Interaction of Freshly Precipitated Silica Gel with Aqueous Silicic Acid Solutions under Ambient and Near Neutral pH-conditions: A Detailed Analysis of Linear Rate Law

Abstract

Interaction of freshly precipitated silica gel with aqueous solutions was studied at laboratory batch experiments under ambient and near neutral pH-conditions. The overall process showed excellent reversibility: gel growth could be considered as an opposite process to dissolution and a linear rate law could be applied to experimental data. Depending on the used rate law form, the resulting rate constants were sensitive to errors in parameters/variables such as gel surface area, equilibrium constants, Si-fluxes, and reaction quotients. The application of an Integrated Exponential Model appeared to be the best approach for dissolution data evaluation. It yielded the rate constants k _dissol ∼ (4.50 ± 0.68) × 10⁻¹² and k _growth ∼ (2.58 ± 0.39) × 10⁻⁹ mol m⁻² s⁻¹ for zero ionic strength. In contrast, a Differential Model gave best results for growth data modeling. It yielded the rate constants k _dissol ∼ (1.14 ± 0.44) × 10⁻¹¹ and k _growth ∼ (6.08 ± 2.37) × 10⁻⁹ mol m⁻² s⁻¹ for higher ionic strength (I ∼ 0.04 to 0.11 mol L⁻¹). The found silica gel solubility at zero ionic strength was somewhat lower than the generally accepted value. Based on the ${}^{298}K_{\rm SS}^{(I=0)} = 10^{-2.754\pm 0.017}$ and $^{298}K_{\rm DBP}^{(I=0)} = 10^{-2.728\pm 0.003},$ standard Gibbs free energy of silica gel formation was calculated as $\Updelta{G}_{\rm f}^0(298\,{\rm K}) \sim -\hbox{850,463} \pm 98\hbox{ J mol}^{-1}$ and  −850,318 ± 20 J mol⁻¹, respectively. Activation energies for silica gel dissolution and growth were determined as $E_{\rm A}^{\rm dissol} \sim 62.0\pm 3.2\,\hbox{kJ mol}^{-1}$ and $E_{\rm A}^{\rm growth} \sim 48.8\pm 4.4\,\hbox{ kJ mol}^{-1},$ respectively. An universal value for growth of any silica polymorph, $E_{\rm A}^{\rm growth} \sim 37.4 \pm 9.4\,\hbox{kJ mol}^{-1},$ is not consistent with the value for silica gel growth, which questions the hypothesis about one unique activated complex controlling the silica polymorph growth.

Appendices

Appendix A

Table 9 Experimental data

Appendix B: Error Propagation

In general, a propagation of errors in a F(x _i ) function with uncorrelated variables is given by

$$ \Updelta {F}^{2}=\left[ {\left({\frac{\partial{F}}{\partial {x}_1 }} \right)\Updelta {x}_1 } \right]^{2}+\left[ {\left({\frac{\partial{F}}{\partial {x}_2 }} \right)\Updelta {x}_2 } \right]^{2}\ldots+\left[ {\left({\frac{\partial{F}}{\partial {x}_n }} \right)\Updelta {x}_n } \right]^{2}=\sum_i {\left[ {\left({\frac{\partial{F}}{\partial {x}_i }} \right)\Updelta {x}_i } \right]^{2}}, $$

(A1)

where $\Updelta{ x}_{i}$ is an absolute error of x _i -variable (see, e.g., Papoulis 1991).

2.1 Saturation Index Error

The uncertainty in saturation index was derived from the expression

$$ {SI} = \log (Q/K). $$

(A2)

A rewriting gives

$$ {SI} = \frac{1}{\hbox{ln} 10}\hbox{ln} \left({\frac{{Q}}{{K}}} \right). $$

(A3)

Developing of A3 according to Eq. A1 with respect to Q a K gives

$$ \Updelta{SI}^{2}=\left({\frac{1}{\hbox{ln} 10}} \right)^{2}\left({\frac{1}{{Q}}} \right)^{2}\Updelta {Q}^{2}+\left({\frac{1}{\hbox{ln} 10}} \right)^{2}\left({\frac{1}{{K}}} \right)^{2}\Updelta {K}^{2}. $$

(A4)

