Question

Find the relation between$Y$(adiabatic constant) and degree of freedom $\left(f\right)$.

• A. $f=2/\gamma -1$
• B. $f=\gamma /\gamma -1$
• C. $f=\gamma -1/2$
• D. $f=\gamma -1/\gamma$

Answer 1

The correct option is A

$f=2/\gamma -1$

Step 1: Given data

Adiabatic constant $=\gamma$

Degree of freedom $=f$

We know the equation of adiabatic constant

$\gamma =\frac{C_P}{C_V}$

Where $C_P=$Specific heat at constant pressure

$C_V=$ Specific heat at constant volume

$U=$Thermal energy

$K_B=$Boltzmann's constant

Step 2 : To find the relation between adiabatic constant and degrees of freedom

We know

$E=U\times N$

$N=$Number of molecules

$$\begin{gathered} R=N{K}_B \\ =\frac{f}{2}RT \end{gathered}$$

Now ,

$$\begin{gathered} C_V={\left(\frac{\partial E}{\partial T}\right)}_V \\ =\frac{f}{2}R \\ {C}_P-C_V=R \\ {C}_P=C_V+R \\ =\frac{R}{2}\left(f+2\right) \\ \gamma =\frac{C_P}{C_V} \\ =\frac{f+2}{f} \\ =1+\frac{2}{f} \\ \gamma f=f+2 \\ f=\frac{2}{\gamma -1} \end{gathered}$$

Hence the relation between $\gamma$(adiabatic constant ) and $f$(degrees of freedom ) is $f=\frac{2}{\gamma -1}$

Therefore the correct answer is Option A