Editing Second law of thermodynamics (section)

=== Equivalence of the Clausius and the Kelvin statements ===
[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]
Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work (the drained heat is fully converted to work) in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the right figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is ''η'' and so the efficiency of the reversed heat engine is 1/''η''. The net and sole effect of the combined pair of engines is to transfer heat <math display="inline">\Delta Q = Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same; <math display="inline"> \text{Input}+\text{Output}=0 \implies (Q + Q_c) - \frac{Q}{\eta} = 0 </math>, so therefore <math display="inline"> Q_c=Q\left( \frac{1}{\eta}-1\right) </math>, where (1) the sign convention of heat is used in which heat entering into (leaving from) an engine is positive (negative) and (2) <math> \frac{Q}{\eta} </math> is obtained by [[Heat engine#Efficiency|the definition of efficiency]] of the engine when the engine operation is not reversed. Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.