Temperature Correction of log K
In the LMA approach, each aqueous species and mineral is characterized by the reaction formula and the corresponding equilibrium constant (K value). The K value, however, depends on temperature T. There are several ways to accomplish this problem, two of which are:
• | Van’t Hoff Equation | for ΔH^{0}(T) = const | |
• | General Approach | for ΔH^{0}(T) ≠ const |
where ΔH^{0} is the enthalpy change (i.e. the heat absorbed or released when the reaction takes place under constant pressure).
Van’t Hoff Equation
Given is the fundamental relationship between the equilibrium constant K and the Gibbs free energy change ΔG^{0}:
(1) |
where R = 8.314 J mol^{-1 }K^{-1} is the gas constant. In classical thermodynamics, the change in the Gibbs energy, ΔG^{0}, results from both enthalpy change ΔH^{0} and entropy change ΔS^{0}:
(2) | ΔG^{0} = ΔH^{0} – T ΔS^{0} |
Inserting 2 into 1 for an arbitrary temperature T (defining K) and for the standard temperature T_{0} (defining K_{0}) yields
(3a) | |
(3b) |
Now subtract the second from the first equation, where the ΔS^{0} term cancels, and you get:
(4) |
This formula is known as the Van’t Hoff equation. Here, the complete temperature dependence of the K is determined by the enthalpy change ΔH^{0} alone.
When the reaction is endothermic (ΔH^{0} > 0), then for high temperatures (T > T_{0}) the K increases, which promotes product formation (the equilibrium reaction ‘shifts right’). Conversely, if the reaction is exothermic (ΔH^{0} < 0), higher temperatures will promote the formation of reactants (the equilibrium reaction ‘shifts left’). This is in full accord with the principle of Le Chatelier.
The relation between the base-10 logarithm and the natural logarithm, ln K = ln 10 · K = 2.303 · K, converts 4 into
(5) |
Approach based on Heat Capacity
The temperature correction a la Van’t Hoff in 4 is based on the assumption that ΔH^{0} does not depend on the temperature at all. In general, however, the enthalpy ΔH^{0} depends on T via the heat capacity C_{P} (at fixed pressure P) as expressed by the fundamental thermodynamic relation:
(6) | or |
Usually, the following parameterization is assumed for the heat capacity:
(7) | C_{P} = a + bT + cT^{2} | (heat capacity polynomial fit) |
which after integration in 6 – and some thermodynamic manipulations (not considered here) – yields the following parametrization of K:
(8) |
Van’t Hoff’s equation is a special case of 8; it emerges if the last three terms are ignored.
Limits. Equation (8) does not provide an overall solution, because it relies on a special assumption about C_{P} in 7. Nonetheless, 8 provides a recipe to construct closed-form expressions from a couple of specific terms of T. The coefficients of those terms are then obtained by fitting measured data.
Application in aqion
In order to define/calculate an equilibrium reaction three pieces of information are required:
- reaction equation (stoichiometry)
- lK value (at 25)
- a parametrization of T correction for K
aqion and PhreeqC are equipped with two options to handle the T-dependency of K:
- Van’t Hoff equation based on the enthalpy ΔH^{0}
- closed-form equation based on five coefficients (of a “generalized polynomial”)
Which option actually is used depends on the data available for each species and mineral. This information is hardwired in the thermodynamic database (that underlies every hydrochemical program).