Temperature Correction of log K
In the LMA approach, each aqueous species and each mineral is defined by the reaction formula and the corresponding equilibrium constant (K value).^{1} The K value, however, strongly depends on temperature T. There are several ways to quantify and parameterize it:
•  van’t Hoff equation  ΔC_{P} = 0  
•  general approach  ΔC_{P} as an (arbitrary) function of T  
•  constant ΔC_{P} approach  ΔC_{P} = const  
•  PhreeqC approach  ΔC_{P} = a + bT – c/T^{2} 
The last two approaches are special cases of the general approach, which all rely on the change of the heat capacity in a reaction:
(1)  ΔC_{P } = C_{P }(products) – C_{P}(educts) 
Note that ΔC_{P} = 0 does not mean that the constantpressure heat capacity C_{P} of a product or educt is zero (which can never be the case, since C_{P} > 0 applies to each component). Moreover, ΔC_{P} can even become negative.
Motivation. The study of the thdyn relationships will shed light on two things: (i) a proper understanding of the PhreeqCparametrization (it’s in use by several hydrochemistry models and databases) and (ii) the reconstruction of thdyn quantities (like ΔS and ΔH) just from these PhreeqCparameters.
van’t Hoff Equation
Given is the fundamental relationship between the equilibrium constant K and the Gibbs energy change ΔG^{0}:^{2}
(1.1)  \(\ln K = \dfrac{\Delta G^{0}}{RT}\) 
where R = 8.314 J mol^{1 }K^{1} is the gas constant. The Gibbs energy itself is a construct of both enthalpy H and entropy S: G = H – TS. This relationship applies also for the Gibbs energy change:
(1.2)  ΔG^{0} = ΔH^{0} – T ΔS^{0} 
where ΔH^{0} is the heat absorbed or released when the reaction takes place under constant pressure. Enthalpy is the energy involved in overcoming the intermolecular forces (breaking and making chemical bonds), while entropy refers to the degree of disorder or “mixedupness” (Gibbs) of the system.
Inserting 1.2 into 1.1, for an arbitrary temperature T (associated with K) and for the standard temperature T_{o} (associated with K_{o}), yields
(1.3a)  \(\ln K \, =  \dfrac{1}{R} \, \left( \dfrac{\Delta H^{0}}{T}  \Delta S^{0} \right)\) 
(1.3b)  \(\ln K_o \,= \dfrac{1}{R} \, \left( \dfrac{\Delta H^{0}}{T_o}  \Delta S^{0} \right)\) 
Subtracting the second from the first equation, and using ln a – ln b = ln (a/b), leads to:
(1.4)  \(\ln \dfrac{K}{K_{o}} \ = \ \dfrac{\Delta H^{0}}{R} \,\left( \dfrac{1}{T_{o}} \dfrac{1}{T} \right )\) 
This formula is known as the van’t Hoff equation, where the Tdependence of K is determined by the enthalpy change ΔH^{0} alone (provided ΔH^{0} itself does not depend on T).
The switch from decadic to natural logarithm, ln K = ln 10 · K = 2.3 · K, converts 1.4 to:
(1.5)  \(\lg K \ = \ \lg K_{o} \,+\, \dfrac{\Delta H^{0}}{2.3 \, R} \,\left( \dfrac{1}{T_{o}} \dfrac{1}{T} \right)\) 
With 1.2 it can also be written as
(1.6)  \(\lg K \ = \ \dfrac{1}{2.3 \, R} \,\left( \Delta S^0  \dfrac{\Delta H^0}{T} \right)\) 
For T = T_{o}, K collapses to K_{o}.
If the reaction is endothermic (ΔH^{0} > 0), then the K value increases with increasing temperature T, which promotes product formation (the equilibrium reaction ‘shifts to the right’). Conversely, if the reaction is exothermic (ΔH^{0} < 0), higher temperatures will promote the formation of educts (the equilibrium reaction ‘shifts to the left’). This is in full accord with the principle of Le Chatelier.
Notation. In the literature, thdyn quantities such as ΔG^{0} are decorated with even more indices than we have done here (for simplicity’s sake). Each symbol and index has its own meaning:

The symbol Δ indicates energy changes between products and educts: ΔG = G_{prod} – G_{educ} (similar to 1). To make it more explicit, instead of ΔG often Δ_{r}G is used, where r stands for “reaction”. The same applies to ΔH and ΔS.

