15.law of Equipartition of Energy
15.law of Equipartition of Energy
15.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
1
system per degree of freedom of the system is kT .
2
where k 1.38 10 23 J / K , T = absolute temperature of the
system.
If the system possess degree of freedom f then
f
Total energy associated with each molecule kT
2
f
Total energy associated with N molecules N kT
2
f
Total energy associated with each mole RT
2
f
Total energy associated with mole RT
2
f
Total energy associated with each gram rT
2
f
Total energy associated with M0 gram M0 rT
2
ILLUSTRATIONS
Ex1.Energy of all molecules of a monoatomic gas having a volume V and pressure P is
3
PV . The total translational kinetic energy of all molecules of a diatomic gas as the
2
same volume and pressure is
f f
Sol. Energy of 1 mole of gas RT PV where f = Degree of freedom
2 2
Monoatomic or diatomic both gases posses equal degree of freedom for translational
motion and that is equal to 3 i.e. f = 3
3
E PV
2
3
Although total energy will be different,For monoatomic gas E total PV [As f = 3]
2
5
For diatomic gas E total PV [As f = 5]
2
Ex2.The temperature of argon, kept in a vessel is raised by 1C at a constant volume.
The total heat supplied to the gas is a combination of translational and rotational
energies. What is their respective share?
Sol. As argon is a monoatomic gas therefore its molecule will possess only
translatory kinetic energy i.e. the share of translational and rotational energies will
be 100% and 0% respectively.
f
Sol. Mean kinetic energy for mole gas . RT
2
7 m 7 1 7 7
E RT NkT NkT NkT [As f = 7 and M = 44 for CO 2 ]
2 M 2 44 2 88
Ex4: At standard temperature and pressure the density of a gas is 1.3 gm/ m3 and
the speed of the sound in gas is 330 m/sec. TFind the degree of freedom of the gas.
m kg
Sol. Given velocity of sound v s 330 , Density of gas 1 .3 , Atomic
sec m3
N
pressure P 1.01 10 5
m2
P
Substituting these value in v sound we get 1.41
2 2 2
Now from 1 we get f 5.
f 1 1 .4 1
Ex5. A gas molecule has 3 translational degree of freedom and 2 rotational degree of
freedom. What is its total energy?
Sol. Total energy= total translational energy +total rotational energy
1 1 5
= 3 kT 2 kT kT
2 2 2
EXERCISES
Q1. Mean kinetic energy per degree of freedom of gas molecules is
3 1 3
(a) KT (b) KT (c) KT (d) RT
2 2 2
Sol.(c) *refer text
Q2.The translatory kinetic energy of a gas per gm is
3 RT 3 RT 3
(a) (b) (c) RT (d) 3 NKT
2 N 2 M 2 2
Sol.(b)*refer text
Q3.A monoatomic gas molecule has
(a)Three degrees of freedom
(b)Four degrees of freedom
(c)Five degrees of freedom
(d)Six degrees of freedom
Sol. (a)
A monoatomice gas molecule has three translational degrees of freedom.
Q4.The degrees of freedom of a triatomic gas is
(a) 2 (b) 4 (c) 6 (d) 8
Sol.(c)
A triatomic gas molecule has 3translational and 3 rotational degrees of freedom.
Q6.A polyatomic gas with n degrees of freedom has a mean energy per molecule
given by
(a) nkT / N A (b) nkT / 2 N A (c) nkT/2 (d) 3kT/2
Sol.(c)
*refer text
Q7.The average thermal energy of a helium atom at room temperature (27 C) is