After dividing by the squared Eq. A3, it becomes

$$ \left({\frac{\Updelta {SI}}{{SI}}} \right)^{2}=\left({\frac{1}{\hbox{ln}10\,{SI}}} \right)^{2}\left({\frac{\Updelta {Q}}{{Q}}} \right)^{2}+\left({\frac{1}{\hbox{ln} 10\,{SI}}} \right)^{2}\left({\frac{\Updelta{ K}}{{K}}} \right)^{2}. $$

(A5)

and, after substituting for the multiplicative terms,

$$ \left({\frac{\Updelta {SI}}{{SI}}} \right)^{2}=\Uplambda^2\left({\frac{{Q}}{{Q}}}\right)^{2}+\Uplambda^2\left({\frac{\Updelta{ K}}{{K}}} \right)^{2}. $$

(A6)

where Λ  = 1/(ln10 SI). The significance of the multiplicative term Λ is enormous close to equilibrium where $\lim\limits_{{Q}\rightarrow {K}} [1/\log ({Q/K})]=\infty.$

Based on the relation $\Updelta{G}_{\rm {gel}{-}{w}} = 2.3{RTSI}_{\rm {gel}{-}{w}}$, the relative error in $\Updelta G_{\rm {gel}{-}{w}}$ is the same as in SI.

2.2 Affinity Term Error

The error in affinity term was derived from the equation

$$ \hbox{AT}=1-\hbox{e}^{\frac{\Updelta {G}}{RT}}=1-\frac{{Q}}{{K}}. $$

(A7)

Based on the evaluation of Eq. A7 according to A1 with respect to Q and K, it is

$$ (\Updelta \hbox{AT})^{2}=\frac{1}{{K}^{2}}\Updelta {Q}^{2}+\frac{{Q}^{2}}{{K}^{4}}\Updelta {K}^{2}. $$

(A8)

Dividing by A ² yields

$$ \left({\frac{\Updelta \hbox{AT}}{\hbox{AT}}} \right)^{2}=\left({\frac{{Q}}{{K}-{Q}}} \right)^{2}\left({\frac{\Updelta {Q}}{{Q}}} \right)^{2}+\left({\frac{{Q}}{{K}-{Q}}} \right)^{2}\left({\frac{\Updelta {K}}{{K}}} \right)^{2}, $$

(A9)

and, after substituting for multiplicative terms,

$$ \left({\frac{\Updelta \hbox{AT}}{\hbox{AT}}} \right)^{2}=\Upphi ^{2}\left({\frac{\Updelta {Q}}{{Q}}} \right)^{2}+\,\Upphi ^{2}\left({\frac{\Updelta {K}}{{K}}} \right)^{2}. $$

(A10)

The share of Q and K errors depends on the Φ multiplicative term. The Q errors are insignificant when Q approaches zero. Their role increases if Φ² > 1, i.e., if Q > K _eq/2. An extreme multiplication effect of the Φ² term is close to equilibrium, as indicated by $\lim \limits_{Q\rightarrow K} \, \Upphi =\infty.$ If $Q \gg K_{\rm eq},$ the multiplicative effect diminishes, as K _eq may be omitted in comparison with Q.

2.3 Normalized Flux Error

The error propagation was derived from the expression

$$ {j}_{\rm norm} ={j}/\{{A}\}. $$

(A11)

Evaluating Eq. A11 according to A1 with respect to j and {A} variables yields

$$ {j}_{\rm norm}^2 =\frac{1}{\{{A}\}^{2}}\Updelta {j}^{2}+\frac{{j}}{\{{A}\}^{2}}\Updelta \{{A}\}^{2}. $$

(A12)

After dividing by the squared Eq. A11 and rewriting, we obtain

$$ \left({\frac{\Updelta {j}_{\rm norm} }{{j}_{\rm norm} }} \right)^{2}=\left({\frac{\Updelta {j}}{{j}}} \right)^{2}+\left({\frac{\Updelta \{{A}\}}{\{{A}\}}} \right)^{2}. $$

(A13)

2.4 DIFFEREL Modeling Error

An uncertainty in the DIFFEREL modeling was derived from Eq. 12. Assuming the term $a_{\rm gel} a_{\rm w}^{2} \cong 1$ and rewriting, it yields

$$ {k}_{\rm d} =\frac{{j}_{\rm norm}{K}}{\left({{K}-{Q}} \right)}. $$

(A14)

