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I don't really understand the basic principles behind this and so I am looking for a good explanation on it.

My lecture notes use inequalities like $dS \geq 0$, yet in a question involving calculating the entropy change when compressing Argon, the total entropy change was negative.

Then there is the notion entropy change of the system and entropy change of the surroundings? Can any of these be negative?

In other words, when can entropy be negative? In what circumstances is it strictly positive?

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  • $\begingroup$ web.mit.edu/16.unified/www/FALL/thermodynamics/notes/… This is a must read. I prefer that you read all mit ocw articles. $\endgroup$ – Mitchell May 19 '17 at 19:12
  • $\begingroup$ Think refrigerator-freezers. When ever you increase order (e.g. ice) you descrese entropy. You'll have to calculate if the compressor which makes the ice is using less energy than the order created by the ice in the insultated compartment. $\endgroup$ – TheDoctor Oct 9 '17 at 22:29
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    $\begingroup$ To be clear, change in entropy (as represented by $dS$ in your equation) can be negative, but entropy itself is always positive. Just like mass. $\endgroup$ – Rococo Oct 10 '17 at 0:30
  • $\begingroup$ At morning when you clean up you room, and make up your bed, you are reducing the entropy in that particular division of your home. $\endgroup$ – J. Manuel Nov 13 '17 at 9:50
  • $\begingroup$ If the entropy change is negative for one part of a system, it must be positive for another, in order to still fulfill the requirement you have shown. A reduction in "chaos" is reduction of entropy, such as when you sort the m&m's in colors, but another part of the system must have an increase in entropy in order for you to do that, such as your body spending energy on the manual motion. $\endgroup$ – Steeven Nov 13 '17 at 9:58
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Basic answers about entropy and its definition can be found elsewhere on this site, so we refrain from providing them here again.

For a closed system undergoing an irreversible process the entropy in the final state must always be greater or equal than the entropy in the initial state. This, however, does not pose any limitation on the value of the entropy for any system at any given state, process and time. In particular non-isolated system need not fulfill the above and isolated system may present negative entropy as long as the final variation after irreversible processes is positive.

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