of temperature at normal temperatures. As we go to low temperatures, however, there is a marked departure from this prediction. Specific heats of all substances approach zero as T→0. This is related to the fact that degrees of freedom get frozen and ineffective at low temperatures. According to classical physics degrees of freedom must remain unchanged at all times. The behavior of specific heats at low temperatures shows the inadequacy of classical physics and can be explained only by invoking quantum considerations, as was first shown by Einstein.

Quantum mechanics requires a minimum, nonzero amount of energy before a degree of freedom comes into play. This is also the reason why vibrational degrees of freedom come into play only in some cases.

Mean Free Path

The average distance travelled by a molecule between two successive collisions is called mean free path (l).

Mean free path is given by

where

d = diameter of the molecule,

p = pressure of the gas,

T = temperature and

kB = Botlzmann constant.

Mean free path is directly proportional to the absolute temperature and inversely proportional to the pressure of the gas.

l ∝ T and

Discussion

The continuous random motion of the particles of microscopic size suspended in air or any liquid, is called Brownian motion.

Consider the molecules of a gas as spheres of diameter d. Focus on a single molecule with the average speed $\stackrel{‾}{\mathrm{v}}$ It will suffer collision with any molecule that comes within a distance d between the centres. In time Δt, it sweeps a volume πd2 $\stackrel{‾}{\mathrm{v}}$ Δt wherein any other molecule will collide with it.

If n is the number of molecules per unit volume, the molecule suffers n πd2 $\stackrel{‾}{\mathrm{v}}$ Δt collisions in time Δt. Thus the rate of collisions is n πd2 $\stackrel{‾}{\mathrm{v}}$ or the time between two successive collisions is on the average, τ = .

The average distance between two successive collisions, called the mean free path l, is

In this derivation, we imagined the other molecules to be at rest. But actually all molecules are moving and the collision rate is determined by the average relative velocity of the molecules. Thus we need to replace by $\stackrel{‾}{{\mathrm{v}}_{\mathrm{r}\mathrm{m}\mathrm{s}}}$.

A more exact treatment gives