Consider the following thermodynamic derivative:

$$ \left(\frac{\partial H}{\partial T}\right)_P $$

Taking the partial derivative w.r.t. $T$ means that all other variables are held constant. Why do we still write it with a subscript like $P$ or $V$?

  • 1
    $\begingroup$ I made some pretty substantial edits to your post -- if I've misrepresented anything you wrote, or if you don't like anything I changed, please feel free to edit it back, or to roll back the edits entirely. $\endgroup$ – hBy2Py Feb 17 '17 at 3:19
  • $\begingroup$ In particular, the use of MathJax in titles is discouraged (here on Chem.SE, at least), as it makes it hard to read search results on, e.g., Google. $\endgroup$ – hBy2Py Feb 17 '17 at 3:32
  • $\begingroup$ Long story short, all other variables aren't and can't be held constant. $\endgroup$ – Ivan Neretin Feb 17 '17 at 5:35
  • $\begingroup$ Because we distinguish f(p,..) and f(V, ...) functions $\endgroup$ – Greg Feb 17 '17 at 16:12
  • 1
    $\begingroup$ its a maths thing not related to thermodynamics per se; if you want to differentiate a function of several variables $f(x, y, z)$ wrt. x then then y, z need to be held constant, hence the curly $\partial$ to indicate a partial derivative and subscripts to indicate what is held constant, $(\partial f / \partial x)_{y,z}$ $\endgroup$ – porphyrin Feb 23 '17 at 16:49

In thermodynamics, the number of degrees of freedom available to describe a system is given by Gibbs' phase rule:

$$ F = C - P + 2 $$

Here, $C$ is the number of components in the system, $P$ is the number of phases in the system, and $F$ is the number of resulting degrees of freedom.

Consider a pure substance in a closed system. One can name a wide variety of thermodynamic parameters to describe its state: temperature, pressure, specific volume, internal energy, enthalpy, entropy, etc. However, Gibbs' phase rule says that we can only independently specify a subset of these parameters before we run out of degrees of freedom:

$$ F = 1 - 1 + 2 = 2 $$

So, in any mathematical description of (in your example) the enthalpy, $H$, it can only properly be a function of two independent variables:

$$ H = H\left(X_1, X_2\right) $$

But: which two variables?

Simply writing the partial derivative $\partial H \over \partial T$ indicates that temperature is to be one of the two available degrees of freedom:

$$ H = H\left(T, X_2\right) $$

Now, though – what is $X_2$? Well, including the subscript on the differential is a way of explicitly indicating what that other independent variable is:

$$ \left({\partial H \over \partial T}\right)_P \quad \Longrightarrow \quad H = H\left(T, P\right) $$

As porphyrin rightly notes in a comment, this applies to multivariate calculus generally, not just the calculus of thermodynamics.

| improve this answer | |
  • $\begingroup$ Ok but what if something is a function of more than two variables ? $\endgroup$ – mathemather Feb 17 '17 at 7:00
  • $\begingroup$ @mathemather For any given first derivative in any given system, if you have $F$ degrees of freedom, you should specify $F-1$ independent variables as subscripts to the derivative. $\endgroup$ – hBy2Py Feb 17 '17 at 12:02

It just means one is holding the experimental pressure constant when determining Cp.

| improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.