The superscript “0” in ΔG^{0} refers to the standard Gibbs energy change, which is a constant value listed in tables and valid for the equilibrium state only. It should not be confused with ΔG = ΔG^{0} + RT ln Q, where Q is the reaction quotient. Similarly, the superscript in ΔH^{0} and ΔS^{0} also indicates standard values (tabulated in books).

The subindex “o” in K_{o} refers to the standard temperature T_{o} = 25 (298 Kelvin). Henceforth, this subindex also applies to ΔG_{o}, ΔH_{o}, and ΔS_{o}.
The notation is further simplified below. Since we focus on equilibrium quantities, we will omit the superscript “0” in ΔG^{0}, ΔH^{0}, and ΔS^{0}.
General Approach based on Heat Capacity Change ΔC_{P}
The temperature correction à la van’t Hoff in 1.4 is based on the assumption that both ΔH^{0} and ΔS^{0} are constants. In general, however, enthalpy and entropy depend on T via the heat capacity ΔC_{P} (at fixed pressure P):
(2.1a)  \(\left( \dfrac{\partial \Delta H}{\partial T}\right)_{\!P} \ = \ \Delta C_P\)  or  \(\Delta H\Delta H_o \, = \, \int\limits^T_{T_o} \, \Delta C_P \, dT\)  
(2.1b)  \(\left( \dfrac{\partial \Delta S}{\partial T}\right)_{\!P} \ = \ \dfrac{\Delta C_P}{T}\)  or  \(\Delta S\Delta S_o \, = \, \int\limits^T_{T_o} \, \dfrac{\Delta C_P}{T} \, dT\) 
The heat capacity itself can also depend on the temperature. At the moment, let us introduce the following abbreviations for the integrals:
(2.2a)  ΔI_{H}  ≡  ΔH – ΔH_{o}  =  \(\int\limits^T_{T_o} \, \Delta C_P \, dT\)  
(2.2b)  ΔI_{S}  ≡  ΔS – ΔS_{o}  =  \(\int\limits^T_{T_o} \, \dfrac{\Delta C_P}{T} \, dT\) 
(2.2c)  \(\Delta I \ \ \equiv \ \ \Delta I_S  \dfrac{\Delta I_H}{T}\) 
The aim is to derive a formula for K that explicitly contains the heat capacity or its integrals. We start with 1.1 and plug 1.2 into it:
(2.3a)  \(R\,\ln \dfrac{K}{K_{o}}\)  =  \(\left( \dfrac{\Delta G}{T}  \dfrac{\Delta G_o}{T_{o}} \right)\)  
(2.3b)  =  \(\left( \dfrac{\Delta H}{T}  \Delta S  \dfrac{\Delta H_o}{T_{o}} + \Delta S_o\right)\) 
Now we replace ΔS – ΔS_{o} by 2.2b and obtain
(2.4)  \(R\,\ln \dfrac{K}{K_{o}} \ = \ \dfrac{\Delta H_o}{T_o}  \dfrac{\Delta H}{T} + \Delta I_S\) 
In the next step, we apply 2.2a and 2.2c to get:
(2.5)  \(\ln \dfrac{K}{K_{o}} \ = \ \dfrac{\Delta H^{0}}{R} \,\left( \dfrac{1}{T_{o}} \dfrac{1}{T} \right ) \ +\ \dfrac{\Delta I}{R}\) 
which can also be written as:
(2.6)  \(\lg K \ = \ \dfrac{1}{2.3 \, R} \,\left( \Delta S^0  \dfrac{\Delta H^0}{T} \right) \, +\, \Delta I\) 
The last two formulas generalize the van’t Hoff equation in (1.4) and (1.6) for nonzero ΔC_{P}. All information about ΔC_{P} is contained in the integral ΔI.
At this point, you cannot get any further unless you provide a formula for ΔC_{P}(T) to calculate the integrals in 2.2a and (2.2b). This will be done for two cases in the next sections.
