Going to thank everyone for the help and suggestions for understanding this.
Ended up coming across a good explanation here for a similar problem.
I'll put what I think is the most straightforward way of getting at my confusion, summarize some of the helpful feedback below, and write out the...
Maybe "ignore" is the wrong word.
I think the confusion on my end is to understand why we aren't treating the partial of volume as an "operator" acting on each term and then applying the product rule to the PdV term.
##\frac{\partial (dU)}{\partial V}_T = \frac{\partial (TdS - PdV)}{\partial...
So this is leading back to my original confusion.
Using the notation this way, why not write (B/dt)*dC or (B*dC)/dt? What permits us to ignore B?
For the first term in the original post, taking T outside the derivative makes sense as we explicitly hold T as constant. But this isn't the case...
I am working through quite a bit of Thermodynamics myself lately, and the impression I get is that the Thermo books aren't doing a great job with explaining the "calculus" of how these things work. As luck would have it, I asked a similar question today.
There is probably more than one way to...
Reading your post got me googling again, and I came across this. Seems like what you are showing here.
I thought that this type of operation was frowned upon, but I will chalk this up to me being mistaken.
There is also a very in-the-weeds derivation given in Physical Chemistry (Engel and Reid).
Hey everyone,
I am a chemical engineer looking to improve their knowledge of physics through self-study.
I hope to ask good questions and help answer a few along the way.
Thanks!
I am having some trouble following the derivation of the partial derivative of internal energy with respect to volume at constant temperature.
The fundamental property relation is given by:
dU = TdS - PdV
The text I have shows the result of taking the partial derivative with respect to volume...