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An important extensive property of all systems in thermodynamics, statistical mechanics, and information theory, quantifying their disorder (randomness), i.e., our lack of information about them. It characterizes the degree to which the energy of the system is *not* available to do useful work.
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A relationship between entropy and temperature
Real equation for entropy of an ideal gas then would look like:
$$\mathrm dS = \frac{1}{T}(P\,\mathrm dV + (3/2)NR\,\mathrm dT) = NR\frac{\mathrm dV}{V} + \frac{3}{2} NR \frac{\mathrm dT}{T}$$
Then $S …