Thermodynamics frequently requires the evaluation of partial derivatives that cannot be measured directly. Fortunately, many derivatives can be transformed into equivalent forms using relationships between state variables.
Consider a system described by three variables related through a constraint
\[ F(x,y,z)=0 \]
Because the variables are related, only two can be varied independently. This allows the third variable to be expressed as a function of the other two:
\[ z=z(x,y) \]
or
\[ y=y(x,z) \]
These relationships lead to several useful transformation rules for thermodynamic partial derivatives.
The reciprocal rule states that
\[ \boxed{ \left( \frac{\partial z} {\partial y} \right)_x = \frac{1} { \left( \frac{\partial y} {\partial z} \right)_x } } \]
Begin with
\[ z=z(x,y) \]
and
\[ y=y(x,z) \]
Their total differentials are
\[ dz = \left( \frac{\partial z} {\partial x} \right)_y dx + \left( \frac{\partial z} {\partial y} \right)_x dy \]
\[ dy = \left( \frac{\partial y} {\partial x} \right)_z dx + \left( \frac{\partial y} {\partial z} \right)_x dz \]
Holding \(x\) constant (\(dx=0\)):
\[ dz = \left( \frac{\partial z} {\partial y} \right)_x dy \]
\[ dy = \left( \frac{\partial y} {\partial z} \right)_x dz \]
Dividing the first equation by the second gives the reciprocal rule.
Begin with
\[ z=z(x,y) \]
So \[ dz = \left( \frac{\partial z}{\partial x} \right)_ y dx + \left( \frac{\partial z}{\partial y} \right)_x dy \]
Divide by \(dz\) and constrain to constant \( x \):
\[ \left. \frac{dz}{dz} \right|_x = \left( \frac{\partial z}{\partial x} \right)_y \left. \frac{dx}{dz} \right|_x + \left( \frac{\partial z}{\partial y} \right)_x \left. \frac{dy}{dz} \right|_x \]
At this stage, we are using a common thermodynamic shortcut. Although differentials are not literally fractions, exact differentials often behave as if they were during algebraic manipulations. This shortcut is not fully mathematically rigorous, but it produces correct results for the exact differentials encountered throughout thermodynamics.
The purpose of this derivation is not to prove the reciprocal rule from first principles, but to provide a quick method that helps explain why the rule works and how it can be used in practice.
Noting that \(dx = 0\) when \(x\) is held constant, and that \[\left. \frac{dz}{dz} \right|_x = 1, \] the expression becomes
\[ 1 = 0 + \left( \frac{\partial z} {\partial y} \right)_x \left( \frac{\partial y}{\partial z} \right)_x \]
Rearranging immediately gives
\[ \left( \frac{\partial z} {\partial y} \right)_x = \frac{1} {\left( \frac{\partial y} {\partial z} \right)_x } \]
\[ \boxed{ \left( \frac{\partial x}{\partial y} \right)_z \left( \frac{\partial y}{\partial z} \right)_x \left( \frac{\partial z}{\partial x} \right)_y = -1 } \]
Or
\[ \boxed{ \left( \frac{\partial z}{\partial y} \right)_x = - \left( \frac{\partial z}{\partial x} \right)_y \left( \frac{\partial x}{\partial y} \right)_z } \]
Conser \( z = z(x, y) \) and \( x = x(y,z) \). We can write the total differentials as
\[ dz = \left( \frac{\partial z}{\partial x} \right)_y dx + \left( \frac{\partial z}{\partial y} \right)_x dy \] and \[ dx = \left( \frac{\partial x}{\partial y} \right)_z dy + \left( \frac{\partial x}{\partial z} \right)_y dz \]
If \( z \) is held constant, the first equation becomes \[ 0 = \left( \frac{\partial z}{\partial x} \right)_y dx + \left( \frac{\partial z}{\partial y} \right)_x dy \] and the second becomes \[ dx = \left( \frac{\partial x}{\partial y} \right)_z dy + 0 \]
Substituting this expression for \( dx \) into the previous expression yields \[ 0 = \left( \frac{\partial z}{\partial x} \right)_y \left( \frac{\partial x}{\partial y} \right)_z dy + \left( \frac{\partial z}{\partial y} \right)_x dy\]
which for \( dy \neq 0 \) can only be true if \[ \left( \frac{\partial z}{\partial y} \right)_x = - \left( \frac{\partial z}{\partial x} \right)_y \left( \frac{\partial x}{\partial y} \right)_z \] or (when rearranged) \[ \boxed{ \left( \frac{\partial x}{\partial y} \right)_z \left( \frac{\partial y}{\partial z} \right)_x \left( \frac{\partial z}{\partial x} \right)_y = -1 } \]
Begin with
\[ z=z(x,y) \]
So
\[ dz = \left( \frac{\partial z} {\partial x} \right)_y dx + \left( \frac{\partial z} {\partial y} \right)_x dy \]
Divide by \(dx\) and constrain to constant \(z\):
\[ \left. \frac{dz}{dx} \right|_z = \left( \frac{\partial z} {\partial x} \right)_y \left. \frac{dx}{dx} \right|_z + \left( \frac{\partial z} {\partial y} \right)_x \left. \frac{dy}{dx} \right|_z \]
At this stage, we are again using a common thermodynamic shortcut. Although differentials are not literally fractions, exact differentials often behave as if they were during algebraic manipulations. This shortcut is not fully mathematically rigorous, but it produces correct results for the exact differentials encountered throughout thermodynamics.
Since \(z\) is held constant, \(dz = 0\), and \(\left. \frac{dx}{dx} \right|_z = 1\). Therefore,
\[ 0 = \left( \frac{\partial z} {\partial x} \right)_y + \left( \frac{\partial z} {\partial y} \right)_x \left( \frac{\partial y} {\partial x} \right)_z \]
Rearranging gives
\[ - \left( \frac{\partial z} {\partial x} \right)_y = \left( \frac{\partial z} {\partial y} \right)_x \left( \frac{\partial y} {\partial x} \right)_z \]
Now divide by \(\left(\frac{\partial z}{\partial y}\right)_x\):
\[ - \frac{ \left( \frac{\partial z} {\partial x} \right)_y } { \left( \frac{\partial z} {\partial y} \right)_x } = \left( \frac{\partial y} {\partial x} \right)_z \]
Using the reciprocal rule,
\[ \frac{1} { \left( \frac{\partial z} {\partial y} \right)_x } = \left( \frac{\partial y} {\partial z} \right)_x \]
gives
\[ - \left( \frac{\partial z} {\partial x} \right)_y \left( \frac{\partial y} {\partial z} \right)_x = \left( \frac{\partial y} {\partial x} \right)_z \]
Rearranging into cyclic form gives
\[ \boxed{ \left( \frac{\partial x} {\partial y} \right)_z \left( \frac{\partial y} {\partial z} \right)_x \left( \frac{\partial z} {\partial x} \right)_y = -1 } \]
The negative sign is the key feature of the cyclic permutation rule.