Differentiating thermodynamic coefficients

[Nikitin]

Homework Statement

In oppgave 1 a) I am supposed to show that the given equality is true (namely that the isoterm compressibility coefficient partial-differentiated with regards to temperature = isobar coefficient differentiated with regards to pressure multiplied by minus one).

http://web.phys.ntnu.no/~stovneng/TFY4165_2014/oving2.pdf

The Attempt at a Solution

I can naturally see that the way to attack this problem is to simply carry out the differentiations, but how does it make sense to partially differentiate $\alpha_V$ with regards to $p$ when $\alpha_V$ assumes constant pressure? And besides, $\alpha_V$ measures the change of relative volume as a function of $T$, yet constant temperature is assumed for $\frac{\partial \alpha_V}{\partial p}$? It's a similar problem with the differentiating the isoterm coefficient with regards to temperature while keeping pressure constant..

 

[BvU]

Not strong in Norwegian. alpha_v may be defined at constant pressure, that doesn't mean its value is independent of the pressure!

 

[Nikitin]

Okay, but alpha_v is the change of relative volume with regards to temperature, yet $\frac{\partial \alpha_V}{\partial p}$ assumes constant temperature? How does it make sense to calculate how $\alpha_V$ (which is a measure of how the volume varies with regards to temperature) varies while the temperature is constant?

 

[voko]

$\alpha_V$ is a function of both temperature and pressure. Unfortunately, many, if not most, texts on thermodynamics do a very bad job at explaining functional dependencies among all the things they operate with. The unconventional notation $\left({\partial X \over \partial Y}\right)_{Z}$ is my personal anti-favorite, because in my experience very few students understand what it really signifies. Just for the record, it means that $X$ is considered to be a function of $Y$ and $Z$, and is differentiated with respect to $Y$. The usually added bit that $Z$ is held constant is redundant because that follows from the definition of partial differentiation. I suspect it is not only redundant, but is severely confusing, because the obvious redundancy makes the inexperienced reader wonder what really is going on.

 

[Chestermiller]

Please write down your defining equations for α_V and κ_T. Once you get a look at these, you will see immediately how to prove what you are trying to prove.