Full Text
Physical Chemistry
V. B. RATINOV and O. M. TODES
ON THE SPONTANEOUS CRYSTALLIZATION OF DISSOCIATED SALTS
(Presented by Academician P. A. Rebinder, January 6, 1960)
The probability \(\alpha\) of the formation, per unit volume and per unit time, of a nucleus of a new phase depends exponentially on the height of the energy barrier associated with the appearance of an interphase boundary possessing surface free energy \(\sigma\). Let \(\gamma\) denote the shape factor of the nucleus, \(v_k\) the molar volume of the new phase, \(\mu_1\) the molar chemical potential of the dissolved substance, and \(\mu_2\) the molar chemical potential of the forming crystal. Then \((1\text{--}3)\)
\[ \alpha=A\exp\left\{-\frac{4\gamma\sigma^3 v_k^2}{RT(\mu_1-\mu_2)^2}\right\}. \tag{1} \]
For nondissociated substances in dilute solutions,
\[ \mu_1=RT\ln\frac{c}{\varphi(p,T)},\qquad \mu_2=RT\ln\frac{c_{\mathrm{ravn}}}{\varphi(p,T)}, \tag{2} \]
where \(c\) is the concentration of the dissolved substance, and \(c_{\mathrm{ravn}}\) is the equilibrium concentration of the saturated solution. Then
\[ \mu_1-\mu_2=RT\ln\frac{c}{c_{\mathrm{ravn}}}, \tag{3} \]
\[ \ln\alpha=\ln A-\frac{4\gamma\sigma^3 v_k^2}{(RT)^3}\, \frac{1}{\ln^2\dfrac{c}{c_{\mathrm{ravn}}}}. \tag{4} \]
By measuring the concentration \(c\) of a supersaturated solution and the probability \(\alpha\) of nucleus formation, one can determine the surface free energy \(\sigma\) from the slope of the straight line (4) in the coordinates
\[ \ln\alpha\;-\;\frac{1}{\ln^2\dfrac{c}{c_{\mathrm{ravn}}}}. \]
For dissociated salts, the initial expressions (2) for the chemical potentials must include not the total concentration \(c\) of the dissolved salt, but only the concentration \(c^*<c\) of undissociated molecules in the solution. The latter, in turn, can be determined from the law of mass action and the balance of dissolved ions.
Let us carry out the calculation for the specific case in which the crystallizing component \(C\) dissociates in solution into two ions \(A\) and \(B\) according to the equation
\[ C=A+B. \tag{5} \]
Using the law of mass action,
\[ c^*=Ka^*b^*, \tag{6} \]
one can represent the driving force of the crystallization process in two equivalent forms
\[ \mu_1-\mu_2=RT\ln\frac{c^*}{c^*_{\mathrm{ravn}}} \tag{7¹} \]
or
\[ \mu_1-\mu_2=RT\ln\frac{K a^* b^*}{c_{\rm eq}}=RT\ln\frac{a^* b^*}{L}, \tag{7′} \]
where
\[ L=c_{\rm eq}/K \tag{8} \]
is the product of the concentrations of dissociated ions in equilibrium with the solid phase.
Owing to the practically instantaneous establishment of dissociation equilibrium, the true concentrations of ions in solution \(a^*\) and \(b^*\) differ from their total concentrations \(a\) and \(b\) and are related to the latter by the balance equations
\[ c_z^*+a^*=a,\qquad c^*+b^*=b. \tag{9} \]
Solving equations (6) and (9) simultaneously, one can find the values of \(c^*\), \(a^*\), and \(b^*\) and substitute them into (7′) or (7²). Then
\[ \mu_1-\mu_2 =RT\ln \frac{1+Ka+Kb-\sqrt{(1+Ka+Kb)^2-4K^2ab}} {2Kc_{\rm eq}} . \tag{10} \]
For a stoichiometric ratio of the ions \(a=b=c\), formula (10) is simplified and assumes the form
\[ \mu_1-\mu_2 =RT\ln \frac{1+2Kc-\sqrt{1+4Kc}} {2Kc_{\rm eq}} . \tag{11} \]
For weakly dissociated salts \(Kc\gg 1\), \(c^*\approx c\), and in the limiting case expression (11) coincides with formula (3) for nondissociated salts. In the opposite limiting case \(Kc\ll 1\), dissociation is practically complete, \(a^*\approx a\), \(b^*\approx b\), and
\[ \mu_1-\mu_2\approx RT\ln\frac{ab}{L}. \tag{12} \]
It should be noted that for strong electrolytes the concentrations \(a\) and \(b\) must be replaced by the activities of the corresponding ions. One may proceed otherwise as well—leaving expression (12) in force, assume that the equilibrium concentration product \(L\) is itself a function of the concentrations of dissolved ions and of the ionic strength of the solution. This method was used in one of the preceding works (\(^4\)), in which a method for calculating the value of \(L\) was developed.
