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If a system can be arranged in W different ways, the entropy is
If a system can be arranged in W different ways, the entropy is
<math> S= k_B \log W </math>
<math> S= k_B \ln W </math>
where <math>k_B</math> is Boltzmann's constant.
where <math>k_B</math> is Boltzmann's constant.<ref name="University Physics with Modern Physics">{{cite book
|author=Hugh D. Young and Roger A. Freedman
|title=University Physics with Modern Physics
|publisher=Pearson
|location=San Francisco
|isbn=
|pages=
|quote=
|language=English}}</ref>
===Example of Boltzmann Entropy===
This example is based on that found in University Physics with Modern Physics.
<ref name="University Physics with Modern Physics">{{cite book
|author=Hugh D. Young and Roger A. Freedman
|title=University Physics with Modern Physics
|publisher=Pearson
|location=San Francisco
|isbn=
|pages=
|quote=
|language=English}}</ref>
To see an example of the statistical nature of this law, consider flipping four coins, the outcome of which can be either heads (H) or tails (T). The [[entropy]] can be calculated using the above.
{| class="wikitable"
|-
!ID
!Combinations
!Number of Combinations
!Entropy <math>S</math>
|-
|1
|HHHH
|1
|<math>\ln{1} =0</math>
|-
|2
|THHH<br/>
HTHH<br/>
HHTH<br/>
HHHT
|4
|<math>\ln{4} \approx 1.39</math>
|-
|3
|TTHH<br/>
THTH<br/>
THHT<br/>
HTTH<br/>
HTHT<br/>
HHTT
|6
|<math>\ln{6} \approx 1.79</math>
|-
|4
|TTTH<br/>
TTHT<br/>
THTT<br/>
HTTT
|4
|<math>\ln{4} \approx 1.39</math>
|-
|5
|TTTT
|1
|<math>\ln{1} =0</math>
|}
Intuitively, the macro-state with the highest [[entropy]] (or "disorder") is the third state as it corresponds the macro-state which has the greatest number of micro-states. In the same way that when you throw four coins, you expect to get two heads and two tails, it is '''not''' impossible to get four heads. It is only less likely.
===Statistical mechanics definition 2 (Gibbs Entropy)===
===Statistical mechanics definition 2 (Gibbs Entropy)===
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