Q.NO
CBSE;Class XI;Physics;Kinetic Theory;Kinetic Theory Specific Heat
Capacity
Question
Solution
The specific heat capacity of a substance is defined as the amount
of heat energy required per unit mass to change its temperature by
one unit. That is,
1 𝑑𝑄
𝑆=
Where, “m” is the mass of the substance and “dQ” is the
𝑚 𝑑𝑇
heat absorbed for a small change “dT” in the temperature.
The amount of heat energy required per one mole of substance to
change its temperature by one unit is called molar specific heat
capacity.
1 𝑑𝑄
𝐶 = 𝑛 𝑑𝑇 , Where “n” is the number of moles of the substance.
According to the First Law of Thermodynamics
Q = U + PdV ,Where delta Q is the heat energy exchanged, U is
the change in internal energy, P is the pressure and d V is the
change in volume
The molar specific heat capacity at a constant volume, C v, is
1 dQ
CV = (
)
n dT V
The subscript “v” on the right side indicates that the heat is
absorbed at a constant volume.
When the volume is constant the work done
P d V = 0 and the heat energy exchanged is equal to the change in
the internal energy.
Q = V
Then the specific heat capacity at a constant volume is
1 dU
CV = (
) − − − − − −(i)
n dV V
Then the molar specific heat capacity at a constant pressure can be
found by using the relationship between C p and C v. That is,
Cp − CV = R  Cp = CV + R − − − (ii)
By using the above relationship, we can find the specific heat
capacities of a mono atomic gas.
In the case of a monatomic gas, a molecule has three degrees of
freedom. The average energy of the molecule according to the Law
of equipartition of energy is
3
Average energy of a molecule in mono atomic gas = k B T
2
One mole of gas contains Avogadro number of molecules.
One mole of gas = NA of molecules
Then the average energy of n moles of monatomic gas:
3
𝑛 × 𝑁𝐴 × 2 k B T
This energy is equal to the internal energy of the gas.
Since the product of the Avogadro number, N A, and Boltzmann’s
constant, k B, is universal gas constant, R
3
U = 2 nRT
According to Equation 1, the molar specific heat capacity of a
monatomic gas at a constant volume, CV = (
1 dU
)
n dV V
By substituting the internal energy of a monatomic gas in the above
equation, we get the specific heat capacity at a constant volume,
3
1 d (2 nRT)
3
CV = [
]= R
n
dT
2
As per Equation 2,
3
5
Cp = CV + R  Cp = 2 R + R  Cp = 2 R,
As Gamma is the ratio of the specific heat capacity, at a constant
pressure, C p, and the specific heat capacity at a constant volume, C
v.
5
R
CP
5
=
  = 2   =   = 1.67, Where 
3
CV
3
2R
= ratio of the heat capacities
Specific heat capacities of a diatomic gas
In the first case, we will consider that the molecule is not vibrating
and that it has only translational and rotational degrees of freedom.
Then the number of degrees of freedom is five.
According to the Law of Equipartition of Energy, the average
5
energy of a molecule is 2 k B T
As one mole of gas contains Avogadro number of molecules
One mole of gas = NA of molecules
Average energy of ‘n’ moles of a diatomic gas:
5
𝑛 × 𝑁𝐴 × 𝐾𝐵 𝑇
2
This energy is equal to the internal energy of the gas.
Since the product of the Avogadro number, N A, and Boltzmann’s
constant, k B, is equal to the universal gas constant, ‘R’
𝑈=
5
2
𝑛𝑅𝑇
The molar specific heat capacity of a diatomic gas at a constant
volume, C v, is
1 𝑑𝑈
𝐶𝑉 = (
) − − − − − − − − − − − (𝑖)
𝑛 𝑑𝑇 𝑉
By substituting the internal energy of a diatomic gas in the above
equation, we get the specific heat capacity at a constant volume, C
v,
5
1 d (2 nRT)
5
CV =
= R
n
dT
2
From Equation 2,
5
7
Cp = CV + R  Cp = R + R  Cp = R
2
2
Thus,
7
R
CP
7
=
  = 2   =   = 1.4, Where 
5
CV
5
2R
= ratio of the heat capacities
Hence, the ratio of specific heat capacities for a diatomic gas, when
the molecule is not vibrating, is 1.4.
Specific heat capacities of Diatomic molecule that has
vibrational motion
Now in the second case, we will consider a diatomic molecule that
has vibrational motion as well. The number of degrees of freedom
in such a case is seven. Then according to the Law of Equipartition
of Energy, the average energy of a molecule
One mole of gas contains Avogadro number of molecules.
Then the average energy of n moles of a diatomic gas is
This energy is equal to the internal energy of the gas.
