Triple product rule

This relation is known as the triple product rule. 

The isothermal compressibility (3.3.25), the thermal pressure coefficient (3.3.5), and the volume expansivity (3.3.6) satisfy a triple product rule (Appendix A) ... 

Implicit function theorem Triple product rules... 

To do so, use a triple product rule to relate this derivative to measurables. For a mixture, is there some constraint which demands that the pressure must always increase or decrease when N is increased ... 

On PT diagrams the lines of constant composition are related to the saturation curves that appear on Fxy and Txy diagrams specifically, these three sets of saturation curves are related through a triple product rule. 

If we substitute (9.3.23) into (9.3.13), we find that critical points occur at extrema on isothermal Px and Py plots. Likewise if we substitute (9.3.23) into (9.3.16), we find that critical points occur at extrema on isobaric Tx and Ty plots. However, the converses of these statements are not true extrema on such plots are not necessarily critical points we have already seen that they could be azeotropes. Further, those extrema mean that the numerator and denominator in the triple product rule (9.3.18) are both zero however, that ratio need not be zero, so on PT diagrams, critical points rarely occur at extrema of constant-composition lines. 

In thermodynamics the implicit function theorem is usually written in the form of a triple product rule ... 

A useful mathematical property is the triple-product rule. We set F = const., fi om which dF=o. The differential then becomes... 

This relationship exists between any three variables that are related by an equation. Since volume, pressure, and temperature are related via the equation of state, we obtain the following result as an immediate consequence of the triple-product rule ... 

The symmetry between F, X, and Fin the triple-product rule indicates that it does not matter which variable is taken to be the function and which are the independent variables since the equation F = F X, Y) can be solved to give X as a function of F and F, or F as a function of F and X. 

Example 5.1 Triple-Product Rule iu the Ideal-Gas State... 

Calculate the three partial derivatives that appear in the triple-product rule between P, T, and Vusing the ideal-gas equation and confirm that their product is equal to -1. 

Comments The example demonstrates that the triple-product rule is satisfied when P, V, and Tare given by the ideal-gas equation. As a general equation, eq. (r.9.) applies regardless of the form of the equation of state. 

To calculate the derivative (9V/9T)p, we have to solve the equation of state for V and perform the differentiation. However, the SRK equation is a cubic polynomial in V and cannot be expressed as an explicit function of P and T. We overcome this difficulty by using the triple-product rule to solve for the required derivative ... 

Now we work on the second integral in eq. (R.d6). We start with the triple-product rule in eq. and split the derivative (dP/dV)T to obtain... 

The following transforming derivatives are widely used in thermodynamic derivations. Triple product rule for F = F(x,y)... 

The mathematical development of Equation (5.22) from Equation (5.4) is actually more complex than this heuristic explanation. However, viewing it in this simplified manner is convenient and, in general, works. We present it here since we will use this method in other places. See Example 5.1 for an alternative development of Equation (5.25). This relation is alternatively termed the triple product rule. 

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