TS Inter 1st Year Physics Notes Chapter 14 Kinetic Theory

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 × 10²³ 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 = P₁ + P₂ +…. where P is total pressure
P₁, P₂, …… 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 = \(\frac{1}{3}\) nmV̄²
Where V̄² 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(\(\frac{1}{2}\) nmV̄²) = \(\frac{3}{2}\) K_BNT
or Average kinetic energy of gas molecule
\(\frac{E}{N}=\frac{3}{2}\) K_BT

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_BT.
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 > = \(\frac{1}{2}\)mV²_x + \(\frac{1}{2}\)mV²_y + \(\frac{1}{2}\)mV²_z = \(\frac{3}{2}\)K_BT
But we assume that molecule is free to move equally in all possible directions
∴ \(\frac{1}{2}\)mV²_x = \(\frac{1}{2}\)mV²_y = \(\frac{1}{2}\)mV²_z = \(\frac{1}{2}\)K_BT
∴ Average kinetic energy for each translational degree of freedom is \(\frac{1}{2}\)K_BT .

→ 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_BT.
∴ U = 3K_BT × 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_BNT
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 = \(\frac{1}{3}\) nmv² where n is number density of molecules, m is the mass of the molecule and v² is the mean of squared speed.

→ Kinetic interpretation of temperature is,
\(\frac{1}{3}\) nmv² = \(\frac{3}{2}\) k_BT
v_rms = (v²)^{\(\frac{1}{2}\)} = \(\sqrt{\frac{3 K_B T}{m}}\)

→ Translational kinetic energy, E = \(\frac{3}{2}\) k_BNT

→ 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 c₁, c₂, c₃ …… 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 .