Question

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

Answer 1

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.