Overview Activity Chem Equilibrium Acids Carbonate System Mineral Phases

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 ΔC_P = 0 does not mean that the constant-pressure 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. 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^-1K^-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 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)	\(\ln K_o \,= -\dfrac{1}{R} \, \left( \dfrac{\Delta H^{0}}{T_o} - \Delta S^{0} \right)\)

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 K_o for T = T_o.

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 = G_prod – G_educ (similar to 1). To make it more explicit, instead of ΔG often Δ_rG 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 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 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 ΔC_P

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 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 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 – 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 non-zero 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:

T-Correction Formula for ΔC_P = const

The simplest assumption about ΔC_P 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 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 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