February 14, 2012  Tagged with: , ,

When I took my process thermodynamics course as an undergraduate student, I was told to use what I had initially considered to be a completely redundant notation for partial derivatives for thermodynamic potentials.  The notation involved wrapping the partial derivative in a set of parentheses and noting which variables were “held constant” as a subscript.

Ex. $\displaystyle\left(\frac{\partial P}{\partial V}\right)_{T,\vec{N}}$

The derivation leading to this was woefully devoid of the mathematical basis for this apparently redundant notation.  Furthermore, so was the general literature on the subject, most of which consisted of fleeting introductions to their respective application.  As I played with the idea, it became clear why this notation was indeed very necessary.  It was only as I was solving a separate problem on my own that such notation yielded valuable information about the nature of the potentials being described.

To see why the notation is necessary, start with a generic equality implicitly relating three variables as interdependent functions of two variables such that any one of the variables is easily isolated.

Eq. 1) $\displaystyle z=x+y\leadsto z\left[x,y\right]=x\left[y,z\right]+y\left[x,z\right]$

Eq. 2) $\displaystyle x=z-y\leadsto x\left[y,z\right]=z\left[x,y\right]-y\left[x,z\right]$

Eq. 3) $\displaystyle y=z-x\leadsto y\left[x,z\right]=z\left[x,y\right]-x\left[y,z\right]$

Then, implicitly differentiate each variable in turn to show their respective total differentials in terms of their respective partial derivative expansions.

Eq. 4) $\displaystyle dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$

Eq. 5) $\displaystyle dx=\frac{\partial x}{\partial y}dy+\frac{\partial x}{\partial z}dz$

Eq. 6) $\displaystyle dy=\frac{\partial y}{\partial x}dx+\frac{\partial y}{\partial z}dz$

And, subsequently, evaluate the partial derivatives using Eq. 1, Eq. 2, and Eq. 3.

Eq. 7) $\displaystyle dz=\frac{\partial\left(x+y\right)}{\partial x}dx+\frac{\partial\left(x+y\right)}{\partial y}dy\leadsto dz=dx+dy$

Eq. 8) $\displaystyle dx=\frac{\partial\left(z-y\right)}{\partial y}dy+\frac{\partial\left(z-y\right)}{\partial z}dz\leadsto dx=dz-dy$

Eq. 9) $\displaystyle dy=\frac{\partial\left(z-x\right)}{\partial x}dx+\frac{\partial\left(z-x\right)}{\partial z}dz\leadsto dy=dz-dx$

This shows the general manner of generating the total differentials.  Now, using Eq. 4, Eq. 5, and Eq. 6, consider the results of holding either of the variables in the partial derivatives expansion as a constant, in turn.  Notice the notation used in still the standard general conditional constraints notation used in pure mathematics.

Eq. 10) $\displaystyle\left. dz\right|_{x=\text{const.}}=\frac{\partial z}{\partial x}\underbrace{\left. dx\right|_{x=\text{const.}}}_{0}+\frac{\partial z}{\partial y}\left. dy\right|_{x=\text{const.}}=\frac{\partial z}{\partial y}\left. dy\right|_{x=\text{const.}}$

Eq. 11) $\displaystyle\left. dz\right|_{y=\text{const.}}=\frac{\partial z}{\partial x}\left. dx\right|_{y=\text{const.}}+\frac{\partial z}{\partial y}\underbrace{\left. dy\right|_{y=\text{const.}}}_{0}=\frac{\partial z}{\partial x}\left. dx\right|_{y=\text{const.}}$

Eq. 12) $\displaystyle\left. dx\right|_{y=\text{const.}}=\frac{\partial x}{\partial y}\underbrace{\left. dy\right|_{y=\text{const.}}}_{0}+\frac{\partial x}{\partial z}\left. dz\right|_{y=\text{const.}}=\frac{\partial x}{\partial z}\left. dz\right|_{y=\text{const.}}$