Substituting the expression (12) obtained into (1), the latter can be brought to the form
\[ \ln\alpha=\ln A-\frac{4\gamma\sigma^3 v_k^2}{(RT)^3}\, \frac{1}{\ln^2\frac{ab}{L}} . \tag{13} \]
This dependence was studied experimentally in the spontaneous crystallization from supersaturated solutions of calcium sulfate dihydrate (gypsum) and monocalcium hydrosilicate. Supersaturation was produced chemically: by mixing the corresponding solutions with vigorous stirring. In obtaining gypsum, these solutions were calcium chloride and potassium sulfate; in obtaining monocalcium hydrosilicate, lime and silicic acid. In different series of experiments, the initial concentrations of the cation and anion, as well as their ratio, were varied.
To measure the decrease in supersaturation with time, 2–3 ml samples were periodically withdrawn through a No. 4 filter from 500 ml of the crystallizing solution. The filtrate was analyzed for calcium-ion content by complexometric titration with Trilon B, and for the content of ...
content of silicic acid—by the colorimetric method from the colored complex with ammonium molybdate in a sulfuric-acid medium. In the crystallization of calcium hydrosilicate, in order to avoid carbonation, the solutions were prepared with boiled water, and the experiments, sampling, and storage of samples were carried out in a nitrogen atmosphere.
Fig. 1. Kinetics of crystallization of gypsum at different supersaturations \(C_{\mathrm{Ca}} = C_{\mathrm{SO_4}}\); \(t = 24^\circ\)
Figure 1 shows the decrease in supersaturation with time during the crystallization of gypsum. The kinetic crystallization curves of monocalcium hydrosilicate have the same S-shaped form, with a clearly expressed induction period \(\tau\).
In one of the preceding works \((^3)\) it was shown that, owing to the very steep decrease of \(\alpha\) with decreasing supersaturation, nuclei of the new phase are formed practically only at the very beginning of the induction period, and then grow with a linear rate \(\lambda = dl/d\tau\), which changes comparatively slowly with supersaturation. The number of nuclei per unit volume is equal to
\[ N_0 = k \left( \frac{\alpha}{\lambda} \right)^{3/4}, \tag{14} \]
where the proportionality coefficient \(k\) also depends only weakly on the supersaturation.
The decrease in supersaturation by the end of the induction period \(\tau\) is equal to
\[ \Delta c = \frac{\rho}{M}\gamma N_0 l^3 = \frac{\rho}{M}\gamma k \lambda^{9/4}(\alpha \tau^4)^{3/4}, \tag{15} \]
where \(\rho\) is the density and \(M\) is the molecular weight of the crystallizing substance.
In the experiments, the induction periods were compared at identical, comparatively small but quite measurable decreases in the initial supersaturation. For gypsum it was taken that \(\Delta c = 6 \cdot 10^{-4}\) mol/l, and for monocalcium hydrosilicate \(\Delta c = 5 \cdot 10^{-5}\) mol/l. Thus, from (15) there follows a relation between \(\alpha\) and \(\tau\) of the form
\[ \alpha \tau^4 = \left( \frac{M}{\rho} \frac{\Delta c}{\gamma k \lambda^{9/4}} \right)^{4/3} \simeq \mathrm{const}. \tag{16} \]
Fig. 2. Probability of occurrence of calcium hydrosilicate crystallites as a function of supersaturation at \(24^\circ\). Series \(I\)—\(C_{\mathrm{SiO_2}} = \mathrm{const}\), \(C_{\mathrm{Ca}} \ne \mathrm{const}\), \(\sigma_1 = 24\) erg/cm\(^2\); series \(II\)—\(C_{\mathrm{SiO_2}} \ne \mathrm{const}\), \(C_{\mathrm{Ca}} = \mathrm{const}\), \(\sigma_2 = 19\) erg/cm\(^2\); series \(III\)—\(C_{\mathrm{Ca}}/C_{\mathrm{SiO_2}} = 6 = \mathrm{const}\), \(C_{\mathrm{SiO_2}}\) and \(C_{\mathrm{Ca}} \ne \mathrm{const}\), \(\sigma_3 = 28\) erg/cm\(^2\)
From (13) and (16) there then follows a dependence of \(\tau\) on supersaturation of the form
\[ \lg \tau = \mathrm{const} + \frac{\gamma \sigma^3 v_k^2}{(2.3\,RT)^3} \frac{1}{\lg^2 \dfrac{c_A c_K}{L}}, \tag{17} \]
in other words, \(\lg \tau\) must be a linear function of
\[ \frac{1}{\lg^2 \dfrac{c_A c_K}{L}}. \]
In Fig. 2 this dependence is plotted for the crystallization kinetics of monocalcium hydrosilicate. As is evident from the figure, the experimental points of the different series fall well on straight lines. From the slope of these straight lines one can find the angular coefficient, equal to \(\gamma \sigma^{3} v_k^{2}/(2.3 RT)^3\).
Taking \(\gamma = {}^{4}/_{3}\pi\) (sphere) and substituting the values of \(v_k\) calculated from literature data, one can from this calculate the effective value of \(\sigma\) for crystal/solution.
The averaged \(\sigma\) calculated in this way for monocalcium hydrosilicate proved to be on the order of 24 erg/cm\(^2\). The mean value of \(\sigma\) for gypsum determined in the same way proved to be on the order of 12 erg/cm\(^2\), which is close to the value of \(\sigma\) for gypsum previously determined by M. L. Chepelevetskii by another method.
The authors express their gratitude to M. L. Chepelevetskii for a valuable discussion.
Received
6 I 1960
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