Second Law of Thermodynamics

Efficiency

General Equation:

$η = \frac{W}{Q_{H}} = 1 - \frac{Q_{L}}{Q_{H}} < 1$

Early statements

Clausius Statement

Created by Rudolf Juilius Emanuel Clausius

It is impossible to construct a device that operates in a cycle and produces no other effects than the transfer of heat from a cooler body to a hotter body.
Layman terms:
- Heat goes from hot to cold
- If you want heat to go from cold to hot, you need to do work.

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Kelvin-Planck Statement

Created by William Thomson, 1st Baron Kelvin and Max Karl Ernst Ludwig Planck

It is impossible to construct a device that operates in a cycle and produce work by exchanging heat with a single reservoir.
Layman terms:
- To produce work, you need a cold and a hot source.
- To produce work, you will lose/waste some heat.

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Proof

Let Clausius' statement be wrong.

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Thus, Kelvin-Plank statement is wrong.
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Based on Irreversibility

Predated by Steam Engine calculations

Carnot Cycle

Created by Lazarre Carnot

Theoretical:
- Cannot be achieved because always some irreversible loss
$η < η_{c a r n o t} = 1 - \frac{T_{L}}{T_{H}}$
$η_{r e v} = 1 - \frac{T_{L}}{T_{H}}$

Reversible $\Rightarrow$ Maximum efficiency $η_{i r r} < η_{r e v}$

Rearranging results:

Reversible Cycle:

$\frac{Q_{H}}{T_{H}} - \frac{Q_{L}}{T_{L}} = 0$

General Case:

$\frac{Q_{H}}{T_{H}} - \frac{Q_{L}}{T_{L}} < 0$

Clausius inequality:

Reversible Cycle:

$\oint \frac{δ Q}{T} = 0$

General Case:

$\oint \frac{δ Q}{T} < 0$

Efficiency

General Equation:

Early statements

Clausius Statement

Kelvin-Planck Statement

Proof

Based on Irreversibility

Predated by Steam Engine calculations

Carnot Cycle

Rearranging results:

Reversible Cycle:

General Case:

Clausius inequality:

Reversible Cycle:

General Case:

Entropy