Kinetic Theory of Gases Class 11 Notes Physics

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231
Kinetic Theory Of Gases
9.1 Kinetic Theory of Gases : Assumption
(1) The molecules of a gas are identical, spherical and perfectly elastic point 
masses.
(2) The volume of molecules is negligible in comparison to the volume of gas.
(3) Molecules of a gas moves randomly in all direction.
(4)	 The 	speed 	of 	gas 	molecules 	lie 	between 	zero 	and 	infinity .
(5) Their collisions are perfectly elastic.
(6) The number of collisions per unit volume in a gas remains constant.
(7) No attractive or repulsive force acts between gas molecules.
9.2 Pressure of an ideal Gas
P = 
Relation between pressure and kinetic energy
?   K.E. per unit volume (E) = 
9.3 Ideal Gas Equation
The equation which relates the pressure (P), volume (V) and temperature 
(T) of the given state of an ideal gas is known as gas equation. 
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231
Kinetic Theory Of Gases
9.1 Kinetic Theory of Gases : Assumption
(1) The molecules of a gas are identical, spherical and perfectly elastic point 
masses.
(2) The volume of molecules is negligible in comparison to the volume of gas.
(3) Molecules of a gas moves randomly in all direction.
(4)	 The 	speed 	of 	gas 	molecules 	lie 	between 	zero 	and 	infinity .
(5) Their collisions are perfectly elastic.
(6) The number of collisions per unit volume in a gas remains constant.
(7) No attractive or repulsive force acts between gas molecules.
9.2 Pressure of an ideal Gas
P = 
Relation between pressure and kinetic energy
?   K.E. per unit volume (E) = 
9.3 Ideal Gas Equation
The equation which relates the pressure (P), volume (V) and temperature 
(T) of the given state of an ideal gas is known as gas equation. 
(1) Universal gas constant (R) : Dimension [ML
2
T
–2
?
–1
]
Thus 	 universal	 gas 	 constant 	 signifies 	 the 	 work	 done 	 by	 (or 	 on) 	 a 	 gas 	 per 	
mole per kelvin.
S.T.P value : 8.31 
(2) Boltzman’s constant (k) : Dimension [ML
2
T
–2
?
–1
]
 k = 1.38 × 10
–23
 Joule/kelvin
9.4 Various Speeds of Gas Molecules
(1) Root wean square speed V
rms
 = 
(2) Most probable speed V
mp
 = 
(3) Average speed V
av
 = 
• V
rms
 > V
av
 > V
mp
 (remembering trick) (RAM)
9.5 Kinetic Energy of Ideal Gas
Molecules of ideal gases possess only translational motion. So they possess 
only translational kinetic energy.
Pv = nrT
?
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231
Kinetic Theory Of Gases
9.1 Kinetic Theory of Gases : Assumption
(1) The molecules of a gas are identical, spherical and perfectly elastic point 
masses.
(2) The volume of molecules is negligible in comparison to the volume of gas.
(3) Molecules of a gas moves randomly in all direction.
(4)	 The 	speed 	of 	gas 	molecules 	lie 	between 	zero 	and 	infinity .
(5) Their collisions are perfectly elastic.
(6) The number of collisions per unit volume in a gas remains constant.
(7) No attractive or repulsive force acts between gas molecules.
9.2 Pressure of an ideal Gas
P = 
Relation between pressure and kinetic energy
?   K.E. per unit volume (E) = 
9.3 Ideal Gas Equation
The equation which relates the pressure (P), volume (V) and temperature 
(T) of the given state of an ideal gas is known as gas equation. 
(1) Universal gas constant (R) : Dimension [ML
2
T
–2
?
–1
]
Thus 	 universal	 gas 	 constant 	 signifies 	 the 	 work	 done 	 by	 (or 	 on) 	 a 	 gas 	 per 	
mole per kelvin.
S.T.P value : 8.31 
(2) Boltzman’s constant (k) : Dimension [ML
2
T
–2
?
–1
]
 k = 1.38 × 10
–23
 Joule/kelvin
9.4 Various Speeds of Gas Molecules
(1) Root wean square speed V
rms
 = 
(2) Most probable speed V
mp
 = 
(3) Average speed V
av
 = 
• V
rms
 > V
av
 > V
mp
 (remembering trick) (RAM)
9.5 Kinetic Energy of Ideal Gas
Molecules of ideal gases possess only translational motion. So they possess 
only translational kinetic energy.
Pv = nrT
?
 Here m = mass of each molecule, M = Molecular weight of gas and  
N
A
 – Avogadro number = 6.023 × 10
23
.
9.6 Degree of Freedom
The total number of independent modes (ways) in which a system can possess 
energy is called the degree of freedom (f).
