1 Objective

The objective of this post is to derive the following mathematical expression:

\[\begin{equation} \left(\frac{\partial x}{\partial y}\right)_z = \frac{\displaystyle\left(\frac{\partial z}{\partial v}\right)_T \left(\frac{\partial x}{\partial T}\right)_v - \left(\frac{\partial z}{\partial T}\right)_v \left(\frac{\partial x}{\partial v}\right)_T}{\displaystyle\left(\frac{\partial z}{\partial v}\right)_T \left(\frac{\partial y}{\partial T}\right)_v - \left(\frac{\partial z}{\partial T}\right)_v \left(\frac{\partial y}{\partial v}\right)_T} \tag{1.1} \end{equation}\]

Equation (1.1) permits deriving an analytical expression for any thermodynamic property from a pressure explicit equation of state. This expression is key to Bridgman’s tables [1].

2 Derivation

First, let’s define a function \(z\) which depends on variables \(x\) and \(y\). Therefore, its differential form writes:

\[\begin{equation} dz\left(x,y\right) = \left(\frac{\partial z}{\partial x}\right)_y dx + \left(\frac{\partial z}{\partial y}\right)_x dy \tag{2.1} \end{equation}\]

Next step is to assume that both \(x\) and \(y\) are functions of variables \(T\) and \(v\). Therefore, differentials of \(T\) and \(v\) write:

\[\begin{equation} \left\{ \begin{array}[l] &dx\left(T,v\right) & = \displaystyle\left(\frac{\partial x}{\partial T}\right)_v dT + \left(\frac{\partial x}{\partial v}\right)_T dv \\ dy\left(T,v\right) & = \displaystyle\left(\frac{\partial y}{\partial T}\right)_v dT + \left(\frac{\partial y}{\partial v}\right)_T dv \end{array} \right. \tag{2.2} \end{equation}\]

Combining equations (2.1) and (2.2), one obtains:

\[\begin{equation} dz\left(x,y\right) = \left(\frac{\partial z}{\partial x}\right)_y \times \left[\left(\frac{\partial x}{\partial T}\right)_v dT + \left(\frac{\partial x}{\partial v}\right)_T dv\right] + \left(\frac{\partial z}{\partial y}\right)_x \times \left[\displaystyle\left(\frac{\partial y}{\partial T}\right)_v dT + \left(\frac{\partial y}{\partial v}\right)_T dv\right] \tag{2.3} \end{equation}\]

Terms in this equations can be rearranged with respect to \(dT\) and \(dv\) to get:

\[\begin{equation} dz\left(x,y\right) = \left[\left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial T}\right)_v + \left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial T}\right)_v\right]dT + \left[\left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial v}\right)_T + \left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial v}\right)_T\right]dv \tag{2.4} \end{equation}\]

Moreover, without loss of generality, it is possible to consider that function \(z\) is depends on variables \(T\) and \(v\). Thus, its differential writes:

\[\begin{equation} dz\left(T,v\right) = \left(\frac{\partial z}{\partial T}\right)_v dT + \left(\frac{\partial z}{\partial v}\right)_T dv \tag{2.5} \end{equation}\]

Because \(z\left(x,y\right) = z\left(T,v\right)\), one gets \(dz\left(x,y\right) = dz\left(T,v\right)\). After equating equations (2.4) and (2.5) and reorganising the terms, one can write:

\[\begin{equation} \left[\left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial T}\right)_v + \left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial T}\right)_v - \left(\frac{\partial z}{\partial T}\right)_v\right]dT + \left[\left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial v}\right)_T + \left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial v}\right)_T- \left(\frac{\partial z}{\partial v}\right)_T\right]dv = 0 \tag{2.6} \end{equation}\]

In order to satisfy (2.6), factors in front of \(dT\) and \(dv\) must both the zero, which is equivalent to:

\[\begin{equation} \left\{ \begin{array}[l] & \displaystyle \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial T}\right)_v + \left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial T}\right)_v - \left(\frac{\partial z}{\partial T}\right)_v = & 0 \\ \displaystyle \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial v}\right)_T + \left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial v}\right)_T- \left(\frac{\partial z}{\partial v}\right)_T =& 0 \end{array} \right. \tag{2.7} \end{equation}\]

