Question

Determine the value of $ {C_p},{C_v} $ and $ \gamma $ for a monatomic, diatomic and polyatomic gas.

Answer 1

Hint: In a thermodynamic system, $ {C_p} $ is the amount of heat energy absorbed or released by a unit mass of a given substance with the change in temperature at a constant pressure. $ {C_v} $ is the amount of heat energy transferred between a system and its surroundings at a constant volume. $ \gamma $ is the ratio of $ {C_p} $ and $ {C_v} $

Complete answer:
The degree of freedom of a molecule will affect the value of all the specific heat constants.
For any molecule the degrees of freedom is given by the equation:
 $ f = 3N - m $
Where f is the degree of freedom
N is the number of particles in the given system/molecule
m is the number of constraints.
So for the given type of molecules we can say that:

Type of molecule	Degrees of freedom
Monoatomic molecule	3 (N=1, m=0)
Diatomic molecule	5 (N=2, m=1)
Polyatomic molecule	>6

Now we can say that the specific heat constants are related to the degree of freedom by the following equations:
 $ \Rightarrow {C_p} = R(1 + \dfrac{1}{2}f) $
 $ \Rightarrow {C_v} = \dfrac{1}{2}f \times R $
 $ \Rightarrow \gamma = 1 + \dfrac{2}{f} $
Therefore we can substitute the values of f in the equations and obtain the value of all the constants for monoatomic, diatomic and polyatomic gas.

Type of Gas molecule	Degree of freedom f	$ {C_p} $	$ {C_v} $	$ \gamma $
Monoatomic	3	$ \dfrac{5}{2}R $	$ \dfrac{3}{2}R $	$ \dfrac{5}{3} $
Diatomic	5	$ \dfrac{7}{2}R $	$ \dfrac{5}{2}R $	$ \dfrac{7}{5} $
Triatomic (A case of polyatomic)	7 (where N=3 and m=2)	$ \dfrac{9}{2}R $	$ \dfrac{7}{2}R $	$ \dfrac{9}{7} $

Note:
For gas with molecules having higher numbers of atomicity, the degree of freedom would vary even more because there are a large number of arrangements that are possible. This will mean that there will be more degrees of freedom with each type of molecule. Thus for such types of gases a known degree of freedom can be used to figure out the specific molar heats and their ratios, which will always be decreasing with increasing atomicity of the molecule.