The entire treatment can be summarized as follows:
TCorrection Formula based on ΔC_{P} = const
The simplest assumption about ΔC_{P} is that it is constant:
(3.1)  ΔC_{P} = a 
The integration according to 2.2c yields:
(3.2)  ΔI = a { ln (T/T_{o}) + (T_{o}/T) – 1 } 
Inserting it into 2.5, we obtain:
(3.3)  \(\ln \dfrac{K}{K_{o}} \ = \ \dfrac{\Delta H^{0}}{R} \,\left( \dfrac{1}{T_{o}} \dfrac{1}{T} \right ) \ +\ \dfrac{a}{R} \,\left( \ln \dfrac{T}{T_o} + \dfrac{T_o}{T} 1 \right)\) 
TCorrection Formula of PhreeqC
Let us adopt the following parametrization of the heat capacity:
(4.1)  ΔC_{P}(T) = a + bT – c /T^{2}  (Maier and Kelley 1932) 
The integration over T in (2.2a) and (2.2b) leads to:
(4.2a)  ΔI_{H}  =  \(\left( aT + \dfrac{bT^2}{2} + \dfrac{c}{T} \right)  \left( aT_o + \dfrac{bT_o^2}{2} + \dfrac{c}{T_o} \right)\)  
(4.2b)  ΔI_{S}  =  \(\left( a \,\ln T + bT  \dfrac{c}{2T^2} \right)  \left( a \,\ln T_o + bT_o  \dfrac{c}{2T_o^2} \right)\) 
Inserting it into 2.6, we obtain — after some algebra — the following parametrization of the K value:
(4.3)  \(\lg K \ = \ A + B\,T + \dfrac{C}{T} + D \,\lg T + \dfrac{E}{T^2}\) 
where the five coefficients (A, B, C, D, E) are constructs of ΔS_{o}, ΔH_{o}, a, b, and c:
(4.4a)  A  =  \(\dfrac{1}{2.3\,R} \,\left( \Delta S_o  a(1+\ln T_o)  b T_o  \dfrac{c}{2T_o^2} \right)\)  
(4.4b)  B  =  \(\dfrac{1}{2.3\,R} \ \dfrac{b}{2}\)  
(4.4c)  C  =  \(\dfrac{1}{2.3\,R} \,\left( a T_o + \dfrac{b T_o^2}{2} + \dfrac{c}{T_o}  \Delta H_o\right)\)  
(4.4d)  D  =  \(\dfrac{a}{R}\)  
(4.4e)  E  =  \(\dfrac{1}{2.3\,R} \ \dfrac{c}{2}\) 
Vice versa, given the five Kparameters (A, B, C, D, E), we are able to retrieve the Tdependence of the involved thdyn quantities:
(4.5a)  ΔH (T)  =  \(2.3\,R\ \left( BT^2  C + \dfrac{DT}{2.3}  \dfrac{2E}{T} \right)\)  
(4.5b)  ΔS (T)  =  \(2.3\,R\ \left( A + 2BT + \dfrac{D}{2.3} \, (1+\ln T)  \dfrac{E}{T^2} \right)\) 
as well as the three parameters
(4.5c)  a  =  \(R\,D\)  
(4.5d)  b  =  \(\ \ \ 2.3\, R \,\cdot\, 2B\)  
(4.5e)  c  =  \(2.3\,R \,\cdot\, 2E\) 
which define — via 4.1 — the Tdependence of the heat capacity:
(4.6)  ΔC_{P} (T) = \(2.3\,R\ \left( \dfrac{D}{2.3} + 2B\cdot T + \dfrac{2E}{T^2} \right)\) 
Equation (4.3) is the parametrization used by PhreeqC and other hydrochemistry programs.
Summary. The ideas behind the K parametrization can be summarized as follows:
In this way, we are able to determine the five parameters (A, B, C, D, E) from the corresponding thdyn quantities:
The inverse task: We extract the thdyn quantities from the five PhreeqC parameters (A, B, C, D, E):
Application in Hydrochemistry Models
To calculate equilibrium reactions at temperatures other than 25 three pieces of information are required:
 reaction equation (stoichiometry)
 K value (at 25)
 a parameterization of the Tcorrection for K
PhreeqC and aqion, for example, are equipped with two options to handle the T correction of K:
 van’t Hoff equation based on constant enthalpy ΔH^{0}
 closedform equation based on five coefficients — see 4.3
The option that is actually used depends on the data available for the species and mineral. This information is hardwired in the thdyn database (which underlies every hydrochemical program).
Remarks & References