Eq. 13) $\displaystyle\left. dx\right|_{z=\text{const.}}=\frac{\partial x}{\partial y}\left. dy\right|_{z=\text{const.}}+\frac{\partial x}{\partial z}\underbrace{\left. dz\right|_{z=\text{const.}}}_{0}=\frac{\partial x}{\partial y}\left. dy\right|_{z=\text{const.}}$

Eq. 14) $\displaystyle\left. dy\right|_{x=\text{const.}}=\frac{\partial y}{\partial x}\underbrace{\left. dx\right|_{x=\text{const.}}}_{0}+\frac{\partial y}{\partial z}\left. dz\right|_{x=\text{const.}}=\frac{\partial y}{\partial z}\left. dz\right|_{x=\text{const.}}$

Eq. 15) $\displaystyle\left. dy\right|_{z=\text{const.}}=\frac{\partial y}{\partial x}\left. dx\right|_{z=\text{const.}}+\frac{\partial y}{\partial z}\underbrace{\left. dz\right|_{z=\text{const.}}}_{0}=\frac{\partial y}{\partial x}\left. dx\right|_{z=\text{const.}}$

Independently rearranging these equations to give the six possible total derivatives and introducing the shorthand that conditional constraints on variables implicitly define them as constants yields interesting results.

Eq. 16) $\displaystyle\left. \left(\frac{dz}{dy}\right)\right|_{x=\text{const.}}=\frac{\partial z}{\partial y}\leadsto\left(\frac{dz}{dy}\right)_{x}=\frac{\partial z}{\partial y}$

Eq. 17) $\displaystyle\left. \left(\frac{dz}{dx}\right)\right|_{y=\text{const.}}=\frac{\partial z}{\partial x}\leadsto\left(\frac{dz}{dx}\right)_{y}=\frac{\partial z}{\partial x}$

Eq. 18) $\displaystyle\left. \left(\frac{dx}{dz}\right)\right|_{y=\text{const.}}=\frac{\partial x}{\partial z}\leadsto\left(\frac{dx}{dz}\right)_{y}=\frac{\partial x}{\partial z}$

Eq. 19) $\displaystyle\left. \left(\frac{dx}{dy}\right)\right|_{z=\text{const.}}=\frac{\partial x}{\partial y}\leadsto\left(\frac{dx}{dy}\right)_{z}=\frac{\partial x}{\partial y}$

Eq. 20) $\displaystyle\left. \left(\frac{dy}{dz}\right)\right|_{x=\text{const.}}=\frac{\partial y}{\partial z}\leadsto\left(\frac{dy}{dz}\right)_{x}=\frac{\partial y}{\partial z}$

Eq. 21) $\displaystyle\left. \left(\frac{dy}{dx}\right)\right|_{z=\text{const.}}=\frac{\partial y}{\partial x}\leadsto\left(\frac{dy}{dx}\right)_{z}=\frac{\partial y}{\partial x}$

As may be seen above, the very definition of partial derivatives is used here to show that a partial derivative is simply a total derivative wherein a single variable is allowed to vary during differentiation.  This elementary concept is learned during  a typical calculus 3 course.  The interesting part of the results is that the resulting partial derivatives retain the invertablility of the constrained total derivatives.  Not all partial derivatives are invertable, yet those that are constructed in this manner are.  This fundamental property of inversion and direct relation to total derivatives is the reason that the notation is used.  The final change in notation to the derivatives of thermodynamic potentials is to allow the constrained total derivatives to remain explicitly as partial derivatives (see the example at the top).  This is done because, typically, not all variables are explicitly noted when holding several of them constant.  As all total derivatives are implicitly related to their respective expanded partial derivative expressions, all total derivatives may be considered partial derivatives with respect to an explicit subset of all possible variables (an infinite number of them).  In this way, the notation is indeed mathematically consistent and, despite the possible confusion introduced by such shorthand, quite necessary for the methods used to manipulate thermodynamic potentials for describing a physical system.

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