Deriving an expression for a partial derivative (Type III)
Thermodynamics involves many variables. But for a single component sample of matter, only two state variables are needed to describe the system and fix all of the thermodynamic properties of the system. As such, it is conceivable that two functions can be specified as functions of the same two variables. In general terms: \(z(x, y)\) and \(w(x, y)\).
So an important question that can be answered is, “What happens to \(z\) if \(w\) is held constant, but \(x\) is changed?” To explore this, consider the total differential of \(z\):
\[dz = \left( \dfrac{\partial z}{\partial x} \right)_y dx + \left( \dfrac{\partial z}{\partial y} \right)_x dy \label{eq5} \]
but \(z\) can also be considered a function of \(x\) and \(w(x, y)\). This implies that the total differential can also be written as
\[dz = \left( \dfrac{\partial z}{\partial x} \right)_w dx + \left( \dfrac{\partial z}{\partial w} \right)_x dy \label{eq6} \]
and these two total differentials must be equal to one another!
\[ = \left( \dfrac{\partial z}{\partial x} \right)_y dx + \left( \dfrac{\partial z}{\partial y} \right)_x dy = \left( \dfrac{\partial z}{\partial x} \right)_w dx + \left( \dfrac{\partial z}{\partial w} \right)_x dw \nonumber \]
If we constrain the system to a change in which \(w\) remains constant, the last term will vanish since \(dw = 0\).
\[ \left( \dfrac{\partial z}{\partial x} \right)_y dx + \left( \dfrac{\partial z}{\partial y} \right)_x dy = \left( \dfrac{\partial z}{\partial x} \right)_w dx \label{eq10} \]
but also, since \(w\) is a function \(x\) and \(y\), the total differential for \(w\) can be written
\[dw = \left( \dfrac{\partial w}{\partial x} \right)_y dx + \left( \dfrac{\partial w}{\partial y} \right)_x dy \nonumber \]
And it too must be zero for a process in which \(w\) is held constant.
\[ 0 = \left( \dfrac{\partial w}{\partial x} \right)_y dx + \left( \dfrac{\partial w}{\partial y} \right)_x dy \nonumber \]
From this expression, it can be seen that
\[dy = - \left( \dfrac{\partial w}{\partial x} \right)_y \left( \dfrac{\partial y}{\partial w} \right)_x dx \nonumber \]
Substituting this into the Equation \ref{eq10}, yields
\[ \left( \dfrac{\partial z}{\partial x} \right)_y dx + \left( \dfrac{\partial z}{\partial y} \right)_x \left[ - \left( \dfrac{\partial w}{\partial x} \right)_y \left( \dfrac{\partial y}{\partial w} \right)_x dx \right] = \left( \dfrac{\partial z}{\partial x} \right)_w dx \label{eq20} \]
which simplifies to
\[ \left( \dfrac{\partial z}{\partial x} \right)_y dx - \left( \dfrac{\partial z}{\partial w} \right)_x \left( \dfrac{\partial w}{\partial x} \right)_y dx = \left( \dfrac{\partial z}{\partial x} \right)_w dx \nonumber \]
So for \(dx \neq 0\), implies that
\[ \left( \dfrac{\partial z}{\partial x} \right)_y - \left( \dfrac{\partial z}{\partial w} \right)_x \left( \dfrac{\partial w}{\partial x} \right)_y = \left( \dfrac{\partial z}{\partial x} \right)_w \nonumber \]
or
\[ \left( \dfrac{\partial z}{\partial x} \right)_y = \left( \dfrac{\partial z}{\partial x} \right)_w + \left( \dfrac{\partial z}{\partial w} \right)_x \left( \dfrac{\partial w}{\partial x} \right)_y \label{final1} \]
As with partial derivative transformation types I and II, this result can be achieved in a formal, albeit less mathematically rigorous method.
Consider \(z(x, w)\). This allows us to write the total differential for \(z\):
\[ dz = \left( \dfrac{\partial z}{\partial x} \right)_w dx + \left( \dfrac{\partial z}{\partial w} \right)_x dw \nonumber \]
Now, divide by \(dx\) and constrain to constant \(y\).
\[\left.\dfrac{dz}{dx} \right\rvert_{y}= \left( \dfrac{\partial z}{\partial x} \right)_w \left.\dfrac{dx}{dx} \right\rvert_{y} + \left( \dfrac{\partial z}{\partial w} \right)_x \left.\dfrac{dw}{dx} \right\rvert_{y} \nonumber \]
noting that \(dx/dx = 1\) and converting the other ratios to partial derivatives yields
\[ \left( \dfrac{\partial z}{\partial x} \right)_y = \left( \dfrac{\partial z}{\partial x} \right)_w + \left( \dfrac{\partial z}{\partial w} \right)_x \left( \dfrac{\partial w}{\partial x} \right)_y \label{final2} \]
which agrees with the previous result (Equation \ref{final1})! Again, the method is not mathematically rigorous, but it works so long as \(w\), \(x\), \(y\), and \(z\) are state functions and the total differentials \(dw\), \(dx\), \(dy\), and \(dz\) are exact.