User:Adwaele/sandbox

Equivalence with other formulations
It is interesting investigate how the mathematical formulation of the second law relates with other well-known formulations of the second law. In order to do that we first look at a heat engine, assuming that $$\dot Q_a=0$$. In other words: the heat flow $$\dot Q_H$$ is completely converted into power. In this case the second law would reduce to


 * $$ 0=\frac{\dot Q_H}{T_H}+\dot S_i.$$

Since $$\dot Q_H\ge 0 $$ and $$T_H>0$$ this would result in $$\dot S_i\leq 0$$ which violates the condition that the entropy production is always positive. Hence: No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work. This is the Kelvin statement of the second law.

Now look at the case of the refrigerator and assume that the input power is zero. In other words: heat is transported from a low temperature to a high temperature without doing work. The first law with P =0 would give


 * $$\dot Q_L=\dot Q_a$$

and the second law


 * $$ 0=\frac{\dot Q_L}{T_L}-\frac{\dot Q_L}{T_a}+\dot S_i$$

or


 * $$ \dot S_i =\dot Q_L\left(\frac{1}{T_a}-\frac{1}{T_L}\right).$$

Since $$\dot Q_L\ge 0 $$ and $$T_a>T_L$$ this would result in $$\dot S_i\leq 0$$ which again violates the condition that the entropy production is always positive. Hence: No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature. This is the Clausius statement of the second law.