The rearrangement of Eq. A14 according to Eq. A1 with respect to j _norm and Q variables gives

$$ \Updelta {k}_{\rm d}^2 =\frac{{K}^{2}}{\left({ {K}-{Q} } \right)^{2}}\Updelta {j}_{\rm norm}^2 + \frac{{j}_{\rm norm}^2 {K}^{2}}{\left({ {K}-{Q} } \right)^{4}}\Updelta {Q}^{2}. $$

(A15)

After dividing by the squared Eq. A14 and rewriting, the equation becomes

$$ \left({\frac{\Updelta {k}_{\rm d} }{{k}_{\rm d} }} \right)^{2}=\left({\frac{\Updelta {j}_{\rm norm} }{{j}_{\rm norm}}} \right)^{2}+ \left({ \frac{{Q}}{{K}-{Q}} } \right)^{2}\left({\frac{\Updelta {Q}}{{Q}}} \right)^{2}, $$

(A16)

or, after substituting for the multiplicative term,

$$ \left({\frac{\Updelta {k}_{\rm d} }{{k}_{\rm d} }} \right)^{2}=\left({\frac{\Updelta {j}_{\rm norm}}{{j}_{\rm norm}}} \right)^{2}+\,\Upphi ^{2}\left({\frac{\Updelta {Q}}{{Q}}} \right)^{2}. $$

(A17)

Because k _g is linked to k _d by the simple relation of k _g = k _d/K _eq, the uncertainty in k _g is the same as in the k _d.

As it can be seen, the share of Q errors depends on the Φ² multiplicative term. The Q errors themselves are insignificant if Q approaches zero. Their role increases if Φ² > 1, i.e., if Q > K _eq/2. An extreme multiplication effect of the Φ² term is close to equilibrium, as indicated by $\lim\limits_{Q\rightarrow K} \, \Upphi =\infty.$ When $Q\gg K_{\rm eq},$ the multiplicative effect diminishes, as K _eq may be omitted in comparison with Q.

2.5 INTEGEXPEL Modeling Error

Error linked to the INTEGEXPEL modeling was derived from Eq. 16. It was rewritten as

$$ {k}_{\rm g} = \frac{{V}}{\{A\}\gamma_{\hbox{Si}}t}\hbox{ln} \left(\frac{K-Q_0}{K-Q}\right). $$

(A18)

An rearrangement according to Eq. A1 with respect to the {A} and Q variables yields

$$ \Updelta {k}_{\rm g}^2 =\frac{{V}^{2}}{\{{A}\}^{4}\gamma_{\rm Si}^2 {t}^{2}}\left[ {\hbox{ln} \left({\frac{{K}-{Q}^{0}}{{K}-{Q}}} \right)} \right]^{2}\Updelta \left\{ {A} \right\}^{2}+\frac{{V}^{2}}{\{{A}\}^{2}\gamma_{\rm Si}^2 {t}^{2}({K}-{Q})^{2}}\Updelta {Q}^{2}. $$

(A19)

After dividing by the squared Eq. A18, it simplifies to

$$ \left({\frac{\Updelta {k}_{\rm g} }{{k}_{\rm g} }} \right)^{2}=\left({\frac{\Updelta \left\{ {A} \right\}}{\{{A}\}}} \right)^{2}+\left({\frac{{Q}}{{K}-{Q}}} \right)^{2}\left({\frac{1}{\hbox{ln} [({K}-{Q}^{0})/({K}-{Q})]}} \right)^{2}\left({\frac{\Updelta {Q}}{{Q}}} \right)^{2}, $$

(A20)

and, after substituting the multiplicative terms, to

$$ \left({\frac{\Updelta {k}_{\rm g} }{{k}_{\rm g} }} \right)^{2}=\left({\frac{\Updelta \left\{ {A} \right\}}{\{{A}\}}} \right)^{2}+\,\Upphi^{2}\Upomega^{2}\left({\frac{\Updelta {Q}}{{Q}}} \right)^{2}. $$

(A21)

Close to equilibrium, the term is of fatal relevance as $\lim\limits_{Q\rightarrow {K}} \Upphi^{2}\Upomega^{2}=\infty$ shows. The term value is enormous also at the start of experiment as indicated by $\lim\limits_{{Q}\rightarrow {Q}^{0}} \Upphi^{2}\Upomega^{2}=\infty.$ Because k _g is linked to k _d by the simple relation of k _g  = k _d/K _eq, the uncertainty in k _g is the same as in k _d.

