Page 18 - Hsslive-Plus-One-Physics-Class-Notes-Part-III (1)

Heat, Thermodynamics & Kinetic Theory {17} Vinodkumar M, St. Aloysius H.S.S, Elthuruth, Thrissur.
m
  2
2  d 
* The smaller the no. of molecules per unit volume of the gas, larger is the mean free path.
* Smaller the diameter, larger is the mean free path.
* Smaller the density, larger is the mean free path.
Degrees of freedon and law of equipartition of energy.

1 2 1 2 1 2
We know that the KE of a single molecule is E  mv  mv y  mv z . For a gas in thermal
t
x
2 2 2
3
equilibrium at a temperature T, the average KE/molecule is K T .
B
2
1 2 1 2 1 2 3
i.e E t  mv x  mv y  mv z  K T
B
2 2 2 2
A molecule free to move in space needs three co-ordinates to specify its location. If it is constrained
to move in a plane, it needs two and if constrained to move along a line, it needs only one co-ordinate to
locate it.
So a molecule has only one degree of freedom for motion in a line, two for motion in a plane and three for
motion in space.
The total number of co-ordinates or independent quantities required to completely specify the
position and configuration of a system is called the degrees of freedom of that system.
So for a mono atomic molecule free to move in space has three translational degrees of freedom.
Certain molecules (O , N ) shows rotational motion also in addition to translational motion. a diatomic
2 2
molecule can’t rotate about the axis connecting the two atoms. Thus the molecule has two rotational degrees
of freedom also. So the total energy of such a molecule is

1 2 1 2 1 2 1 2 1 2 5
E  E  mv  mv  mv  I  1  I  2  K T ; where  and  are
x
y
z
r
1
2
B
t
2 2 2 2 2 2 1 2
the angular speeds and I and I are the corresponding moment of inertia.
1 2
Some molecules like Co has another mode of vibration also and contribute a vibrational energy term to the
total energy.
2
1 1 1 1 1 1  dy  1 7
2
2
2
2
E  E  E  m v  m v  m v  I  2  I  2  m    k y  K T
t r P x y z 1 1 2 2 B
2 2 2 2 2 2 B  dt  2 2
where k is the force constant of the oscillator and y the vibrational co-ordinate.
Each term involving the square of a variable of motion occuring in the expression for energy is a mode
of absorption of energy by the molecule.
According to Maxwell, “in equilibrium, the total energy is equally distributed in all possible energy
1
modes with each mode having an average energy = K T ”. This is known as law of equipartition of energy..
B
2
Specific heat capacity
1. Mono atomic gas
The molecule of a monoatomic gas has three translational degrees of freedom. The average energy of a
3
molecule at temperature T is K T . The total internal energy of one mole of such a gas is
B
2
3 3
U  K T N A  R T
B
2 2
3
U  R T
2