The Second Law of Thermodynamics. Entropy

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  • Heat engine efficiency: $$ \eta = \frac{A}{Q_1} = 1 - \frac{Q_2^\prime}{Q_1},$$ where $Q_1$ is the heat obtained by the working substance, $Q_2^\prime$ is the heat released by the working substance.
  • Efficiency of a Carnot cycle: $$\eta = \frac{T_1 - T_2}{T_1},$$ where $T_1$ and $T_2$ are the temperatures of the hot and cold bodies respectively.
  • Clausius inequality: $$\oint \frac{\delta Q}{T} \leqslant 0,$$ where $\delta Q$ is the elementary amount of heat transferred to the system ($\delta Q$ is an algebraic quantity).
  • Entropy increment of a system: $$ \Delta S \geqslant \int \frac{\delta Q}{T}.$$
  • Fundamental relation of thermodynamics: $$ T dS \geqslant dU + pdV.$$
  • Relation between the entropy and the statistical weight $\Omega$ (the thermodynamic probability): $$S = k \ln \Omega,$$ where $k$ is the Boltzmann constant.


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