# 3.4.3: The Second Law of Thermodynamics

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Why do we pay so much attention to entropy? Why is it so important? Well, it’s because the entropy is – let’s use such a metaphor – the “leading character” in the Second law of Thermodynamics.

One scientist once counted all non-identical formulations of the Second Law he was able to find – and his result was about one hundred. But non- identical does not mean that they are different: no, no, they are all equivalent! They all convey exactly the same information, only the differently (sometimes, very differently!) formulated. In some of them entropy is explicitly mentioned, and in some of them it is not. For instance, one of this author’s favorite formulation of the Second Law is the following: *It’s not possible to build a Perpetual Motion Machine of the Second Kind*.

In fact, what the second law states, is expressed in the last paragraph of the preceding Section. In a more compact form, one can say: *In a closed thermodynamic (adiabatic) system, the entropy may either remain constant, or increase*. Or, even in a compacter form:

In a closed thermodynamic (adiabatic) system, \( \dfrac{dS\left ( t \right )}{dt} \geqslant 0 \)

If the time derivative of a function of time is zero, it means that the function does not change in time; and if the derivative is greater than zero, it means that its value is increasing in time.

Why does the Author so stubbornly stress each time that the Second Law applies only to thermally isolated, i.e., adiabatic systems? Because he knows from his long teaching practice that students often forget to add this clause. The Author himself forgot about it at the thermodynamics exam at his senior year. Fortunately, other answers were correct, so the exam ended up with a B. Some instructors treat the omission of this clause as a cardinal error (not the Author, remembering his own fault – he believes that giving the student “another chance” is a better way of dealing with the problem).

In a system which is not thermally isolated, the entropy may decrease. Take a glass of water and on a cold winter day leave it outside. Water will freeze, its entropy will lower considerably. It’s all OK.