## 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 ΔCP = 0 • General Approach ΔCP   as an (arbitrary) function of T • Constant ΔCP Approach ΔCP = const • PhreeqC Approach ΔCP = a + bT – c/T2

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) ΔCP   =  CP (products) – CP(educts)

Note that ΔCP = 0 does not mean that the constant-pressure heat capacity CP of a product or educt is zero (which can never be the case, since CP > 0 applies to each component). Moreover, ΔCP can even become negative.

Motivation. By investigating and deriving the thermodynamic relationships, two birds were killed with one stone: (i) a proper understanding of the PhreeqC-parameterization (it’s in use by several hydrochemistry models and databases) and (ii) the reconstruction of thermodynamic quantities (like ΔS and ΔH) just from these PhreeqC-parameters.

Van’t Hoff Equation

Given is the fundamental relationship between the equilibrium constant K and the Gibbs energy change ΔG0:2

 (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) ΔG0  =  ΔH0 – T ΔS0

where ΔH0 is the heat absorbed or released when the reaction takes place under constant pressure. Recall that enthalpy is the energy involved in overcoming the inter-molecular forces (breaking and making chemical bonds), while entropy refers to the degree of disorder or “mixed-up-ness” (Gibbs) of the system.

Inserting 1.2 into 1.1, for an arbitrary temperature T (associated with K) and for the standard temperature To (associated with Ko), yields

 (1.3a) $$\ln K \, = - \dfrac{1}{R} \, \left( \dfrac{\Delta H^{0}}{T} - \Delta S^{0} \right)$$ (1.3b) $$\ln K_o \,= -\dfrac{1}{R} \, \left( \dfrac{\Delta H^{0}}{T_o} - \Delta S^{0} \right)$$

Now subtract the second from the first equation and you get (using ln a – ln b = ln (a/b)):

 (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. Here, the T-dependence of K is determined by the enthalpy change ΔH0 (provided ΔH0 itself does not depend on T).

Using the conversion between decadic and natural logarithm, ln K = ln 10 · K = 2.3 · K, we get from 1.4:

 (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) $$\lg K \ = \ \dfrac{1}{2.3 \, R} \,\left( \Delta S^0 - \dfrac{\Delta H^0}{T} \right)$$

In fact, K becomes Ko for T = To.

If the reaction is endothermic (ΔH0 > 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 (ΔH0 < 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 literature, thermodynamic quantities such as ΔG0 are decorated with even more indices than we have done here (for simplicity’s sake). Each symbol and each index has its own meaning:

• The symbol Δ indicates energy changes between products and educts: ΔG = Gprod – Geduc (similar to 1). To make it more explicit, instead of ΔG often ΔrG is used in texts, where r stands for reaction. This also applies for ΔH and ΔS.

• The superscript “0” in ΔG0 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 = ΔG0 + RT ln Q, where Q is the reaction quotient. Similarly, the superscript in ΔH0 and ΔS0 also indicates standard values (tabulated in books).

• The subindex “o” in Ko refers to the standard temperature To = 25 (298 Kelvin). Henceforth, this subindex also applies to ΔGo, ΔHo, and ΔSo.

In the following, we further simplify the notation. Since we only deal with equilibrium quantities, we will omit the superscript “0” in ΔG0, ΔH0, and ΔS0.

General Approach based on Heat Capacity Change ΔCP

The temperature correction à la van’t Hoff in 1.4 is based on the assumption that both ΔH0 and ΔS0 are constant. In general, however, the enthalpy and entropy changes depend on T via the heat capacity ΔCP (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 may depend on temperature. At the moment, let’s introduce the following abbreviations for the integrals:

 (2.2a) ΔIH ≡ ΔH – ΔHo = $$\int\limits^T_{T_o} \, \Delta C_P \, dT$$ (2.2b) ΔIS ≡ ΔS – ΔSo = $$\int\limits^T_{T_o} \, \dfrac{\Delta C_P}{T} \, dT$$
 (2.2c) $$\Delta I \ \ \equiv \ \ \Delta I_S - \dfrac{\Delta I_H}{T}$$

The aim is to now 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) $$R\,\ln \dfrac{K}{K_{o}}$$ = $$-\left( \dfrac{\Delta G}{T} - \dfrac{\Delta G_o}{T_{o}} \right)$$ (2.3b) = $$-\left( \dfrac{\Delta H}{T} - \Delta S - \dfrac{\Delta H_o}{T_{o}} + \Delta S_o\right)$$

Now we replace ΔS – ΔSo 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 non-zero ΔCP. All information about ΔCP is contained in the integral ΔI.

At this point, you cannot get any further unless you provide a formula for ΔCP(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: T-Correction Formula for ΔCP = const

The simplest assumption about ΔCP is that it is constant:

 (3.1) ΔCP  =  a

The integration according to 2.2c yields:

 (3.2) ΔI  =  a { ln  (T/To) + (To/T) – 1 }

Inserting it into 2.5 we obtain:

 (3.3) $$\ln \dfrac{K}{K_{o}} \ = \ \dfrac{\Delta H^{0}}{R} \,\left( \dfrac{1}{T_{o}} -\dfrac{1}{T} \right ) \ +\ \dfrac{a}{R} \,\left( \ln \dfrac{T}{T_o} + \dfrac{T_o}{T} -1 \right)$$

T-Correction Formula of PhreeqC

Let’s assume the following parameterization of the heat capacity:

 (4.1) ΔCP(T)   =   a + bT – c /T2 (Maier and Kelley 1932)

The integration over T in (2.2a) and (2.2b) leads to:

 (4.2a) ΔIH = $$\left( aT + \dfrac{bT^2}{2} + \dfrac{c}{T} \right) - \left( aT_o + \dfrac{bT_o^2}{2} + \dfrac{c}{T_o} \right)$$ (4.2b) ΔIS = $$\left( a \,\ln T + bT - \dfrac{c}{2T^2} \right) - \left( a \,\ln T_o + bT_o - \dfrac{c}{2T_o^2} \right)$$

Inserting it into 2.6 we obtain – after some algebra – the following parameterization 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 simple constructs of ΔSo, ΔHo, 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 K-parameters (A, B, C, D, E), we are able to retrieve the T-dependence of the involved thermodynamic 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 T-dependence of the heat capacity:

 (4.6) ΔCP (T)   =   $$2.3\,R\ \left( \dfrac{D}{2.3} + 2B\cdot T + \dfrac{2E}{T^2} \right)$$

Equation (4.3) is the parameterization used by PhreeqC and other hydrochemistry programs. In fact, these are all really non-trivial relationships that cannot be guessed at in advance.

Summary. The ideas behind the K parameterization can be summarized as follows: In this way, we are able to determine the five parameters (A, B, C, D, E) from the corresponding thermodynamic quantities: The inverse task: We extract the thermodynamic quantities from the five PhreeqC parameters (A, B, C, D, E): Application in Hydrochemistry Models

In order 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 T-correction 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 ΔH0
• closed-form equation based on five coefficients – see 4.3

The option that 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

1. In the text outside the formulas we use the term “K value” for the decadic logarithm K = log10 K (base-10 logarithm). This somewhat inconsistent notation results from the fact that in many countries K and K are synonyms.

2. ln K is the abbreviation for the natural logarithm loge K.