I'm confused about the Second Law of Thermodynamics. The Second Law of Thermodynamics prohibits a decrease in the entropy of a closed system and states that the entropy is unchanged during a reversible process.
Then why do we say that $\Delta S = \int_a^b{\frac{dQ}{T}}$ for a reversible process? Doesn't the second law simply state that $\Delta S = 0$?
(I'm a high school student teaching himself the principles of thermodynamics, but I am struggling with more challenging material due to my poor understanding of these basics)
Best Answer
To say the same thing David Zaslavsky said in slightly different words, the second law implies that entropy cannot be destroyed, but it doesn't prevent you from moving it around from place to place. When we write the equation $\Delta S = \int_a^b \frac{dQ}{T}$, we're assuming that this $dQ$ represents a flow of heat into or out of the system from somewhere else. Therefore $S$ (which, by convention, represents only the entropy of some particular system) can either increase or decrease. Since we're talking about a reversible process, the entropy of some other system must change by an equal and opposite amount, in order to keep the total constant. That is, $\Delta S + \Delta S_\text{surroundings} = 0$.
One other thing: in thermodynamics, "closed" and "isolated" mean different things. "Isolated" means neither heat nor matter can be exchanged with the environment, whereas "closed" means that matter cannot be exchanged, but heat can. In your question you say the second law "prohibits a decrease in the entropy of a closed system," but actually this only applies to isolated systems, not closed ones. When we apply the equations above, we're not talking about an isolated system, which is why its entropy is allowed to change. I mention this because you said you're teaching yourself, and in that case it will be important to make sure you don't get confused by subtleties of terminology.