The entropy $($S$)$ of a system can be defined as:

$S=k_{B}lnW$

where $k_B$ is the Boltzmann constant $(k_B=\frac{R}{N_A})$ and $W$ is the number of energetically equivalent microstates of the system.

For a system of $N$ independent particles occupying $x$ spatial cells, the number of microstates is:

$W=x^N$

Consider a system with a constant number of particles $($i$.$e$.,$ constant $N)$ where the volume of the system increases such the number of accessible spatial cells, $x,$ increases. Call

the initial number of accessible spatial cells $x_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}}$ and the final number of accessible spatial cells after the volume expansion $x_{\text{f}\text{i}\text{n}\text{a}\text{l}}$ where $x_{\text{f}\text{i}\text{n}\text{a}\text{l}}>x_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}}.$

Derive an expression for the change in entropy $(S)$ for this system due to the volume expansion in terms of $N,x_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}},$ and $x_{\text{f}\text{i}\text{n}\text{a}\text{l}}$ and select an equivalent expression from the

list below.

Hints:

$S=S_{\text{f}\text{i}\text{n}\text{a}\text{l}}-S_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}}$

$ln(x)-ln(y)=ln(\frac{x}{y})$

$ln(x^y)=yln(x)$

$S=N{k}_{B}ln(\frac{x_{\text{f}\text{i}\text{n}\text{a}\text{l}}}{x_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}}})$

$S=\frac{k_B}{N}ln(\frac{x_{\text{f}\text{i}\text{n}\text{a}\text{l}}}{x_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{e}\text{l}}})$

$S=-N{k}_{B}ln(\frac{x_{\text{f}\text{i}\text{n}\text{e}\text{l}}}{x_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}}})$

$S=N{k}_{B}ln(\frac{x_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}}}{x_{\text{f}\text{i}\text{n}\text{a}\text{l}}})$

$S=k_{B}ln(\frac{x_{\text{f}\text{i}\text{n}\text{a}\text{l}}}{x_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{e}\text{l}}})$

$S=-\frac{k_B}{N}ln(\frac{x_{\text{f}\text{i}\text{n}\text{e}\text{l}}}{x_{\text{i}\text{n}\text{s}\text{t}\text{i}\text{a}\text{l}}})$

$S=\frac{k_B}{N}ln(\frac{x_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}}}{x_{\text{f}\text{i}\text{n}\text{a}\text{l}}})$