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:
 $\Rightarrow C_p=R(1+{\displaystyle \frac{1}{2}}f)$
 $\Rightarrow C_v={\displaystyle \frac{1}{2}}f\times R$
 $\Rightarrow \gamma =1+{\displaystyle \frac{2}{f}}$
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	${\displaystyle \frac{5}{2}}R$	${\displaystyle \frac{3}{2}}R$	${\displaystyle \frac{5}{3}}$
Diatomic	5	${\displaystyle \frac{7}{2}}R$	${\displaystyle \frac{5}{2}}R$	${\displaystyle \frac{7}{5}}$
Triatomic (A case of polyatomic)	7 (where N=3 and m=2)	${\displaystyle \frac{9}{2}}R$	${\displaystyle \frac{7}{2}}R$	${\displaystyle \frac{9}{7}}$

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.