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Added classical entropy example
\Delta S=\int_{i}^{f}\frac{dQ}{T}
\Delta S=\int_{i}^{f}\frac{dQ}{T}
</math>
</math>
====Example of Thermodynamic Entropy====
Consider a cup of [[coffee]], of mass <math>m = 0.1 \mbox{kg}</math> and since it is mostly water, [[Specific heat|specific heat capacity]] <math>c = 4.2 \times 10^{3} \ \mbox{J} \ \mbox{kg}^{-1} \ \mbox{K}^{-1}</math>. We shall assume that both <math>m</math> and <math>c</math> are constant. If we leave the coffee for a while, it shall cool from <math>T_i = 50^{\circ} \mbox{C}</math> to <math>T_f = 20^{\circ} \mbox{C}</math>. The change in entropy is, from above:
<math>
\Delta S=\int_{i}^{f}\frac{dQ}{T}
</math>
We can relate the change in [[heat]], <math>dQ</math>, to the change in [[temperature]], <math>dT</math> by <math>dQ = mc \, dT</math>. Then we can write:
<math>
\Delta S=\int_{T_i}^{T_f}\frac{mc \, dT}{T}= mc \ln{\frac{T_f}{T_i}}
</math>
Plugging our numbers we find that the change in entropy is <math>\Delta S = -40.9 \ \mbox{J} \ \mbox{K}^{-1}</math>. The change in entropy is negative. This does not disagree with the [[second law of thermodynamics]], as it our cup of coffee is an open system, not an [[isolated system]].
===Statistical mechanics definition 1 (Boltzmann Entropy)===
===Statistical mechanics definition 1 (Boltzmann Entropy)===
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