# Theories of dissolution

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## 1.5.1 . Theories of dissolution

Some workers ^{48,49 }have reviewed the factors which can affect the dissolution of tablets and these include the stirring speed, temperature, viscosity, pH, composition of the dissolution medium and the presence or absence of wetting agents.

Physical models have been set up to account for the observed dissolution of tablets. According to Higuchi ^{50, }there are three models which either alone or in combination, can be used to describe the dissolution mechanisms. These are:

### (i) The Diffusion layer model

This model (Fig 2) assumes that a layer of liquid, H cm thick, adjacent to the solid surface remains stagnant as the bulk liquid passes over the surface with a certain velocity. The reaction at the solid/liquid interface is assumed to be instantaneous forming a saturated solution, C _{s }, of the solid in the static liquid film. The rate of dissolution is governed entirely by the diffusion of the solid molecules from the static liquid film to the bulk liquid according to Fick’s first law:

J = - D _{f }dc / dx (4)

where J is the amount of substance passing perpendicularly through a unit surface area per time, D _{f },is the diffusion coefficient and dc / dx, is the concentration gradient. After a time t, the concentration between the limit of the static liquid layer and the bulk liquid becomes C _{t }. Once the solid molecules pass into the bulk liquid, it is assumed that there is rapid mixing and the concentration gradient disappears.

The theory predicts that if the concentration gradient is always constant i. e. C _{s }- C _{t }is constant because C _{s }>> C _{t }(“sink” conditions which usually mean C _{s }> 10 C _{t }) then a uniform rate of dissolution is obtained.

**Fig. 2 . Diffusion Layer Model **

### (ii) The Interfacial Barrier Model

In the interfacial barrier model (Fig 3), it is assumed that the reaction at the solid/liquid interface is not instantaneous due to a high activation free energy barrier which has to be surmounted before the solid can dissolve. Thereafter the dissolution mechanism is essentially the same as in (i) above, with the concentration at the limit of the static layer of liquid becoming C _{t }after time t.

The rate of diffusion in the static layer is relatively fast in comparison with the surmounting of the energy barrier, which therefore becomes rate limiting in the dissolution process.

**Fig. ****3 ****.** **Diagrammatic representation of the free energy barrier to dissolution **

### (iii) The Danckwert’s Model

The Danckwert’s model (Fig 4) assumes that macroscopic packets of solvent reach the solid/liquid interface by eddy diffusion in some random fashion.

**Fig. ****4 ****. ****The Danckwert’s Model. **

At the interface, the packet is able to absorb solute according to the laws of diffusion and is then replaced by a new packet of solvent. This surface renewal process is related to the solute transport rate and hence to the dissolution rate.

The rate laws predicted by the different mechanisms both alone and in combination, have been discussed by Higuchi ^{50 }. However, the earliest equation expressing dissolution rate in a quantitative manner was proposed by Noyes and Whitney ^{51 }as:-

dc / dt = k (C _{s }- C _{t }) (5)

where dc / dt is the rate of change in concentration with respect to time, and k is the rate constant. The integrated form of the equation is:

In [C _{s }/ (C _{s }- C _{t }) ] = kt (6)

The equation in resemblance to the other rate law equations ^{50 }, predicts a first order dependence on the concentration gradient (i.e. C _{s }- C _{t }) between the static liquid layer next to the solid surface and the bulk liquid. Noyes and Whitney explained their dissolution data using a concept similar to that used for the diffusion model ^{50 }. This considerations relate to conditions in which there is no change in the shape of the solid during the dissolution process ( i. e. the surface area remains constant). However, for pharmaceutical tablets, disintegration occurs during the dissolution process and the surface area generated therefore varies with time.

Aguiar et al ^{52 }proposed a scheme which holds that dissolution occurs only when the drug is in small particles. Wagner ^{53 }modified this idea and showed that dissolution occurs from both the intact tablet and the aggregates and/or granules produced after disintegration by using a plot of the percentage of drug dissolved versus time on logarithmic - probability graph papers.

A modification of this approach was proposed by Kitazawa et al ^{54,55 }. Employing the integrated form of Noyes and Whitney equation (equation 6), they determined the dissolution rate constant of uncoated caffeine tablets. The Kitazawa equations have been used to determine the dissolution rates of some pharmaceutical tablet formulations ^{7,25,41,42 }.