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I was under the impression that the idea of taking a partial derivative of some function $f(x,y)$ with, for example, respect to $x$ is where you take a deriviatve of the function with respect to the variable in question while holding the rest of the variables as constants.

My thermodynamaics proff introduced the following notation $$ \left(\frac{\partial f}{\partial x}\right)_y $$ he verbally described it as

The partial derivative of $f$, with respect to $x$, while holding $y$ constant.

and he kept enforcing the point that $$ \left(\frac{\partial f}{\partial x}\right)_y \neq \frac{\partial f}{\partial x} $$ but I really do not understand why this is. They fundamentally say, and accomplish, the exact same thing, do they not?

Kalcifer
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  • that verbal description is literally the definition of a partial derivative. I assume what he might have meant is something like $$\left. \frac{\partial f}{\partial x}\right|_{y=y_0}$$ – Henry Lee Oct 05 '22 at 17:33
  • You can find this discussed in various posts. Here is one example. In the simple case of a function $f(x,y)$ of independent variables $x,y$, there is no difference from the usual notation. Henry's assumption is wrong. – Ted Shifrin Oct 05 '22 at 18:37

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in strict mathematical notation this is the definition for partial derivative

In thermodynamics however we need a method to remember what variables we use.

Physically any thermodynamic function $f(x,y)$ can be converted using Legendre transform to $f(u,v)$ so i guess what the prof. meant was something like this:

Take for example the the state function for internal energy:

$du=dq-pdv$

Now for an ideal gas:

$dq=C_vdT$

where $C_v=\frac{\partial u}{\partial T}$ and we need to remember that the process is with respect to constant volume.

but we can also look at the process for constant pressure:

$C_pdT$ where now the pressure variable is the second variable.

how is this done?

$dh=du+d(pv)$ this is the Legendre transform to enthalpy.

So now the $C_p=\frac{\partial h}{\partial T}$

Only now $p$ is held constant.

The notation is simply a mnemonic to remember our state variables and therefore not a mathematical statement.

ryaron
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