I was reading the second step of Carnot cycle in which the system undergoes adiabatic expansion doing work & thus decreasing the internal energy of itself. The entropy didn't change as no further heat energy was supplied or taken out; it remained the same as it was after the first step of the cycle that is $\Delta S = \frac{Q_\text{isothermal}}{T}.$

However, I wonder why wouldn't the disorder of the system decrease after the adiabatic expansion; the system is doing work & is losing its internal energy with fall in temperature . Wouldn't this decrease the randomness or disorder of the system after-all it is now less energetic? And doesn't less energetic mean lesser number of microstates & lesser disorder in the system? So, why should the entropy which is the measure of the disorder changes?

Could anyone please help me shun my confusion?

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    $\begingroup$ It is true that T decreases, but the volume V increases. The increase in disorder due to the increase in V exactly compensates the decrease due to the change in internal energy $\endgroup$ – user83548 Nov 10 '15 at 21:13
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    $\begingroup$ The very simple answer to this question is that entropy is not a measure of disorder in the usual sense. Trying to understand entropy as disorder might be the source of confusion here. Of course, if by "disorder" you actually mean "number of microstates" (which is not the meaning of the word disorder), then bruce smitherson's argument is entirely correct. $\endgroup$ – Mark Mitchison Nov 10 '15 at 21:14
  • $\begingroup$ @MarkMitchison I am just curious, what definition of disorder is the universally accepted one? $\endgroup$ – user83548 Nov 10 '15 at 21:21
  • $\begingroup$ @brucesmitherson Well, according to the dictionary it could refer to confusion, leprosy, or rioting in the streets, but the entry about phase-space volume seems to be missing... ;) Actually, following Humpty-Dumpty, I believe that one should use any word for whatever meaning one likes if it enables communication. And the heuristic use of disorder in physics to refer to entropy is commonplace and perfectly acceptable. But it's worth occasionally reminding beginners that entropy is not necessarily disorder in any intuitive sense. $\endgroup$ – Mark Mitchison Nov 10 '15 at 21:29
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    $\begingroup$ @Mark Mitchison: Entropy is not disorder. I'm not using the word disorder like that attributed to messy desks or shuffled cards etc.; what I mean is simply the number of microstates; that's it. $\endgroup$ – user36790 Nov 11 '15 at 2:56

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