Kinetic theory is one of the most important chapters from the IIT- JEE & CBSE perspective. So make make sure that you go through these notes to score the most.

**(a) Solids:-** It is the type of matter which has got fixed shape and volume. The force of attraction between any two molecules of a solid is very large.

**(b) Liquids:-** It is the type of matter which has got fixed volume but no fixed shape. Force of attraction between any two molecules is not that large as in case od solids.

**(c) Gases:-** It is the type of matter does not have any fixed shape or any fixed volume.

**Ideal Gas:-** A ideal gas is one which has a zero size of molecule and zero force of interaction between its molecules.

**Ideal Gas Equation:-** A relation between the pressure, volume and temperature of an ideal gas is called ideal gas equation.

PV/T = Constant or PV = nRT

Here, n is the number of moles and R is the universal gas constant.

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**Gas Constant:-**

** (a) Universal gas constant (R):- **

R= P0 V0/T0

=8.311 J mol^{-1}K^{-1}

**(b) Specific gas constant (r):-**

PV= (R/M) T = rT,

Here, r = R/M

**Real Gas:-**The gases which show deviation from the ideal gas behavior are called real gas.

**Vander wall’s equation of state for a real gas:- **

Here n is the number of moles of gas.

**Avogadro’s number (N):-** Avogadro’s number (N), is the number of carbon atoms contained in 12 gram of carbon-12.

N = 6.023×10^{23}

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**(a) To calculate the mass of an atom/molecule:-**

Mass of one atom = atomic weight (in gram)/N

Mass of one molecule = molecular weight (in gram)/N

**(b) To calculate the number of atoms/molecules in a certain amount of substance:- **

Number of atoms in m gram = (N/atomic weight)×m

Number of molecules in m gram = (N/molecular weight)×m

**(b) To calculate the number of atoms/molecules in a certain amount of substance:- **

Number of atoms in m gram = (N/atomic weight)×m

Number of molecules in m gram = (N/molecular weight)×m

**(c) Size of an atom:- **

Volume of the atom, V = (4/3)πr^{3}

Mass of the atom, m = A/N Here, A is the atomic weight and N is the Avogadro’s number.

Radius, r =[3A/4πNρ]^{1/3\}

Here ρ is the density.

**Gas laws:-**

(a) Boyle’s law:- It states that the volume of a given amount of gas varies inversely as its pressure, provided its temperature is kept constant.

PV = Constant

(b) Charlers law or Gey Lussac’s law:- It states that volume of a given mass of a gas varies directly as its absolute temperature, provided its pressure is kept constant.

V/T= Constant

V–V_{0}/V_{0}t = 1/273 = γ_{p}

Here γ_{p}(=1/273) is called volume coefficient of gas at constant pressure.

Volume coefficient of a gas, at constant pressure, is defined as the change in volume per unit volume per degree centigrade rise of temperature.

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**(c) Gay Lussac’s law of pressure:-** It states that pressure of a given mass of a gas varies directly as its absolute temperature provided the volume of the gas is kept constant.

P/T = P_{0}/T_{0} or P – P_{0}/P_{0}t = 1/273 = γ_{p}

Here γ_{p} (=1/273) is called pressure coefficient of the gas at constant volume. Pressure coefficient of a gas, at constant volume, is defined as the change in pressure per unit pressure per degree centigrade rise of temperature.

**(d) Dalton’s law of partial pressures:-** Partial pressure of a gas or of saturated vapors is the pressure which it would exert if contained alone in the entire confined given space.

P= p_{1}+p_{2}+p_{3}+……..

nRT/V = p_{1}+p_{2}+p_{3}+……..

**(e) Grahm’s law of diffusion:-** Grahm’s law of diffusion states that the rate of diffusion of gases varies inversely as the square root of the density of gases.

R∝1/√ρ or R_{1}/R_{2} =√ρ_{2}/ ρ_{1}

So, a lighter gas gets diffused quickly.

**(f) Avogadro’s law:-** It states that under similar conditions of pressure and temperature, equal volume of all gases contain equal number of molecules.

For m gram of gas, PV/T = nR = (m/M) R

Pressure of a gas (P):- P = 1/3 (M/V) C^{2} = 1/3 (ρ) C^{2}

** Root mean square (r.m.s) velocity of the gas:-** Root mean square velocity of a gas is the square root of the mean of the squares of the velocities of individual molecules.

