On exact and inexact differentials

The CarnotCycle blog recently posted a nice little article titled “On Exact And Inexact Differentials” and I’m bringing it to people’s attention because its the sort of thing which would have been extremely useful to me at a time when I was reading calculus-heavy texts that just assumed you knew what exact differentials were, without being aware that you probably missed the day in intro differential equations when they were explained. (That was by far my worst performance in a class. I have no excuse.)

So this isn’t going to be the most accessible article you run across on my blog here, until I finish making the switch to a full-on advanced statistical mechanics course. But if you start getting into, particularly, thermodynamics and wonder where this particular and slightly funky string of symbols comes from, this is a nice little warmup. For extra help, CarnotCycle also explains what makes something an inexact differential.


From the search term phrases that show up on this blog’s stats, CarnotCycle detects that a significant segment of visitors are studying foundation level thermodynamics  at colleges and universities around the world. So what better than a post that tackles that favorite test topic – exact and inexact differentials.

When I was an undergraduate, back in the time of Noah, we were first taught the visual approach to these things. Later we dispensed with diagrams and got our answers purely through the operations of calculus, but either approach is equally instructive. CarnotCycle herewith presents them both.

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The visual approach

Ok, let’s start off down the visual track by contemplating the following pair of pressure-volume diagrams:

visual track

The points A and B have identical coordinates on both diagrams, with A and B respectively representing the initial and final states of a closed PVT system, such as an…

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