91Ó°ÊÓ

Eight molecules are given, n=8 to 1

With the help of the multiplicity of central configurations formula and the corresponding entropy formula, we can generate the required table.

Formulae:

The formula of multiplicity of microstates according to multiplicity of central configurations,

${\mathrm{W}}_{\left({\mathrm{n}}_1,\mathrm{N}\right)}=\frac{\mathrm{N}!}{{\mathrm{n}}_1!{\mathrm{n}}_2!}$ (1)

The formula of entropy according to multiplicity of central configurations,

$\mathrm{S}=\mathrm{klnW}$ (2)

Where, k = Boltzmann constant, that is $1.38×{10}^{-23}\mathrm{J}/\mathrm{K}$particles.

For Label I, N = 8, ${\mathrm{n}}_1=8$

The multiplicity of microstates using equation (1):

$$\begin{gathered} \mathrm{W}=\frac{8!}{8!0!} \\ =1 \end{gathered}$$

Therefore, the entropy using equation (2):

$$\begin{gathered} \mathrm{S}=\left(1.38×{10}^{-23}\mathrm{J}\right)\mathrm{In}\left(1\right) \\ =0 \end{gathered}$$

For Label II, N = 8, ${\mathrm{n}}_1=7$

The multiplicity of microstates using equation (1):

$$\begin{gathered} \mathrm{W}=\frac{8!}{7!1!} \\ =8 \end{gathered}$$

Therefore, the entropy using equation (2):

$$\begin{gathered} \mathrm{S}=\left(1.38×{10}^{-23}\mathrm{J}\right)\mathrm{In}\left(8\right) \\ =2.9×{10}^{-23}\mathrm{J} \end{gathered}$$

For Label III,$\mathrm{N}=8,{\mathrm{n}}_1=6$

The multiplicity of microstates using equation (1):

$$\begin{gathered} \mathrm{W}=\frac{8!}{6!2!} \\ =28 \end{gathered}$$

Therefore, the entropy using equation (2):

$$\begin{gathered} \mathrm{S}=\left(1.38×{10}^{-23}\mathrm{J}\right)\mathrm{In}\left(28\right) \\ =4.6×{10}^{-23}\mathrm{J} \end{gathered}$$

For Label IV,$\mathrm{N}=8,{\mathrm{n}}_1=5$

The Multiplicity of microstates using equation (1):

$$\begin{gathered} \mathrm{W}=\frac{8!}{5!3!} \\ =56 \end{gathered}$$

Therefore, the entropy using equation (2):

$$\begin{gathered} \mathrm{S}=\left(1.38×{10}^{-23}\mathrm{J}\right)\mathrm{In}\left(56\right) \\ =5.6×{10}^{-23}\mathrm{J} \end{gathered}$$

For Label V,$\mathrm{N}=8,{\mathrm{n}}_1=4$

Multiplicity of microstates using equation (1):

$$\begin{gathered} W=\frac{8!}{4!4!} \\ =70 \end{gathered}$$

Therefore, the entropy using equation (2):

$$\begin{gathered} \mathrm{S}=\left(1.38×{10}^{-23}\mathrm{J}\right)\mathrm{In}\left(70\right) \\ =5.9×{10}^{-23}\mathrm{J} \end{gathered}$$

For Label VI, $\mathrm{N}=8,{\mathrm{n}}_1=3$

Multiplicity of microstates using equation (1):

$$\begin{gathered} \mathrm{W}=\frac{8!}{3!5!} \\ =56 \end{gathered}$$

Therefore, the entropy using equation (2):

$$\begin{gathered} \mathrm{S}=\left(1.38×{10}^{-23}\mathrm{J}\right)\mathrm{In}\left(56\right) \\ =5.6×{10}^{-23}\mathrm{J} \end{gathered}$$

For Label VII,$\mathrm{N}=8,{\mathrm{n}}_1=2$

Multiplicity of microstates using equation (1):

$$\begin{gathered} \mathrm{W}=\frac{8!}{2!6!} \\ =28 \end{gathered}$$

Therefore, the entropy using equation (2):

$$\begin{gathered} \mathrm{S}=\left(1.38×{10}^{-23}\mathrm{J}\right)\mathrm{In}\left(28\right) \\ =4.6×{10}^{-23}\mathrm{J} \end{gathered}$$

For Label VIII,$\mathrm{N}=8,{\mathrm{n}}_1=1$

Multiplicity of microstates using equation (1):

$\begin{array}{l}\mathrm{W}=\frac{8!}{1!7!}\\ =8\end{array}$

Therefore, the entropy using equation (2):

$$\begin{gathered} \mathrm{S}=\left(1.38×{10}^{-23}\mathrm{J}\right)\mathrm{In}\left(8\right) \\ =2.9×{10}^{-23}\mathrm{J} \end{gathered}$$

For Label IX,$\mathrm{N}=8,{\mathrm{n}}_1=1$

Multiplicity of microstates using equation (1):

$$\begin{gathered} \mathrm{W}=\frac{8!}{0!8!} \\ =1 \end{gathered}$$

Therefore, Entropy using equation (2):

$$\begin{gathered} \mathrm{S}=\left(1.38×{10}^{-23}\mathrm{J}\right)\mathrm{In}\left(1\right) \\ =0\mathrm{J} \end{gathered}$$

Hence, the above table represents the central configuration of eight molecules.