2.6 INTEGLOGEL Modeling Error

Error linked to the INTEGLOGEL modeling was derived from Eq. 16. It can be rewritten as

$$ {k}_{\rm g} =\frac{{V}}{\{{A}\}\gamma_{\rm Si} {t}}\hbox{ln} \left({\frac{{K}-{Q}^{0}}{{K}-{Q}}} \right). $$

(A22)

Assuming {A}, K _eq, and Q as variables, we can write

$$ \begin{aligned} \Updelta {k}_{\rm g}^2 =\frac{{V}^{2}}{\{{A}\}^{4}\gamma_{\rm Si}^2 {t}^{2}}\left[ {\hbox{ln} \left({\frac{{K}-{Q}^{0}}{{K}-{Q}}} \right)} \right]^{2}\Updelta \left\{ {A} \right\}^{2}&+\frac{{V}^{2}({Q}^{0}-{Q})^{2}}{\{{A}\}^{2}\gamma_{\rm Si}^2 {t}^{2}({K}-{Q})^{2}({K}-{Q}^{0})^{2}}\Updelta {K}^{2} \\ &+\frac{{V}^{2}}{\{{A}\}^{2}\gamma_{\rm Si}^2 {t}^{2}({K}-{Q})^{2}}\Updelta {Q}^{2} \\ \end{aligned}. $$

(A23)

Dividing by the squared Eq. A22 and a rewriting yield

$$ \begin{aligned} \left({\frac{\Updelta {k}_{\rm g}}{{k}_{\rm g} }} \right)^{2}&=\left({\frac{\Updelta \{{A}\}}{\{{A}\}}} \right)^{2}+\left({\frac{{K}}{{K}-{Q}}} \right)^{2}\left({\frac{{Q}^{0}-{Q}}{{K}-{Q}^{0}}} \right)^{2}\left({\frac{1}{\ln[({K}-{Q}^{0})/({K}-{Q})]}} \right)^{2}\left({\frac{\Updelta {K}}{{K}}} \right)^{2} \\ &+\left({\frac{{Q}}{{K}-{Q}}} \right)^{2}\left({\frac{1}{\ln[({K}-{Q}^{0})/({K}-{Q})]}} \right)^{2}\left({\frac{\Updelta {Q}}{{Q}}} \right)^{2} \\ \end{aligned}, $$

(A24)

and, after substituting for the error multiplicative terms, it gives

$$ \left({\frac{\Updelta {k}_{\rm g} }{{k}_{\rm g} }} \right)^{2}=\left({\frac{\Updelta \{{A}\}}{\{{A}\}}} \right)^{2}+\,\Upgamma ^{2}\Uppsi^{2}\Upomega^{2}\left({\frac{\Updelta {K}}{{K}}} \right)^{2}+\,\Upphi^{2}\Upomega^{2}\left({\frac{\Updelta {Q}}{{Q}}} \right)^{2}. $$

(A25)

The analysis of a share of K _eq and Q error shows that the both Γ²Ψ²Ω² and Φ²Ω² multiplicative terms effect is at maximum close to equilibrium, as indicated by $\lim\limits_{Q\rightarrow K} \Upgamma \Uppsi \Upomega =\infty$ and $\lim \limits_{Q\rightarrow K} \Upphi \Upomega =\infty.$

2.7 The AFFINEL Modeling

Error propagation linked to the AFFINEL modeling was derived from Eqs. 12 and A14. In contrast to the DIFFEREL case, the equation was evaluated with respect to j _norm, Q, and K. The rearrangement of Eq. A14 according to Eq. A1 gives

$$ \Updelta {k}_{\rm d}^2 =\frac{{K}^{2}}{({K}-{Q})^{2}}\Updelta {j}_{\rm norm}^2 +\frac{{j}_{\rm norm}^2 {Q}^{2}}{({K}-{Q})^{4}}\Updelta {K}^{2}+\frac{{j}_{\rm norm}^2 {K}^{2}}{({K}-{Q})^{4}}\Updelta {Q}^{2} $$

(A26)

Dividing by the squared Eq. A14 and a rewriting yield

$$ \left({\frac{\Updelta {k}_{\rm d} }{{k}_{\rm d} }} \right)^{2}=\left({\frac{\Updelta {j}_{\rm norm} }{{j}_{\rm norm} }} \right)^{2}+\left({\frac{{Q}}{{K}-{Q}}} \right)^{2}\left({\frac{\Updelta {K}}{{K}}} \right)^{2}+\left({\frac{{Q}}{{K}-{Q}}} \right)^{2}\left({\frac{\Updelta {Q}}{{Q}}} \right)^{2}, $$

