Boltzmann's entropy formula

Equation in statistical mechanics

In statistical mechanics, Boltzmann's entropy formula (also known as the Boltzmann–Planck equation, not to be confused with the more general Boltzmann equation, which is a partial differential equation) is a probability equation relating the entropy S {\displaystyle S} , also written as S B {\displaystyle S_{\mathrm {B} }} , of an ideal gas to the multiplicity (commonly denoted as Ω {\displaystyle \Omega } or W {\displaystyle W}), the number of real microstates corresponding to the gas's macrostate: where k B {\displaystyle k_{\mathrm {B} }} is the Boltzmann constant (also written as simply k {\displaystyle k}) and equal to 1.380649 × 10−23 J/K, and ln {\displaystyle \ln } is the natural logarithm function (or log base e, as in the image above). In short, the Boltzmann formula shows the relationship between entropy and the number of ways the atoms or molecules of a certain kind of thermodynamic system can be arranged. What is important to note is that W is not all possible states of the system, but ways the system can be arranged and still have the same properties from perspective of external observer.

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