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The Second Law of Thermodynamics. Entropy

  • 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.

Keywords: 
second law, entropy, heat engine, efficiency, Carnot, Carnot cycle, Clausius, Clausius inequality
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