Temperature Correction of log K
In the LMA approach, each aqueous species and mineral is defined by the reaction formula and the corresponding equilibrium constant (K value).^{1} The K value, however, depends on temperature T. There are several ways to quantify and parameterize the Tdependence:
•  Van’t Hoff Equation  ΔC_{P} = 0  
•  General Approach  ΔC_{P} as a 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. Here, the central quantity is the change of the heat capacity in a reaction:
(1)  ΔC_{P } = C_{P }(products) – C_{P}(reactants) 
Be aware that ΔC_{P} = 0 does not mean that the constantpressure heat capacity C_{P} of a product or reactant is zero (which is never the case because C_{P} > 0 applies for each constituent). Moreover, ΔC_{P} can even become negative.
Van’t Hoff Equation
Given is the fundamental relationship between the equilibrium constant K and the Gibbs energy change ΔG^{0}:^{2}
(1.1) 
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. Recall that 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)  
(1.3b) 
Now subtract the second from the first equation and you get (using ln a – ln b = ln (a/b)):
(1.4) 
This formula is known as the van’t Hoff equation. Here the temperature dependence of K is determined by the enthalpy change ΔH^{0} (provided ΔH^{0} does not depend on T itself).
Using the conversion between the decadic and natural logarithm, ln K = ln 10 · K = 2.303 · K, 1.4 can be converted to:
(1.5) 
Using 1.2 it can be rewritten as
(1.6) 
In fact, K becomes K_{o} for T = T_{o}.
When the reaction is endothermic (ΔH^{0} > 0), then for high temperatures (T > T_{o}) the K value increases, 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 reactants (the equilibrium reaction ‘shifts to the left’). This is in full accord with the principle of Le Chatelier.
Notation. In literature, thermodynamic quantities such as ΔG^{0} are decorated even with more indices than we have done here (for simplicity’s sake). Each symbol and index have their own meaning:

The symbol Δ indicates energy changes between products and reactants: ΔG = G_{prod} – G_{reac} (similar to 1). To make it more explicit, instead of ΔG often Δ_{r}G is used in text books, where r stands for reaction. This applies also for ΔH and ΔS.

The superscript “0” in ΔG^{0} refers to the standard Gibbs energy change, which is a constant value listed in tables. 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}.
In the following we simplify the notation. Since the subsequent discussion refers to equilibrium quantities only, we 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 constant. In general, however, the enthalpy and entropy changes depend on T via the heat capacity ΔC_{P} (at fixed pressure P):
(2.1a)  or  
(2.1b)  or 
The heat capacity itself may depend on temperature. At the moment let’s introduce the following abbreviations for the integrals:
(2.2a)  ΔI_{H}  ≡  ΔH – ΔH_{o}  =  
(2.2b)  ΔI_{S}  ≡  ΔS – ΔS_{o}  = 
(2.2c) 
The aim is now to derive a K formula that explicitly contains the heat capacity or its integrals. We start with 1.1 and plug 1.2 into it:
(2.3a)  =  
(2.3b)  = 
Now we replace ΔS – ΔS_{o} by 2.2b and obtain
(2.4) 
In the next step we apply 2.2a and 2.2c to get
(2.5) 
which can also be written as:
(2.6) 
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.
TCorrection Formula for 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) 
TCorrection Formula of PhreeqC
Let’s assume the following parameterization of the heat capacity:
(4.1)  ΔC_{P}(T) = a + bT – c/T^{2}  (Maier and Kelley 1932) 
The integration in (2.2a) and (2.2b) leads to:
(4.2a)  ΔI_{H}  =  
(4.2b)  ΔI_{S}  = 
Inserting it into 2.6 we obtain – after some algebra – the following parameterization of the K value:
(4.3) 
where the five coefficients (A, B, C, D, E) are simple constructs of ΔS_{o}, ΔH_{o}, a, b, and c:
(4.4a)  A  =  
(4.4b)  B  =  
(4.4c)  C  =  
(4.4d)  D  =  
(4.4e)  E  = 
Conversely, given the five Kparameters (A, B, C, D, E), we are able to retrieve the thermodynamic quantities:
(4.5a)  ΔH  =  
(4.5b)  ΔS  =  
(4.5c)  a  =  
(4.5d)  b  =  
(4.5e)  c  = 
The last three equations define the heat capacity:
(4.6)  ΔC_{P}  = 
Equation (4.3) is the parameterization used by PhreeqC and other hydrochemistry programs.
Application in Hydrochemistry Models
In order to calculate equilibrium reactions at temperatures different from the the standard temperature 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 temperature correction of K:
 Van’t Hoff equation based on constant enthalpy ΔH^{0}
 closedform equation based on five coefficients – see 4.3
Which option is actually used depends on the data available for each species and mineral. This information is hardwired in the thermodynamic database (which underlies every hydrochemical program).
Remarks & References