Equation (2.7) can be recast in the matrix form:

\[\begin{equation} \underbrace{ \begin{bmatrix} \displaystyle\left(\frac{\partial x}{\partial T}\right)_v & \displaystyle\left(\frac{\partial y}{\partial T}\right)_v \\ \displaystyle\left(\frac{\partial x}{\partial v}\right)_T & \displaystyle\left(\frac{\partial y}{\partial v}\right)_T \end{bmatrix} }_A \times \underbrace{ \begin{bmatrix} \displaystyle\left(\frac{\partial z}{\partial x}\right)_y \\ \displaystyle\left(\frac{\partial z}{\partial y}\right)_x \end{bmatrix} }_X - \underbrace{ \begin{bmatrix} \displaystyle\left(\frac{\partial z}{\partial T}\right)_v \\ \displaystyle\left(\frac{\partial z}{\partial v}\right)_T \end{bmatrix} }_B =0 \tag{2.8} \end{equation}\]

In equation (2.8) elements of matrices \(A\) and \(B\) can be estimated from pressure explicit equation of state because they are derivatives of functions expressed with respect to variables \(T\) and \(v\). Thus, only elements of vector \(X\) are unknown and remain to be expressed with the partial derivatives expressed in matrices \(A\) and \(B\). To do so, one can rewrite (2.8) this way:

\[\begin{equation} \underbrace{ \begin{bmatrix} \displaystyle\left(\frac{\partial z}{\partial x}\right)_y \\ \displaystyle\left(\frac{\partial z}{\partial y}\right)_x \end{bmatrix} }_X = \underbrace{ \begin{bmatrix} \displaystyle\left(\frac{\partial x}{\partial T}\right)_v & \displaystyle\left(\frac{\partial y}{\partial T}\right)_v \\ \displaystyle\left(\frac{\partial x}{\partial v}\right)_T & \displaystyle\left(\frac{\partial y}{\partial v}\right)_T \end{bmatrix}^{-1} }_{A^{-1}} \times \underbrace{ \begin{bmatrix} \displaystyle\left(\frac{\partial z}{\partial T}\right)_v \\ \displaystyle\left(\frac{\partial z}{\partial v}\right)_T \end{bmatrix} }_B \tag{2.9} \end{equation}\]

From basic calculus, matrix \(A^{-1}\) has the following expression:

\[\begin{equation} A^{-1} = \frac{1}{\left(\frac{\partial x}{\partial T}\right)_v \left(\frac{\partial y}{\partial v}\right)_T - \left(\frac{\partial y}{\partial T}\right)_v \left(\frac{\partial x}{\partial v}\right)_T} \times \begin{bmatrix} \displaystyle\left(\frac{\partial y}{\partial v}\right)_T & \displaystyle -\left(\frac{\partial y}{\partial T}\right)_v \\ \displaystyle -\left(\frac{\partial x}{\partial v}\right)_T & \displaystyle\left(\frac{\partial x}{\partial T}\right)_v \end{bmatrix} \tag{2.10} \end{equation}\]

Therefore, equation (2.9) rewrites:

\[\begin{equation} \underbrace{ \begin{bmatrix} \displaystyle\left(\frac{\partial z}{\partial x}\right)_y \\ \displaystyle\left(\frac{\partial z}{\partial y}\right)_x \end{bmatrix} }_X = \underbrace{ \frac{1}{\left(\frac{\partial x}{\partial T}\right)_v \left(\frac{\partial y}{\partial v}\right)_T - \left(\frac{\partial y}{\partial T}\right)_v \left(\frac{\partial x}{\partial v}\right)_T} \times \begin{bmatrix} \displaystyle\left(\frac{\partial y}{\partial v}\right)_T & \displaystyle -\left(\frac{\partial y}{\partial T}\right)_v \\ \displaystyle -\left(\frac{\partial x}{\partial v}\right)_T & \displaystyle\left(\frac{\partial x}{\partial T}\right)_v \end{bmatrix} }_{A^{-1}} \times \underbrace{ \begin{bmatrix} \displaystyle\left(\frac{\partial z}{\partial T}\right)_v \\ \displaystyle\left(\frac{\partial z}{\partial v}\right)_T \end{bmatrix} }_B \tag{2.11} \end{equation}\]

