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In the course reader, the lnW in the Boltzmann equation is said to be statistical and that in there being a large error, then it would result in a small error in the thermodynamic property, S. I'm not entirely sure what the statistical aspect of this equation means or represents.
The quantity W in the Boltzmann equation stands for the number of different ways that atoms in molecules in a sample can be arranged and yet each still amounting to the same total energy (Chemical Principles: The Quest for Insight, 6th edition). Every different arrangement of the molecules in a sample is called a microstate. Therefore, W stands for the amount of varied microstates, each amounting to the same total energy. The entropy S calculated from the Boltzmann equation (S=k lnW) is called the statistical entropy. For example, we know that there is only one arrangement of molecules in a perfect crystal; therefore, W=1 and S = k lnW = k ln(1) = 0 J K-1.
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