## 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 ΔH0(T) = const • General Approach for ΔH0(T) ≠ const

where ΔH0 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 ΔG0:

 (1) $log \, K = -\dfrac{\Delta G^{0}}{2.303 \,\cdot RT}$

where R = 8.314 J mol-1 K-1 is the gas constant. In classical thermodynamics, the change in the Gibbs energy, ΔG0, results from both enthalpy change ΔH0 and entropy change ΔS0:

 (2) ΔG0  =  ΔH0 – T ΔS0

Inserting 2 into 1 for an arbitrary temperature T (defining K) and for the standard temperature T0 (defining K0) yields

 (3a) $log \, K = - \dfrac{\Delta H^{0}}{2.303 \, RT} - \dfrac{\Delta S^{0}}{2.303 \, R}$ (3b) $log \, K_{0} = - \dfrac{\Delta H^{0}}{2.303 \, RT_0} - \dfrac{\Delta S^{0}}{2.303 \, R}$

Now subtract the second from the first equation, where the ΔS0 term cancels, and you get:

 (4) $log \, K \ = \ log \, K_{0} + \dfrac{\Delta H^{0}}{2.303 \, R} \left ( \dfrac{1}{T_{0}} -\dfrac{1}{T} \right )$

This formula is known as the Van’t Hoff equation. Here, the complete temperature dependence of the K is determined by the enthalpy change ΔH0 alone.

When the reaction is endothermic (ΔH0 > 0), then for high temperatures (T > T0) the K increases, which promotes product formation (the equilibrium reaction ‘shifts right’). Conversely, if the reaction is exothermic (ΔH0 < 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) $ln\,\dfrac{K}{K_{0}} \ = \ \dfrac{\Delta H^{0}}{R} \left ( \dfrac{1}{T_{0}} -\dfrac{1}{T} \right )$

Approach based on Heat Capacity

The temperature correction a la Van’t Hoff in 4 is based on the assumption that ΔH0 does not depend on the temperature at all. In general, however, the enthalpy ΔH0 depends on T via the heat capacity CP (at fixed pressure P) as expressed by the fundamental thermodynamic relation:

 (6) $\left( \dfrac{\partial H}{\partial T}\right)_P \ = \ C_P$ or $H-H_0 \, = \, \int\limits^T_{T_0} \, C_P \, dT$

Usually, the following parameterization is assumed for the heat capacity:

 (7) CP   =   a + bT + cT2 (heat capacity polynomial fit)

which after integration in 6 – and some thermodynamic manipulations (not considered here) – yields the following parametrization of K:

 (8) $ln \,K \ = \ const - \dfrac{\Delta H^0_0}{R} \dfrac{1}{T} + \dfrac{\Delta a}{R} \,ln \, T + \dfrac{\Delta b}{R} \,T + \dfrac{\Delta c}{R} \,T^2$

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 CP 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 ΔH0
• 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).