It is now possible to get the expressions for \(\left(\frac{\partial z}{\partial x}\right)_y\) and \(\left(\frac{\partial z}{\partial y}\right)_x\):

\[\begin{equation} \left\{ \begin{array}[l] &\displaystyle\left(\frac{\partial z}{\partial x}\right)_y = & \frac{\displaystyle\left(\frac{\partial y}{\partial v}\right)_T \left(\frac{\partial z}{\partial T}\right)_v - \left(\frac{\partial y}{\partial T}\right)_v \left(\frac{\partial z}{\partial v}\right)_T}{\displaystyle\left(\frac{\partial y}{\partial v}\right)_T \left(\frac{\partial x}{\partial T}\right)_v - \left(\frac{\partial y}{\partial T}\right)_v \left(\frac{\partial x}{\partial v}\right)_T}\\ \displaystyle\left(\frac{\partial z}{\partial y}\right)_x = & \frac{\displaystyle\left(\frac{\partial x}{\partial T}\right)_v \left(\frac{\partial z}{\partial v}\right)_T - \left(\frac{\partial x}{\partial v}\right)_T \left(\frac{\partial z}{\partial T}\right)_v}{\displaystyle\left(\frac{\partial x}{\partial T}\right)_v \left(\frac{\partial y}{\partial v}\right)_T - \left(\frac{\partial x}{\partial v}\right)_T \left(\frac{\partial y}{\partial T}\right)_v} \end{array} \tag{2.12} \right. \end{equation}\]

Because variables \(x\), \(y\) and \(z\) are dummy variables, both equations in system (2.12) are equivalent and only one can be considered. This allows to write:

\[\begin{equation} \left(\frac{\partial x}{\partial y}\right)_z = \frac{\displaystyle\left(\frac{\partial z}{\partial v}\right)_T \left(\frac{\partial x}{\partial T}\right)_v - \left(\frac{\partial z}{\partial T}\right)_v \left(\frac{\partial x}{\partial v}\right)_T}{\displaystyle\left(\frac{\partial z}{\partial v}\right)_T \left(\frac{\partial y}{\partial T}\right)_v - \left(\frac{\partial z}{\partial T}\right)_v \left(\frac{\partial y}{\partial v}\right)_T} \tag{2.13} \end{equation}\]

which is equation (1.1).

3 Jacobian notation

One can observe that the numerator and the denominator of equation (2.13) can be expressed as determinants:

\[\begin{equation} \left(\frac{\partial x}{\partial y}\right)_z = \frac{ \begin{vmatrix} \displaystyle\left(\frac{\partial x}{\partial T}\right)_v & \displaystyle\left(\frac{\partial x}{\partial v}\right)_T \\ \displaystyle\left(\frac{\partial z}{\partial T}\right)_v & \displaystyle\left(\frac{\partial z}{\partial v}\right)_T \end{vmatrix} } { \begin{vmatrix} \displaystyle\left(\frac{\partial y}{\partial T}\right)_v & \displaystyle\left(\frac{\partial y}{\partial v}\right)_T \\ \displaystyle\left(\frac{\partial z}{\partial T}\right)_v & \displaystyle\left(\frac{\partial z}{\partial v}\right)_T \end{vmatrix} } \tag{3.1} \end{equation}\]

Such determinants are commonly written:

\[\begin{equation} \left\{ \begin{array}[l] &\left(\partial x\right)_z &= \begin{vmatrix} \displaystyle\left(\frac{\partial x}{\partial T}\right)_v & \displaystyle\left(\frac{\partial x}{\partial v}\right)_T \\ \displaystyle\left(\frac{\partial z}{\partial T}\right)_v & \displaystyle\left(\frac{\partial z}{\partial v}\right)_T \end{vmatrix} \\ \left(\partial y\right)_z &= \begin{vmatrix} \displaystyle\left(\frac{\partial y}{\partial T}\right)_v & \displaystyle\left(\frac{\partial y}{\partial v}\right)_T \\ \displaystyle\left(\frac{\partial z}{\partial T}\right)_v & \displaystyle\left(\frac{\partial z}{\partial v}\right)_T \end{vmatrix} \end{array} \right. \tag{3.2} \end{equation}\]

with \(\left(\partial x\right)_z\) and \(\left(\partial y\right)_z\) being the determinant of the Jacobian matrices.

Finally, equation (3.1) writes in a simpler form:

\[\begin{equation} \left(\frac{\partial x}{\partial y}\right)_z = \frac{\left(\partial x\right)_z}{\left(\partial y\right)_z} \tag{3.3} \end{equation}\]

Bridman’s tables provide \(\left(\partial x\right)_z\) expressions for most commonly encountered \(x\) and \(z\) properties in thermodynamics.