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 T-dependence:

  Van’t Hoff Equation   ΔCP = 0
  General Approach   ΔCP as a 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. Here, the central quantity is the change of the heat capacity in a reaction:

(1) ΔCP   =  CP (products) – CP(reactants)

Be aware that ΔCP = 0 does not mean that the constant-pressure heat capacity CP of a product or reactant is zero (which is never the case because CP > 0 applies for each constituent). Moreover, ΔCP can even become negative.

Van’t Hoff Equation

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

(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) 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)
(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 ΔH0 (provided ΔH0 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 Ko for T = To.

When the reaction is endothermic (ΔH0 > 0), then for high temperatures (T > To) the K value increases, 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 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 ΔG0 are decorated even with more indices than we have done here (for simplicity’s sake). Each symbol and index have their own meaning:

In the following we simplify the notation. Since the subsequent discussion refers to equilibrium quantities only, we 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)     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)     ΔIH ΔH – ΔHo =
(2.2b)     ΔIS ΔS – ΔSo =
(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 – ΔSo 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 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.

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)

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 in (2.2a) and (2.2b) leads to:

(4.2a)   ΔIH =
(4.2b)   ΔIS =

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 ΔSo, ΔHo, a, b, and c:

(4.4a)   A =
(4.4b)   B =
(4.4c)   C =
(4.4d)   D =
(4.4e)   E =

Conversely, given the five K-parameters (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)   ΔCP =

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:

PhreeqC and aqion, for example, are equipped with two options to handle the temperature correction of K:

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

  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. 

[last modified: 2018-11-20]