Linear growth rate, crystals solution

Concentration at crystal surface in equilibrium with T, kg/kg. Slope of solution freezing curve at the freezing point, kg/(kg-h). linear growth rate, nun/h. 

Figure 16.5. Supersaturation behavior, (a) Schematic plot of the Gibbs energy of a solid solute and solvent mixture at a fixed temperature. The true equilibrium compositions are given by points b and e, the limits of metastability by the inflection points c and d. For a salt-water system, point d virtually coincides with the 100% salt point e, with water contents of the order of 10-6 mol fraction with common salts, (b) Effects of supersaturation and temperature on the linear growth rate of sucrose crystals [data of Smythe (1967) analyzed by Ohara and Reid, 1973],...

Here, Cp is the concentration of the dissolved solute in the bulk of the liquid, Cp is the concentration of the solute at the liquid-crystal interface, and Cp is the solubility. Note that the nucleation rate (Jn) and the linear growth rate (G) have been transformed into molar units by using appropriate multiplying factors. It should be emphasized that, while these equations capture the phenomena under consideration, to be correct, they should be expressed in terms of activities in stead of concentrations. 

M the mass of one mole of the crystal, p the density of the crystal, and c the concentration of the solute. Equation (31) assumes that the change in concentration of the solution can be solely attributed to the growth of a single characterised solid. The mean linear growth rate of the suspension, R, may then be defined as... 

If the volume, V, of a solution changes at a rate dV/dt because of solvent evaporation, the linear growth rate of a crystal of area A is ... 

The linear growth rates of crystal faces vary enormously. Some approximate examples of the average rate for crystallization from solution are... 

When studying crystal growth in a dispersion of crystals in a supersaturated solution, it can be observed that the linear growth rate is greater for small than for large crystals. In other cases, the opposite is observed. Can you propose explanations ... 

In practice, one is generally not so much interested in the linear growth rate Lc as in the amount (mass or volume) of crystals formed per unit volume and unit time. In principle, the latter can be given as Lc x Ac, where Ac is the specific surface area of the crystals. However, both factors tend to vary with time. Generally, Lc decreases because crystallization implies depletion of solute and hence a decrease of supersaturation. Moreover, release of the heat of fusion may cause the temperature to increase significantly, hence the solubility to increase, hence In [1 to decrease. Ac increases because (a) each crystal increases in size, and (b) more crystals are formed if nucleation goes on. Several, often complicated, growth rate theories have been worked out for various conditions. We will only touch on a few aspects. 

In Section 15.2.2 it was argued that the linear growth rate of a-lactose hydrate crystals is very much smaller than that of sucrose crystals, because in the former case fierce competition with 3-lactose occurs, and nothing like that can occur for sucrose. Can you now give additional causes for the crystallization rate of a lactose solution being far smaller than that of a sucrose solution at the same supersaturation, especially at low temperature Tip also consult Section 2.2. 

The results in Figure 6.16 show the effects of both solution supersaturation and velocity on the linear growth rates of the (111) faces of potash alum crystals at 32 °C. This hydrated salt [K2SO4 Al2(S04)3 24H2O] grows as almost perfect octahedra, i.e. eight (111) faces. 

The work of Schoeman et al. [22] demonstrated this conclusion more convincingly by use of a chronomal analysis of the conversion with respect to time, a technique suggested previously by Nielsen [62]. In such an analysis, the linear growth rate of the population of crystals is hypothesized to depend on certain driving forces, and the time dependence of the crystal size function is then derived for the circumstance when new crystal nucleation does not occur, as was observed in these experiments. For example, if diffusional transport from the bulk fluid phase to the surface of uniformly sized spherical particles is assumed to be rate limiting, then the change of the particle radius is given by the solution of ... 

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