Question

What is meant by degree of freedom. Explain the specific heats for monoatomic, diatomic and polyatomic gases.

Answer 1

A degree of freedom is an independent physical parameter in the formal description of the state of a physical system.
In three-dimensional space, three degrees of freedom are associated with the movement of a particle.
A diatomic gas molecule thus has six degrees of freedom. This set may be split in terms of translations, rotations, and vibrations of the molecule. The centre of mass motion of the whole molecule accounts for three (3) degrees of freedom. Also, the molecule has two rotational degrees of motion and one vibrational degree of freedom. Since in a monoatomic gas, each molecule of the gas have only translatory motion and hence its degree of freedom is $3$, i. e., $f=3$, Then from equations (14.78), (14.79) and (14.81)
$C_v=\frac{f}{2}R=\frac{5}{2}R$
$C_p=(\frac{fr}{2}+1)R=(\frac{5}{2}+1)R=\frac{7}{2}R$
and $\gamma =(1+\frac{2}{f})=(1+\frac{2}{5})=\frac{7}{5}=1.4$
We know that each molecule of a diatomic gas is made up of two molecules, in which degree of freedom of each molecule is is $5$, i.e., $f=5$
From equations (14.78), (14.79) and (14.81),
$C_v=\frac{f}{2}R=\frac{5}{2}R$
$C_p=(\frac{fr}{2}+1)R=(\frac{5}{2}+1)R=\frac{7}{2}R$
and $\gamma =(1+\frac{2}{f})=(1+\frac{2}{5})=\frac{7}{5}=1.4$
Since a polyatomic has three degrees of freedom each of translation and rotation. Also, there is some definite number of vibrational degrees of freedom (fv). Then, total degree of freedom per molecule (f),
$f=3+3+fv=6+fv$
From equations (14.78), (14.79) and (14.81),
$C_v=\frac{f}{2}R=\frac{(6+fv)}{2}R=(3+\frac{fv}{2})R$
$C_p=(\frac{f}{2}+1)R=(\frac{6+fv}{2}+1)R=(4+\frac{fv}{2})R$
and $\gamma =(1+\frac{2}{5})=(1+\frac{2}{6+frv})=\frac{8+fv}{6+fv}$
If $fv=0,$ then $C_v=3R;C_p=4R$ and $\gamma =1.33$
The graph plotted between Cv and T for diatomic gases, is shown in Fig. 14.7. We see that the temperature increases, the gas absorbs heat through the translational, rotational and vibrational ways.