Mobility and Interaction of Heavy Metals in a Natural Soil

Abstract

We study the mobility and interaction under competing conditions observed for copper ($\text{ Cu}^{2+}$) and zinc ($\text{ Zn}^{2+}$) ions in the context of laboratory-scale experiments performed in natural soil columns. The experiments focus on the analysis of solute breakthrough curves (BTCs) obtained after injection of an aqueous solution containing similar concentrations of the two metal ions into a soil column fully saturated with double deionized water. Transport of the competing ions is tested for the same soil under aerobic and anaerobic conditions. Measurements show that the species with lower affinity for the soil, $\text{ Zn}^{2+}$, migrates occupying all available adsorption sites, and is then progressively replaced by the ion with higher affinity, $\text{ Cu}^{2+}$. The two ions are displaced in the system with different effective retardation. The slowest species replaces the sorbed ions, resulting in observed $\text{ Zn}^{2+}$ concentrations that display a non-monotonic behavior in time and which, for a certain period, are larger than the concentration supplied continuously at the inlet. In the absence of a complete geochemical characterization of the system, we show that the measured concentrations of both metals can be interpreted through simple models based on a set of coupled partial differential and algebraic equations, involving a small subset of aqueous and adsorbed species that are present in the system. Depending on the model considered, the relationship between aqueous and adsorbed ion concentrations is described at equilibrium by a Gaines–Thomas (GT) formulation, a competitive Sheindorf–Rebhun–Sheintuch (SRS) isotherm, or an Extended Langmuir (EL) isotherm, respectively. The GT formulation provides the best interpretation of the observed behavior among the models tested. We find that employing these simple models, which account only for the main governing reactive processes, allows reasonable estimation of the observed BTCs in experiments where only partial geochemical datasets are available.

Appendices

Appendix 1

Each experiment was performed in duplicate to ensure reproducibility. Each measurement was repeated six times to allow error quantification. Figure 7 shows the two replicates illustrating the degree of reproducibility of the experiment. The total experimental and analysis error is represented in terms of error bars computed on the basis of the performed replicates.

Fig. 7

Time dependence of concentration of metal ions eluted from the soil column under aerobic condition. The constant inlet concentrations, $C_{0,\mathrm{Cu}} = 1.55\, \text{ mmol} \text{ L}^{-1},\, C_{0,\mathrm{Zn}} = 1.77\, \text{ mmol} \text{ L}^{-1}$, are shown for reference ($dotted$ $lines$). $Error$ $bars$ corresponding to $\pm 2\sigma$ are reported, $\sigma$ being the estimated standard deviation of the performed measurement replicates. The two replicates of the same experiment are indicated as I and II

Appendix 2

Note that, by way of (4), Eq. (1) for $i=3$ can be written as

$$\begin{aligned} \left( {1+\frac{\gamma }{z_3}K_a} \right)\frac{\partial C_3}{\partial t}=L(C_3);\quad L(C_3)=-v\frac{\partial C_3}{\partial x}+D\frac{\partial ^{2} C_3}{\partial x^{2}} \end{aligned}$$

(12)

Equation (12) is a classical one-dimensional ADE with constant (in space and time) retardation and can be solved independently of the other equations. It provides the space-time dependence of $C_3$, i.e., $C_3 (x,t)$, once a value of $z_3$ (e.g., $z_3 = 2$) is assumed. We then replace $C_3 (x,t)$ into (4) to obtain $\beta _3 (x,t)$. This allows rewriting (2) as

$$\begin{aligned} \beta _1 (x,t)+\beta _2 (x,t)=1-\beta _3 (x,t)=A(x,t) \end{aligned}$$

(13)

Considering (1), we can introduce

$$\begin{aligned} u^{*}=\sum _{i=1}^3 {C_i} \end{aligned}$$

(14)

The quantity $u^{*}$ is a conservative component and satisfies a one-dimensional conservative ADE. We then define $u=u^{*}-C_3 =C_1 +C_2$. Considering (2), (3), and (14), and noting that $z_1 =z_2 =2$, we can write $\beta _1$ as a function of $C_1$ according to (for simplicity, we drop the space-time dependence from quantities)

$$\begin{aligned}&K_{12}^2 = \left( {\frac{\beta _1}{C_1}} \right)\left({\frac{u-C_1}{A-\beta _1}} \right);\quad \quad K_{12}^2 A-K_{12}^2 \beta _1 =\left( {\frac{u-C_1}{C_1}} \right)\beta _1 \nonumber \\&K_{12}^2 A=\left( {\frac{u-C_1}{C_1}+K_{12}^2} \right)\beta _1;\quad \beta _1 =K_{12}^2 A\left( {\frac{u-C_1}{C_1}+K_{12}^2 } \right)^{-1} \nonumber \\&\beta _1 =K_{12}^4 A\frac{C_1}{u-C_1 +C_1 K_{12}^2} \end{aligned}$$

(15)

From the last of (15) it follows that

$$\begin{aligned} \frac{\partial \beta _1}{\partial t}&= \frac{\partial \beta _1}{\partial C_1}\frac{\partial C_1}{\partial t}+\frac{\partial \beta _1}{\partial u}\frac{\partial u}{\partial t}\end{aligned}$$

(16)

$$\begin{aligned} \frac{\partial \beta _1}{\partial C_1}&= K_{12}^4 A\frac{u-C_1 +C_1 K_{12}^2 -(K_{12}^2 -1)}{\left( {u-C_1 +C_1 K_{12}^2 } \right)^{2}}\end{aligned}$$

(17)

$$\begin{aligned} \frac{\partial \beta _1}{\partial u}&= \frac{-K_{12}^4 AC_1}{\left( {u-C_1 +C_1 K_{12}^2 } \right)^{2}} \end{aligned}$$

(18)

Introducing (16)–(18) into (1) allows writing the partial differential equation governing the evolution of $C_1$

$$\begin{aligned}&\left[ {1+\frac{\gamma }{2}K_{12}^4 A\frac{u-C_1 +C_1 K_{12}^2 -(K_{12}^2 -1)}{\left( {u-C_1 +C_1 K_{12}^2 } \right)^{2}}} \right]\frac{\partial C_1 }{\partial t}-\left[ {\frac{K_{12}^4 AC_1}{\left( {u-C_1 +C_1 K_{12}^2 } \right)^{2}}} \right]\frac{\partial u}{\partial t}\nonumber \\&\quad =-\frac{v}{\phi }\frac{\partial C_1}{\partial x}+D\frac{\partial ^{2} C_1}{\partial x^{2}} \end{aligned}$$

(19)

Equation (19) is nonlinear in $C_1$ and is solved numerically. Note that (19) has the format of a classical ADE with a retardation and decay term (the second term on the left hand side). The retardation factor is highly non-linear. The decay term can be either positive or negative, depending on the sign of $\partial u/\partial t$. Once $C_1$ is known, one can calculate $C_2$ as

$$\begin{aligned} C_2 =u-C_1 \end{aligned}$$

(20)

Appendix 3

See Tables 5 and 6.

Table 5 Saturation indices associated with the input solution

Table 6 Saturation indices associated with the solution corresponding to the peak value of $\text{ Zn}^{2+}$ concentration