The degree of freedom are of three types :
(i) Translational degree of freedom
(ii) Rotational degree of freedom
(iii) Vibrational degree of freedom
General expression for degree of freedom
 f = 3N – R, where N = Number of independent particles, R = Number of 
independent restriction
(1) Monoatomic gas : It can have 3 degrees of freedom (all translational).
(2) Diatomic gas : A diatomic molecule has 5 degree of freedom : 3 
translational and 2 rotational.
(3) Triatomic gas (Non-linear) : It has 6 degrees of freedom : 3 translational 
and 3 rotational.
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231
Kinetic Theory Of Gases
9.1 Kinetic Theory of Gases : Assumption
(1) The molecules of a gas are identical, spherical and perfectly elastic point 
masses.
(2) The volume of molecules is negligible in comparison to the volume of gas.
(3) Molecules of a gas moves randomly in all direction.
(4)	 The 	speed 	of 	gas 	molecules 	lie 	between 	zero 	and 	infinity .
(5) Their collisions are perfectly elastic.
(6) The number of collisions per unit volume in a gas remains constant.
(7) No attractive or repulsive force acts between gas molecules.
9.2 Pressure of an ideal Gas
P = 
Relation between pressure and kinetic energy
?   K.E. per unit volume (E) = 
9.3 Ideal Gas Equation
The equation which relates the pressure (P), volume (V) and temperature 
(T) of the given state of an ideal gas is known as gas equation. 
(1) Universal gas constant (R) : Dimension [ML
2
T
–2
?
–1
]
Thus 	 universal	 gas 	 constant 	 signifies 	 the 	 work	 done 	 by	 (or 	 on) 	 a 	 gas 	 per 	
mole per kelvin.
S.T.P value : 8.31 
(2) Boltzman’s constant (k) : Dimension [ML
2
T
–2
?
–1
]
 k = 1.38 × 10
–23
 Joule/kelvin
9.4 Various Speeds of Gas Molecules
(1) Root wean square speed V
rms
 = 
(2) Most probable speed V
mp
 = 
(3) Average speed V
av
 = 
• V
rms
 > V
av
 > V
mp
 (remembering trick) (RAM)
9.5 Kinetic Energy of Ideal Gas
Molecules of ideal gases possess only translational motion. So they possess 
only translational kinetic energy.
Pv = nrT
?
 Here m = mass of each molecule, M = Molecular weight of gas and  
N
A
 – Avogadro number = 6.023 × 10
23
.
9.6 Degree of Freedom
The total number of independent modes (ways) in which a system can possess 
energy is called the degree of freedom (f).
The degree of freedom are of three types :
(i) Translational degree of freedom
(ii) Rotational degree of freedom
(iii) Vibrational degree of freedom
General expression for degree of freedom
 f = 3N – R, where N = Number of independent particles, R = Number of 
independent restriction
(1) Monoatomic gas : It can have 3 degrees of freedom (all translational).
(2) Diatomic gas : A diatomic molecule has 5 degree of freedom : 3 
translational and 2 rotational.
(3) Triatomic gas (Non-linear) : It has 6 degrees of freedom : 3 translational 
and 3 rotational.
(4) Tabular display of degree of freedom of different gases
• The above degrees of freedom are shown at room temperature. Further
at high temperature the molecule will have an additional degrees of
freedom, due to vibrational motion.
9.7 Law of Equipartition of Energy
For any system in thermal equilibrium, the total energy is equally distributed 
among its various degree of freedom. And the energy associated with each 
 molecule of the system per degree of freedom of the system is 
9.8 Mean Free Path
The average distance travelled by a gas molecule is known as mean free path.
 Let ?
1
, ?
2
, ?
3 
......... ?
n 
be the distance travelled by a gas molecule during n 
collisions respectively, then the mean free path of a gas molecule is given 
by 
? = 
?
1
 =  where d = Diameter of the molecule, n = Number of molecules 
per unit volume.
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231
Kinetic Theory Of Gases
9.1 Kinetic Theory of Gases : Assumption
(1) The molecules of a gas are identical, spherical and perfectly elastic point 
masses.
(2) The volume of molecules is negligible in comparison to the volume of gas.
(3) Molecules of a gas moves randomly in all direction.
(4)	 The 	speed 	of 	gas 	molecules 	lie 	between 	zero 	and 	infinity .
(5) Their collisions are perfectly elastic.
(6) The number of collisions per unit volume in a gas remains constant.
(7) No attractive or repulsive force acts between gas molecules.
9.2 Pressure of an ideal Gas
P = 
Relation between pressure and kinetic energy
?   K.E. per unit volume (E) = 
9.3 Ideal Gas Equation
The equation which relates the pressure (P), volume (V) and temperature 
(T) of the given state of an ideal gas is known as gas equation. 
