1.1 Reaction thermodynamics

Reaction equilibrium constants $K_{eq}$ . Here we briefly cover fundamental concepts in reaction thermodynamics. For a more comprehensive coverage, readers are referred to books listed in reading materials [e.g., (Langmuir et al., 1997)]. A chemical reaction transforms one set of chemical species to another. Here is an example,

$\begin{equation}\alpha A+\beta B \text { É } \quad \chi C+\delta D\end{equation}$

The reactants A and B combine to form the products C and D. The symbols $\alpha,\beta,\chi,\text{ and }\delta$ are the stoichiometric coefficients that quantify the relative quantity of different chemical species during the reaction in mole basis. That is, $\alpha$ mole of A and $\beta$ mole of B transform into $\chi$ mole of C and $\delta$ mole of D. The Symbol “ $\text { É }$ ” indicates that the reaction is reversible and can occur in both forward and backward directions. According to reaction thermodynamics, the change in the Gibbs free energy $\Delta G_{R}$ during the reaction can be expressed as:

$\begin{equation}\Delta G_{R}=\Delta G_{R}^{0}+R T \ln \left(\frac{a_{C}^{\chi} a_{D}^{\delta}}{a_{A}^{\alpha} a_{B}^{\beta}}\right)\end{equation}$

Here, $\Delta G_{R}^{0}$ is the change in reaction free energy under the standard condition (293.15 K and $10^5$ Pa); R is the ideal gas constant (1.987 cal/(mol·K)); T is the absolute temperature (K); and $a_{A}, a_{B}, a_{C},a_{D}$ are the activities of the species A, B, C and D, respectively. The derivation of this equation goes to the heart of reaction thermodynamics theory. In any particular aqueous solution, we define ion activity product (IAP) for the reaction (1) as follows:

$\begin{equation}I A P=\frac{a_{C}^{\chi} a_{D}^{\delta}}{a_{A}^{\alpha} a_{B}^{\beta}}\end{equation}$

The reaction reaches equilibrium when the forward and backward reactions occur at the same rate, the point at which the change in the reaction Gibbs free energy $\Delta G_{R}$ reaches zero. At reaction equilibrium, the IAP equals to the equilibrium constant $K_{eq}$ :

$\begin{equation}K_{e q}=\left(\frac{a_{C}^{\chi} a_{D}^{\delta}}{a_{A}^{\alpha} a_{B}^{\beta}}\right)_{e q}\end{equation}$

where $a_{A}, a_{B}, a_{C}, a_{D}$ are the activities of the species A, B, C and D at equilibrium, respectively. In general, larger $K_{eq}$ means larger reaction tendency to the right direction in the equation as written in (1). The $K_{eq}$ is a constant for a particular reaction at any specific temperature and pressure conditions. Although they look very similar, IAP is the ion activity product under any point along the reaction progress, while $K_{eq}$ is the IAP only at one particular point during the reaction, i.e., at equilibrium. We use the saturation index (SI) to compare IAP and $K_{eq}$ :

$\begin{equation}\S I=\log _{10}\left(\frac{I A P}{K_{e q}}\right)\end{equation}$

When SI =0, the reaction is at equilibrium; when SI < 0, the reaction proceeds to the right (forward); when SI > 0, the reaction proceeds to the left (backward).

Dependence of $K_{eq}$ on temperature and pressure. Values of $K_{eq}$ are a function of temperature and pressure. According to reaction thermodynamics, the $K_{eq}$ dependence on temperature follows the Van’t Hoff equation under constant pressure:

$\begin{equation}\left(\frac{\partial\left(\ln K_{eq}\right)}{\partial T}\right)_P=\frac{\Delta H_r^o}{RT^2}\end{equation}$

Here

$\Delta H_{r}^{o}$ is the standard state enthalpy change of the reaction (cal/mol). If the reaction is exothermic (

$\left.\Delta H_{r}^{o}<0\right), K_{e q}$ decreases with increasing T; if the reaction is endothermic

$\left(\Delta H_{r}^{o}>0\right), K_{e q}$ , increases with increasing T. For example,

$K_{eq}$ of quartz dissolution increases with temperature, meaning its solubility increases with temperature increase. The heat capacity at constant pressure,

