Four years after taking a thermo course and I finally understand what the hell the partial derivative notation in thermodynamics means I think
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i asked early on .. it just means to hold the subscripts constant .. but isn't that what a partial derivative means anyways? it's kind of an over-clarification that makes it somewhat confusing tho i understand why they do it
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It’s the mixture of different disciplines and their notations. I’m still learning to speak engineer mathematically. As with your partials point, my assumptions never seem clear to non-physics people. Also I never groked thermo UNTIL I learned stat mechanics.
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I have a little page of lecture notes in which I explain to students that Maxwell relations (which I struggled with) are just identities of partial derivatives. It is delightful to see the dawn of understanding in their eyes.
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The notation for partial derivatives is actually ambiguous. Yes, it implies that all the other variables are held constant, but what ARE the other variables? Consider f(x,y)=(x+y)²; what is ∂f/∂x? It depends, are you holding y constant or z=x+y constant?
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why would you hold x+y constant? once you define f(x,y) then ∂f/∂x is unambiguous
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It certainly is necessary in a sense. Not all variables are independent of each other. You basically wind up doing calculus on some level surface (the equation of state) instead of a flat coordinate grid.
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I just taught this the other day. I stressed that specifying what is held constant is an important reminder of what it is a function of.
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Are you working with temperature volume? Use thermal energy, U(V,T). Working with temperature and pressure, as chemists typically do? Use enthalpy, H(p,T).
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Never understood this either. I think it has something to do with the derivatives of implicit function from analysis
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The confusion comes about because the symbols in thermo which we differentiate are typically physical quantities instead of mathematical functions. The former can be expressed as different functions depending on the variables you have in mind.
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You would never need this notation if the symbols represented functions. Then it would be redundant.
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