## Temperature Correction of log K

In the LMA approach each aqueous species and mineral is characterized by the reaction formula and the equilibrium constant (log K value). The log K value, however, depends on temperature. Two ways to accomplish this problem are:

 • Van’t Hoff Gleichung (for ΔH0(T) = const) • General Approach (for ΔH0(T) ≠ const)

Van’t Hoff Equation

Given is the relation between the equilibrium constant K and Gibbs energy ΔG0:

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

where ΔG0 refers to standard conditions at T0 = 298 K (i.e. 25 °C). In order to exhibit the effect of temperature we exploit classical thermodynamics which relates Gibbs energy ΔG0 and enthalpy ΔH0 by

 (2) ΔG0   =   ΔH0 – T ΔS0

where ΔS0 abbreviates the entropy. Inserting Eq.(2) into Eq.(1) for an arbitrary temperature T (defining log K) and for the standard temperature T0 (defining log K0) we get

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

Substracting the second from the first equation the ΔS0 term cancels out leaving the relation

 (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. The main message is that the temperature dependence of log K is determined by the enthalpy ΔH0 rather than by the Gibbs energy ΔG0.

The relation between the base-10 logarithm and the natural logarithm, ln K = ln 10 · log K = 2.303 · log K, converts Eq.(4) into

 (5) $ln\,\dfrac{K}{K_{0}} \ = \ \dfrac{\Delta H^{0}}{R} \left ( \dfrac{1}{T_{0}} -\dfrac{1}{T} \right )$

When the reaction is endothermic (ΔH0>0) then an increase of temperature increases K, which promotes product formation (the reaction ‘shifts right’). Conversely, if the reaction is exothermic (ΔH0<0), higher temperatures will promote the formation of reactants (the equilibrium ‘shifts left’). This is in full accord with the principle of Le Chatelier.

Approach based on Heat Capacity

The temperature correction a la Van’t Hoff in Eq.(4) or (5) is based on the assumption that ΔH0 does not depend on 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 ansatz is assumed for the T dependence of the heat capacity:

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

which after integration in Eq.(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 Eq.(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 Eq.(7). Nonetheless, Eq.(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)
• log K value (at 25°C)
• a parametrization of T correction for log K

aqion and PhreeqC are equipped with two options to handle the T-dependency of log 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).