$\Delta C p_{r}^{o}$ (cal/mol K), is defined as follows:

$\begin{equation}\Delta C p_{r}^{o}=\left|\frac{\partial\left(\Delta H_{r}^{o}\right)}{\partial T}\right|_{P}\end{equation}$

$\Delta H_{r}^{o}$ is independent of temperature, the integrated form of equation (6) is the following:

$\begin{equation}-\log K_{e q, 2}=-\log K_{e q, 1}+\frac{\Delta H_{r}^{o}}{4.576}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)\end{equation}$

Here

$K_{e q, 1}$ and

$K_{e q, 2}$ are the equilibrium constants at

$\text{T}_1$ and

$\text{T}_2$ , respectively. Typically

$\text{T}_1$ is

$25^{\circ} \mathrm{C}$ because the equilibrium constants of many reactions are measured and available at

$25^{\circ} \mathrm{C}$ . For reactions only involve aqueous ions, this equation is applicable if

$\text{T}_2$ is within 10 to 15 degree C difference from

$25^{\circ} \mathrm{C}$ . If the reaction enthalpy is not a constant and

$\Delta C p_{r}^{c}$ is non-zero however is constant, we have a more general integrated form as follows:

$\begin{equation}-\log K_{e q, 2}=-\log K_{e q, 1}+\frac{\Delta H_{r}^{o}}{4.576}\left[\frac{1}{T_{2}}-\frac{1}{T_{1}}\right]-\frac{\Delta C_{p}^{o}}{1.987}\left[\frac{1}{2.303}\left(\frac{T_{1}}{T_{2}}-1\right)-\log \frac{T_{1}}{T_{2}}\right]\end{equation}$

Similarly, pressure dependence of

$K_{eq}$ can be expressed by:

$\begin{equation}\left(\frac{d\left(\ln K_{e q}\right)}{d p}\right)_{T}=\frac{\Delta V_{r}^{o}}{R T^{2}}\end{equation}$

Here, $\Delta V_{r}^{o}$ is the molar volume change of the reactions under the standard condition (cm³/mol). Typically, the effect of pressure is important when the reaction involves gas (e.g. $\mathrm{CO}_{2(g)}$ ) and when there is a significant difference in the molar volume of reactants and products.

If the $\Delta V_{r}^{o}$ is a constant and does not depend on pressure, the integrated form the above equation is:

$\begin{equation}\frac{\ln K_{eq,2}}{\ln K_{eq,1}}=\frac{\Delta V_r^o\left(P_2-P_1\right)}{RT}\end{equation}$

Where $K_{eq,1}$ and $K_{eq,2}$ are the equilibrium constants at P1 and P2, respectively. Typically, P1 is at atmospheric condition at 1 bar.

The standard thermodynamic properties (e.g. $H_{r}^{o}$ and $V_{r}^{o}$ ) of most compounds can be found or calculated from CRC handbooks (e.g., Handbook of Chemistry and Physics (Haynes, 2012), and the NIST Chemistry WebBook(link is external). Readers are also referred to the standard geochemical database Eq3/6 for values of equilibrium constants (Wolery et al., 1990).

Biogeochemical reaction systems typically include both slow reactions with kinetic rate laws and fast reactions that are governed by reaction thermodynamics. Kinetic reactions include, for example, mineral dissolution and precipitation and redox reactions. Thermodynamically-controlled reactions are those with rates so fast that the kinetics does not matter for the problem of interest. In geochemical systems, these include, for example, aqueous complexation reactions that reach equilibrium at the time scales of milli-seconds to seconds. For these reactions, the activities of reaction species are algebraically related through their equilibrium constants, or laws of mass action, as shown in equation (4). As such, their concentrations are not independent of each other and should not be numerically solved independently. This necessitates the classification of aqueous species into primary and secondary species. The primary species are essentially the building blocks of chemical systems, whereas the concentrations of secondary species depend on those of primary species through the laws of mass action. As such, with the definition of primary and secondary species, a reactive transport code only needs to numerically solve the number of equations for the species that are independent of each other, which is essentially the number of primary species. The number of primary species is equivalent to the number of components in a system. The concentrations of secondary species can be calculated based on the concentrations of primary species using laws of mass action.

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