# Why do we still use the 'P' subscript in a thermodynamic derivative such as (dH/dT)P?

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 $V$?

• 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. – hBy2Py Feb 17 '17 at 3:19
• 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. – hBy2Py Feb 17 '17 at 3:32
• Long story short, all other variables aren't and can't be held constant. – Ivan Neretin Feb 17 '17 at 5:35
• Because we distinguish f(p,..) and f(V, ...) functions – Greg Feb 17 '17 at 16:12
• 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, $(\partial f / \partial x)_{y,z}$ – 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.

• Ok but what if something is a function of more than two variables ? – mathemather Feb 17 '17 at 7:00
• @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. – hBy2Py Feb 17 '17 at 12:02

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