(1) Universal gas constant (R) : Dimension [ML
2
T
–2
?
–1
]
Thus 	 universal	 gas 	 constant 	 signifies 	 the 	 work	 done 	 by	 (or 	 on) 	 a 	 gas 	 per 	
mole per kelvin.
S.T.P value : 8.31 
(2) Boltzman’s constant (k) : Dimension [ML
2
T
–2
?
–1
]
 k = 1.38 × 10
–23
 Joule/kelvin
9.4 Various Speeds of Gas Molecules
(1) Root wean square speed V
rms
 = 
(2) Most probable speed V
mp
 = 
(3) Average speed V
av
 = 
• V
rms
 > V
av
 > V
mp
 (remembering trick) (RAM)
9.5 Kinetic Energy of Ideal Gas
Molecules of ideal gases possess only translational motion. So they possess 
only translational kinetic energy.
Pv = nrT
?
 Here m = mass of each molecule, M = Molecular weight of gas and  
N
A
 – Avogadro number = 6.023 × 10
23
.
9.6 Degree of Freedom
The total number of independent modes (ways) in which a system can possess 
energy is called the degree of freedom (f).
The degree of freedom are of three types :
(i) Translational degree of freedom
(ii) Rotational degree of freedom
(iii) Vibrational degree of freedom
General expression for degree of freedom
 f = 3N – R, where N = Number of independent particles, R = Number of 
independent restriction
(1) Monoatomic gas : It can have 3 degrees of freedom (all translational).
(2) Diatomic gas : A diatomic molecule has 5 degree of freedom : 3 
translational and 2 rotational.
(3) Triatomic gas (Non-linear) : It has 6 degrees of freedom : 3 translational 
and 3 rotational.
(4) Tabular display of degree of freedom of different gases
• The above degrees of freedom are shown at room temperature. Further
at high temperature the molecule will have an additional degrees of
freedom, due to vibrational motion.
9.7 Law of Equipartition of Energy
For any system in thermal equilibrium, the total energy is equally distributed 
among its various degree of freedom. And the energy associated with each 
 molecule of the system per degree of freedom of the system is 
9.8 Mean Free Path
The average distance travelled by a gas molecule is known as mean free path.
 Let ?
1
, ?
2
, ?
3 
......... ?
n 
be the distance travelled by a gas molecule during n 
collisions respectively, then the mean free path of a gas molecule is given 
by 
? = 
?
1
 =  where d = Diameter of the molecule, n = Number of molecules 
per unit volume.
235
9.9 Specific heat or Specific Heat Capacity
(1)	 Gram 	 specific 	 heat 	 : 	 It 	 is 	 defined 	 as 	 the 	 amount 	 of 	 heat 	 required 	 to 	 raise 	
the temperature of unit gram mass of the substance by unit degree. Gram 
specific 	heat 	c = 
.
(2)	 Molar 	 specific 	 heat 	 : 	 It 	 is 	 defined 	 as 	 the 	 amount 	 of 	 heat 	 required 	 to 	 raise 	
the temperature of one gram mole of the substance by a unit degree, it 
is represented by capital (C)
C = 
C = Mc = 
9.10 Specific Heat of Gases
(i) In adiabatic process i.e., ?Q = 0,
 ? C =  = 0 i.e., C = 0
(ii) In isothermal process i.e., ?T = 0
? C = i.e., C = 8
Specific 	 heat 	 of 	 gas 	 can 	 have 	 any 	 positive 	 value 	 ranging 	 from 	 zero 	 to 	 infinity . 	
Further 	 it 	 can 	 even 	 be 	 negat ive. 	 Out	 of 	 many 	 values 	 of 	 specific 	 heat 	 of 	 a 	 gas, 	
two	 are 	of 	special 	significance.
(1)	 Specific 	 heat 	 of 	 a	 gas 	 at 	 constant 	 volume 	 (C
v
) : 	 It 	 is 	 defined 	 as 	 the 	
quantity of heat required to raise the temperature of unit mass of gas 
through 1 K when its volume is kept constant.
(2)	 Specific 	 heat 	 of 	 a	 gas 	 at 	 constant 	 pr essur e	 (C
p
) : It 	 is 	 defined 	 as 	 the 	
quantity of heat required to raise the temperature of unit mass of gas 
through 1 K when its pressure is kept constant.
9.11 Mayer’s Formula
C
p
 – C
v
 = R
This relation is called Mayer’s formula and shows that C
p
 > C
v
 i.e., molar 
specific 	heat 	at 	constant 	pressure 	is 	greater than that at constant volume.