(A27)

and, after substituting for the multiplicative terms, it gives

$$ \left({\frac{\Updelta {k}_{\rm d} }{{k}_{\rm d} }} \right)^{2}=\left({\frac{\Updelta {j}_{\rm norm} }{{j}_{\rm norm} }} \right)^{2}+\Upphi^{2}\left({\frac{\Updelta {K}}{{K}}} \right)^{2}+\Upphi^{2}\left({\frac{\Updelta {Q}}{{Q}}} \right)^{2}. $$

(A28)

The analysis of K _eq and Q error shares shows that the maximum effect of the Φ multiplicative term is close to equilibrium, as indicated by $\lim \limits_{Q\rightarrow K} \Upphi =\infty.$

2.8 The Equilibrium Constant Based on Detailed Balancing Principle

The uncertainty in the $K_{\rm eq}^{\rm DBP}$ was calculated from the equation ${K}_{\rm eq}^{\rm \rm DBP}\, =\, {k}_{\rm d}/{k}_{\rm g}.$ Because k _d and k _g values are correlated, we must use a modified form of the function A1,

$$ \Updelta {F}^{2}=\left[ {\left({\frac{\partial {F}}{\partial {x}_1 }} \right) \Updelta {x}_1 } \right]^{2}+2 C\, \left({\frac{\partial {F}}{\partial {x}_1 }} \right) \left({\frac{\partial {F}}{\partial {x}_2 }} \right)\, \Updelta {x}_1 \Updelta {x}_2 +\, \left[ {\left({\frac{\partial {F}}{\partial {x}_2 }} \right) \Updelta {x}_2 } \right]^{2}, $$

(A29)

where C is correlation coefficient (Papoulis 1991).

Developing the equation ${K}_{\rm eq}^{\rm DBP}\, =\, {k}_{\rm d}/{k}_{\rm g}$ according to Eq. A29 with respect to k _d and k _g gives

$$ \left({\Updelta {K}_{\rm eq}^{\rm KDP}} \right)^{2}=\left({\frac{1}{{k}_{\rm g}}} \right)^{2}\left({ \Updelta {k}_{\rm d}} \right)^{2}+2 C\, \left({\frac{1}{{k}_{\rm g}}} \right) \left({-\frac{{k}_{\rm d}}{{k}_{\rm g}^2 }} \right) \Updelta {k}_{\rm d} \Updelta {k}_{\rm g} +\left({-\frac{{k}_{\rm d}}{{k}_{\rm g}^2 }} \right)^{2}\left({ \Updelta {k}_{\rm g}} \right)^{2}. $$

(A30)

After dividing by $\left({{K}_{\rm eq}^{\rm DBP}} \right)^{2}\, =\, ({k}_{\rm d})^{2}/({k}_{\rm g})^{2}$ and rewriting, Eq. A30 yields

$$ \left({\frac{\Updelta {K}_{\rm eq}^{\rm KDP} }{{K}_{\rm eq}^{\rm KDP} }} \right)^{2}=\left({\frac{\Updelta {k}_{\rm d}}{{k}_{\rm d} }} \right)^{2}\, -\, 2 C\, \left({\frac{\Updelta {k}_{\rm d} }{{k}_{\rm d} }} \right) \left({\frac{\Updelta {k}_{\rm g} }{{k}_{\rm g} }} \right) \, +\, \left({\frac{\Updelta {k}_{\rm g} }{{k}_{\rm g} }} \right)^{2}. $$

(A31)

When C ∼ 1, the Eq. A31 simplifies to

$$ \left({\frac{\Updelta {K}_{\rm eq}^{\rm KDP} }{{K}_{\rm eq}^{\rm KDP} }} \right)^{2}=\left[ {\left({\frac{\Updelta {k}_{\rm d} }{{k}_{\rm d} }} \right)-\, \left({\frac{\Updelta {k}_{\rm g} }{{k}_{\rm g} }} \right)} \right]^{2}. $$

(A32)

In the case that $\Updelta {k}_{\rm d}/{k}_{\rm d} \approx \Updelta {k}_{\rm g}/{k}_{\rm g},$ the uncertainty stemming from error propagation approaches zero.