Since the product of the Avogadro number, N A, and Boltzmann’s
constant, k B, is equal to the universal gas constant, R, we get
The molar specific heat capacity of a diatomic gas at a constant
volume, C v, is
1 𝑑𝑈
𝐶𝑉 = (
)
𝑛 𝑑𝑇 𝑉
By substituting the internal energy of a diatomic gas in the above
equation, we get the specific heat capacity at a constant volume, C
v,
7
1 𝑑 (2 𝑛𝑅𝑇)
7
𝐶𝑉 =
= 𝑅
𝑛
𝑑𝑇
2
From the relationship among C p, C v and R,
7
9
Cp = CV + R  Cp = 𝑅 + 𝑅  Cp = 𝑅
2
2 9
C
Thus, substituting the values,  = CP   =
V
R
2
7
R
2
9
 = 7   =
1.29
Hence, the ratio of specific heat capacities of a diatomic gas, when
the molecule is vibrating, is 1.29.
Specific heat capacities of polyatomic gases
In the case of polyatomic gases, the molecule has three translational
degrees of freedom and three rotational degrees of freedom and a
certain number (2f) of vibration degrees of freedom since the
molecule can move in the space in three dimensions and can rotate
about the three axes of rotation. And for “f” modes of vibrational
motion, we have f degrees of freedom associated with elastic
potential energy and f degrees of freedom due to the kinetic energy
of vibration, which totals “2f” degrees of freedom.
One mole of gas contains Avogadro number of molecules.
According to the Law of Equipartition of Energy, the aaverage
energy of a molecule of a polyatomic gas:
(6+2f)
= 2 k B T = (3 + f)k B T
If we have “n” moles of gas, the average energy of all the molecules
of the gas is equal to the internal energy of the gas.
Average energy of n molecules = n × NA (3 + f)k B T
Average energy = Internal energy and the product of the Avogadro
number, N A, and Boltzmann’s constant, k B, is equal to the
universal gas constant, R, we get
U = (3 + f)nRT
By substituting the internal energy of a polyatomic gas in the above
equation, we get the specific heat capacity at a constant volume, C
v,
Molar specific heat capacity of poly atomic gas at constant volume
1 dU
= CV = ( n dT)
V
1 d((3 + f)nRT)
𝐶𝑉 = (
) = (3 + 𝑓)𝑅
𝑛
𝑑𝑇
𝑉
From the relationship between C p, C v and R, C p is equal to C v
plus R.
Cp = CV + R  Cp = (3 + 𝑓)𝑅 + R = (4 + f)R
Thus, gamma is equal to C p by C v or 4 plus f whole into R by 3
plus f whole into R.
Gama is equal to C p by C v.
Thus,
C
Substituting the values,  = CP   =
V
(4+f)R
(3+𝑓)𝑅
Hence, the ratio of specific heat capacities of a polyatomic gas is
(4 + 𝑓)
 =
(3 + 𝑓)
At ordinary temperatures, the predicted values of CP and CV are in
good agreement with the actual values, according to the Law of
Equipartition of Energy.
But at higher temperatures, there are some polyatomic gases such as
ethane and methane for which the predicted values are not the same
as the actual values. In such cases, we should include the degrees of
freedom in the vibration mode.
Specific heat capacity of solids
In the case of solids molecules don’t have translational motion and
rotational motion.
The molecules simply vibrate in the three dimensions.
When a molecule is vibrating, it has two degrees of freedom for
each mode, one associated with potential energy and the other
associated with kinetic energy.
Then, according to the Law of Equipartition of Energy, the average
energy of each molecule for each mode is
1
2 × 2 kBT
Then in three dimensions, the total average energy of the vibrating
molecule of a solid is
1
3 × 2 kBT
For one mole of a solid, the number of molecules is equal to the
Avogadro number and the average energy is
NA × 3k B T
Since the product of the Avogadro number, N A, and Boltzmann’s
constant, k B, is equal to the universal gas constant, R, we get the
total average energy of the molecules as:
3RT.
This average energy of the molecules is equal to the internal energy
of the molecules.
The change in the volume of solids is negligible, and hence the heat
energy exchanged is equal to the change in the internal energy
Then the specific heat capacity of the solid can be found by
substituting the internal energy in the specific heat capacity
formula,
Average energy = Internal energy Q = U
U
3RT
Specific heat capacity of solid,
C=
C =
 C = 3R
T
T
In the case of solids, the predicted values are in agreement with the
actual values of the solids.
Specific heat capacity of water
To calculate the specific heat capacity of water, we will treat water
as a solid. A water molecule has three atoms and every atom has an
average energy of 3 k B T. The total energy of the water molecule is
equal to 9 k B T.
The internal energy of one mole of water is:
NA × 9k B T  U = 9RT, Because NA × k B = R
 𝑈 = NA × 9 k B T  U = 9RT
By substituting the value of internal energy in the specific heat
capacity formula, it is equal to 9 R. This value is in agreement with
the observed value.
That is, one mole of water contains 18 grams the observed value of
the specific heat capacity of water is 4.2 joules per gram per kelvin.
So the molar specific heat capacity of water is approximately equal
to 75 joules per mole per kelvin.
That is, the predicted value and the observed value are the same.
All these predictions of specific heat capacities are based on the
classical Law of Equipartition of Energy the specific heat capacities
are independent of temperature.
But practically this is not true. As the absolute temperature tends
towards zero, the specific heat capacity of all substances approaches
zero.
This is because at low temperatures, the degrees of freedom get
frozen.
This was explained by quantum mechanics because it requires
minimum non-zero energy before the degrees of freedom comes
into play.