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 $V$?




In thermodynamics, the number of degrees of freedom available to describe a system is given by [Gibbs' phase rule](https://en.wikipedia.org/wiki/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](https://chemistry.stackexchange.com/questions/68676/why-do-we-still-use-the-p-subscript-in-a-thermodynamic-derivative-such-as-dh/68677#comment120567_68676), this applies to multivariate calculus generally, not just the calculus of thermodynamics.


 11

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


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