Here students can locate TS Inter 1st Year Physics Notes 14th Lesson Kinetic Theory to prepare for their exam.

## TS Inter 1st Year Physics Notes 14th Lesson Kinetic Theory

→ Avogadro’s law: Equal volumes of all gases at equal temperature and pressure have the same number of molecules.

→ Avogadro number (N_{A}) : At S. T.P 22.4 liters of any gas contains 6.02 × 10^{23} atoms. This is known as “Avogadro’s number (N_{A})”.

→ Mean free path : The average distance that a gas molecule can travel without colliding is called “mean free path”.

→ Dalton’s law of partial pressures : For a mixture of non interacting ideal gases at same temperature and volume total pressure in the vessel is the sum of partial pressures of individual gases.

i. e., P = P_{1} + P_{2} +…. where P is total pressure

P_{1}, P_{2}, …… etc. are individual pressures of each gas.

→ Assumptions of kinetic theory :

- Gas is a collection of large number of mole-cules.
- Gas molecules are always in random motion.
- The interaction between gas molecules is negligible.
- They will always move in straight lines.
- Molecules collide with each other and also with walls of the container.
- These collisions are considered as totally elastic collisions.
- During collisions kinetic energy and momentum are totally conserved.

Note: From kinetic theory pressure of ideal gas P = \(\frac{1}{3}\) nmV̄^{2}

Where V̄^{2} denotes the mean of the squared speed.

→ Average kinetic energy of gas molecules: Internal energy E’ of an ideal gas is purely kinetic.

∴ E = N(\(\frac{1}{2}\) nmV̄^{2}) = \(\frac{3}{2}\) K_{B}NT

or Average kinetic energy of gas molecule

\(\frac{E}{N}=\frac{3}{2}\) K_{B}T

Note: Average kinetic energy of gas mole cule is proportional to absolute temperature

\(\frac{E}{N}\) ∝ T

→ Law of equipartition of energy : The total energy of a gas is equally distributed in all possible energy modes, with each mode having an average energy equal to \(\frac{1}{2}\) K_{B}T.

This is known as “law of equipartition of energy.”

Explanation: A gas molecule is free to move in space in all t he three directions (x, y & z). At a given temperature T the average kinetic energy

< Er > = \(\frac{1}{2}\)mV^{2}_{x} + \(\frac{1}{2}\)mV^{2}_{y} + \(\frac{1}{2}\)mV^{2}_{z} = \(\frac{3}{2}\)K_{B}T

But we assume that molecule is free to move equally in all possible directions

∴ \(\frac{1}{2}\)mV^{2}_{x} = \(\frac{1}{2}\)mV^{2}_{y} = \(\frac{1}{2}\)mV^{2}_{z} = \(\frac{1}{2}\)K_{B}T

∴ Average kinetic energy for each translational degree of freedom is \(\frac{1}{2}\)K_{B}T .

→ Specific heat predictions (From law of equipartition of energy)

→ Solids : In a solid the atoms are free to vibrate in all three dimensions. Energy for each degree of freedom of vibration is K_{B}T.

∴ U = 3K_{B}T × N_{A} = 3RT

∴ Specific heat C = \(\frac{\mathrm{dU}}{\mathrm{dT}}\) = 3R

→ Specific heat of water: Water (H20) contains three atoms. So specific heat of water

U = 3 × 3RT = 9RT

∴ Specific heat of water ^ ^

= 9 × 8.31 = 75 Jmol^{-1} K^{-1}

Note: Specific heat predictions from law of equipartition of energy are not applicable at low temperature because at nearly ‘O’ kelvin the degrees of freedom gets frozen.

→ Ratio of specific heats of gas (γ ): In gases the ratio of molar specific heats of a gas \(\frac{C_P}{C_V}\) is called “ratio of specific heats” (γ).

∴ γ = \(\frac{C_P}{C_V}\)

- For monoatomic gas γ = \(\frac{5}{3}\) = 1.66̄ =1.67
- For diatomic gas γ = \(\frac{7}{5}\) = 1.4
- For tri or polyatomic gas γ =1.33.

→ The ideal gas equation connecting pressure (P), volume (V) and absolute temperature (T) is

PV = µRT = K_{B}NT

Where µ is the number of moles and N is the number of molecules. R and K_{B} are universal constants.

R = 8.314 J mol^{-1} K^{-1}; K_{B} = \(\frac{\mathrm{R}}{\mathrm{N}_{\mathrm{A}}}\)

= 1.38 × 1o^{-23} JK^{-1}

→ Kinetic theory of an ideal gas gives the relation, P = \(\frac{1}{3}\) nmv^{2} where n is number density of molecules, m is the mass of the molecule and v^{2} is the mean of squared speed.

→ Kinetic interpretation of temperature is,

\(\frac{1}{3}\) nmv^{2} = \(\frac{3}{2}\) k_{B}T

v_{rms} = (v^{2})^{\(\frac{1}{2}\)} = \(\sqrt{\frac{3 K_B T}{m}}\)

→ Translational kinetic energy, E = \(\frac{3}{2}\) k_{B}NT

→ Mean free path, l = \(\frac{1}{\sqrt{2} \mathrm{n} \pi \mathrm{d}^2}\)

Where n is the number density and d is the diameter of the molecule.

→ Root mean square (rms) speed of a gas at temperature ‘T’ is, c_{rms} = \(\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}}\)

Where ‘M’ is the molecular weight of molar mass of the gas.

→ If n’ molecules of a gas have speeds c_{1}, c_{2}, c_{3} …… c_{n} respectively then rms speed is given by,

c_{rms} = \(\sqrt{\frac{c_1^2+c_2^2+c_3^2+\ldots \ldots \ldots \ldots+c_n^2}{n}}\)

→ If a gas has ‘f degrees of freedom then,

γ = \(\frac{c_p}{c_v}\) = 1 + \(\frac{2}{f}\)

→ The relation between rms velocity and absolute temperature of a gas is c ∝ √T .