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.
Pv = nrT
◻
(1) Universal gas constant (R) : Dimension [ML2T–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 [ML2T–2–1]
k = 1.38 × 10–23 Joule/kelvin
9.4 Various Speeds of Gas Molecules
(1) Root wean square speed Vrms =
(2) Most probable speed Vmp =
(3) Average speed Vav =
• Vrms > Vav > Vmp (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.
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Here m = mass of each molecule, M = Molecular weight of gas and
NA – Avogadro number = 6.023 × 1023.
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.
Kinetic Theory Of Gases 287
(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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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 =
Specific heat of gas can have any positive value ranging from zero to infinity.
Further it can even be negative. Out of many values of specific heat of a gas,
two are of special significance.
(1) Specific heat of a gas at constant volume (Cv) : 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 pressure (Cp) : 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
Cp – Cv = R
This relation is called Mayer’s formula and shows that Cp > Cv i.e., molar
specific heat at constant pressure is greater than that at constant volume.
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9.12 Specific Heat in Terms of Degree of Freedom
Specific heat and kinetic energy for different gases
Monoatomic Diatomic Triatomic Triatomic
non-linear linear
Atomicity A 1 2 3 3
Restriction B 0 1 3 2
Degree of f = 3A – B 3 5 6 7
freedom
Molar
specific
heat at
3R
constant
volume
Molar Cp =
specific
heat at
constant 4R
pressure
Ratio of Cp
and Cv
Kinetic
energy of 3RT
1 mole
Kinetic
energy of 1 3kT
molecule
Kinetic 3rT
energy of
1 gm
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