C= √[c1^{2}+ c2^{2}+ c3^{2}+…..+ cn^{2} ]/n = √3P/ ρ

**Pressure in terms of kinetic energy per unit volume:-** The pressure of a gas is equal to twothird of kinetic energy per unit volume of the gas.

P= 2/3 E

**Kinetic interpretation of temperature:-** Root mean square velocity of the molecules of a gas is proportional to the square root of its absolute temperature.

C= √3RT/M

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

At, T=0, C=0

Thus, absolute zero is the temperature at which all molecular motion ceases.

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**Kinetic energy per mole of gas:- **

K.E. per gram mol of gas = ½ MC^{2} = 3/2 RT

**Kinetic energy per gram of gas:-**

½ C^{2} = 3/2 rt

Here, ½ C^{2} = kinetic energy per gram of the gas and r = gas constant for one gram of gas.

** Kinetic energy per molecule of the gas:-**

Kinetic energy per molecule = ½ mC^{2} = 3/2 kT

Here, k (Boltzmann constant) = R/N

Thus, K.E per molecule is independent of the mass of molecule. It only depends upon the absolute temperature of the gas.

**Regnault’s law:-** P∝T Graham’s law of diffusion:- R_{1}/R_{2} = C_{1}/C_{2} = √ρ_{2}/ ρ_{1}

**Distribution of molecular speeds:-**

**(a) Number of molecules of gas possessing velocities between v and v+dv**

**(b) Number of molecules of gas possessing energy between u and u+dv:-**

**(c) Number of molecules of gas possessing momentum between p and p+dp :-**

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**(d) Most probable speed:-** It is the speed, possessed by the maximum number of molecules of a gas contained in an enclosure.

V_{m}= √[2kT/m]

** (e) Average speed (Vav):-** Average speed of the molecules of a gas is the arithmetic mean so the speeds of all the molecules.

V_{av}= √[8kT/πm]

** (f) Root mean square speed (Vrms):-** It is the square root of the mean of the squares of the individual speeds of the molecules of a gas.

V_{rms} = √[3kT/m]

**V _{rms} > V_{av} > V_{m} **

**Degree of Freedom (n):-** Degree of freedom, of a mechanical system, is defined as the number of possible independent ways, in which the position and configuration of the system may change.

In general, if N is the number of particles, not connected to each other, the degrees of freedom n of such a system will be,

n = 3N

If K is the number of constraints (restrictions), degree of freedom n of the system will be,

n = 3N –K

**Degree of freedom of a gas molecule:-**

**(a) Mono-atomic gas:- Degree of freedom of monoatomic molecule, n = 3 **

**(b) Di-atomic gas:-**

At very low temperature (0-250 K):- Degree of freedom, n = 3

At medium temperature (250 K – 750 K):- Degree of freedom, n = 5 (Translational = 3, Rotational = 2)

At high temperature (Beyond 750 K):- Degree of freedom, n = 6 (Translational = 3, Rotational = 2, Vibratory =1), For calculation purposes, n = 7

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**Law of equipartition of energy:-** In any dynamical system, in thermal equilibrium, the total energy is divided equally among all the degrees of freedom and energy per molecule per degree of freedom is ½ kT.

E = ½ kT

**Mean Energy:-** K.E of one mole of gas is known as mean energy or internal energy of the gas and is denoted by U.

U = n/2 RT

Here n is the degree of freedom of the gas.

(a) Mono-atomic gas(n = 3):- U = 3/2 RT

(b) Diatomic gas:-

At low temperature (n=3):- U = 3/2 RT

At medium temperature (n=5):- U = 5/2 RT

At high temperature (n=7):- U = 7/2 RT

**Relation between ratio of specific heat capacities (γ) and degree of freedom (n):-**

γ = C_{p}/C_{v} = [1+(2/n)]

(a) For mono-atomic gas (n=3):- γ = [1+(2/n)] = 1+(2/3) = 5/3=1.67

(b) For diatomic gas (at medium temperatures (n=5)):- γ = [1+(2/5)] = 1+(2/5) = 7/5=1.4

(c) For diatomic gas (at high temperatures (n=7)):- γ = [1+(2/7)] = 9/7 = 1.29

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