State the law of equipartition of energy. Using this law find the ratio of Cp and Cy for monoatomic gas and diatomic gas.
Answers
Answer:
The Law of equipartition of energy states that for a dynamical system in thermal equilibrium the total energy of the system is shared equally by all the degrees of freedom. The energy associated with each degree of freedom per molecule is
2
1
kT, where k is the Boltzmann’s constant.
For example, for a monoatomic molecule, each molecule has 3 degrees of freedom. According to the kinetic theory of gases, the mean kinetic energy of a molecule is
2
3
kT.
Specific heat capacity of Monatomic gas:
The molecules of a monatomic gas have 3 degrees of freedom.
The average energy of a molecule at temperature T is
2
3KBT
The total internal energy of a mole is:
2
3KBT
×N
A
The molar specific heat at constant volume C
v
is
For an ideal gas,
C
v
(monatomic gas) =
dT
dU
=
2
3RT
For an ideal gas, C
p
−C
v
=R
where C
p
is molar specific heat at constant pressure.
Thus, C
p
=
2
5R
Specific heat capacity of Diatomic gas:
The molecules of a monatomic gas have 5 degrees of freedom, 3 translational, and 2 rotational.
The average energy of a molecule at temperature T is
2
5K
B
T
The total internal energy of a mole is:
2
5K
B
T
×N
A
The molar specific heat at constant volume C
v
is
For an ideal gas,
Cv (monatomic gas) =
dT
dU
=
2
5RT
For an ideal gas, C
p
−C
v
=R
where C
p
is molar specific heat at constant pressure.
Thus, C
p
=
2
7R