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In a physical chemistry class, the professor derived entropy $S$ as a function of concentration starting from $S=k\ln(W)$. Somehow he made a mistake and ended up with the wrong sign. This remains unsolved until the end of the class.

So I would like to figure out the correct solution before the next class. The final answer is

$$S=-k\ln(\text{conc.}) + S^\circ,$$

where $S^\circ$ is the entropy of the substance in standard state. I know that concentration is inversely proportional to the multiplicity should be used somewhere, but I cannot complete the whole derivation.

So far I have got that

$$S = -k\ln(\text{conc.}) + k\ln(N),$$

where $N$ is the number of particles. But why is the $k\ln(N)$ part $S^\circ$?

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