Advanced Pharmacy

By Martin Masuelli and Mauricio Filippa

Advanced Pharmacy is a textbook dedicated to advanced applications in pharmacy. The book balances information by including chapters that give basic knowledge and inform readers on the latest insights in pharmaceutical science. Authored by pharmacology experts and academics, each chapter highlights current knowledge in the field, presenting research in a didactic and educational manner for academics, researchers, and students who need to understand pharmacy. Additional features of the book include chapter summaries and references for advanced readers.
Topics:
Physical Pharmacy: Covers foundations of physical pharmacy, providing a solid understanding of the subject.
Preformulation Studies: examines active pharmaceutical ingredient-excipent compatibility studies, a crucial aspect of drug formulation.
Medicinal Chemistry Applications: explores medicinal chemistry applications using QSAR/QSPR theory.
Computer-assisted Study: explains computer-assisted studies with the example of garlic organosulfur as an antioxidant agent.
Enzymes in Biocatalysis: sheds light on enzyme characteristics, kinetics, production, and applications in biocatalysis.
Antifungal Agents: provides insights into antifungal agents and their significance.
Polysaccharides in Drug Delivery: explores the use of naturally and chemically sulfated polysaccharides in drug delivery systems.
Immunomodulatory Plant Extracts: covers the evaluation of safety and benefits of immunomodulatory plant extracts.
Biofilms and Persistent Cells: examines the development, causes, and consequences of biofilms and persistent cells.
Analytical Methods for Drug Analysis: focuses on the development of analytical methods for analyzing drugs of abuse using experimental design.
Steric Exclusion Chromatography: discusses steric exclusion chromatography, including chromatography of polymers in aqueous solutions.
Intrinsic Viscosity Methods: explores intrinsic viscosity methods in natural polymer pharmaceutical excipients.
Extraction Techniques in Green Analytical Chemistry: highlight environment-firendly techniques in analytical chemistry.

Advanced Pharmacy is a comprehensive resource that bridges the gap between pharmaceutical research and practice, offering invaluable insights into the latest developments in the field. This textbook serves as an essential reference for both learners and scholars in basic and advanced courses in pharmacy and pharmaceutical science.

• Language: English
• Publisher: Bentham Science Publishers
• Release date: May 22, 2000
• ISBN: 9789815049428

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Physical Pharmacy

Mauricio Filippa¹, *

¹ Área de Química Física, Departamento de Química, Facultad de Química, Bioquímica y Farmacia, Universidad Nacional de San Luis. Ejercito de los Andes 950, ZC: 5700, San Luis, Argentina

Abstract

In this chapter, we focus on solutions. In the introduction, general and descriptive aspects are defined, such as the classification of solutions and the addition of solids. Considering the properties of these systems, we focus on definitions related to colligative properties, and on the use of this property for the adjustment of isotonic solutions considering the selective capacity of the membranes, differentiation of tonicity and osmolarity. We also introduce the calculations necessary for the preparation of isotonic solutions with blood plasma, using the mass and volume adjustment method. The solutions require different methods of expressing their concentration, and in order to develop this point, we present the different forms of expression, with extensive detail on one of the variables, i.e., normality, very important in the formulation of parenteral solutions. Since the preparation of solutions is an important aspect, we detail the existing interactions between the solute and the solvent, specify the thermodynamic aspects that condition the solubility of the solute in the solvent, and develop variables such as polar interactions, capacities to accept or give Hydrogen bridge junctions and the energy requirement to generate space within the solvent, giving a rational look at the process of improving the capacity of the solvent to contain the solute. The dissolution rate is another variable developed through simple equations, which make an analysis of the factors that modify it, such as agitation, temperature, particle size, and diffusion coefficient. We also describe variables such as the pH and the dielectric constant of the solvent to modify solubility.

Keywords: Cohesion force, Colligative properties, Concentration gradient, Co-solvent, Descriptive terms, Dissolution rate, Hildebrand coefficient, Hydrogen bridge, Intrinsic factors, Isosmotic, Isotonic, Melting point, pH, Polarizability, Solubility, Solute-solvent interaction, Solvatochromic properties, Staverman coefficient, Stagnant film, Tonicity.

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* Corresponding author Mauricio Filippa: Área de Química Física, Departamento de Química, Facultad de Química, Bioquímica y Farmacia, Universidad Nacional de San Luis. Ejercito de los Andes 950, ZC: 5700, San Luis, Argentina; E-mails: [e-mail]; [e-mail]

INTRODUCTION

Numerous physical variables affect a pharmaceutical product and its pharmacological action. In the API (Active Principle Ingredient) development process, which also includes the configuration of the finished product, along with its manufacture and dispensing, many physical variables, such as temperature, transport, process fluidity, inter and intra-molecular interactions, pore size, permeation speed, dissolution speed, light effects, thicknesses and characteristics of the packaging materials in contact with the product, pressures and torque forces, and particle sizes play a transcendent role in the results.

Analyzing all these variables in this chapter is not feasible, for this requires a special space. However, there are responsibilities that for pharmacists —a professional included within health systems— are specific to their profession. One of them is the vehicle of APIs, i.e., any process that requires conforming a mixture of API with other ingredients, commonly called excipients, and which may result in a product in solid, liquid, or gaseous form, which facilitates access and dispensing for the patient or the health team.

The formation of mixtures between two or more substances may lead to two very different results. On the one hand, mixing can generate what we commonly know as a solution. On the other, simply mixing may result in a spread of one material over another. Depending on the materials that are mixed, the formation of a system that is the sum of the two options described above may also occur. Such is the case of suspensions of a substance that has low solubility in the extensive medium.

Simply put, a solution can be defined as the portion of the system in which when two components are mixed, a single phase is observed. Normally, one of the components is in a lower proportion than the other. This is called a solute. The other component that is in a higher proportion is called a solvent. By definition, a phase is that portion of a system that has the same chemical and physical properties in all its parts, or, in other words, it is chemically and physically homogeneous. For this, it is necessary that the minority component, that is, the solute, is evenly distributed within the solvent. The best way to ensure this is for the solute to distribute each of its molecules within the solvent, each of them interacting independently. These solutions can be classified according to the physical state of the materials that give rise to them, as shown in Table 1.

Table 1 Classification of solutions considering the state of their components.

The independence of the behavior of the solute molecules is conditioned not only by the concentration of the solute but also by its molecular size. When the size of the solute molecule has significant values, the solvent —which normally has a small molecular size— presents difficulties in ensuring this independence and the necessary stability of the solution. This problem is observed for solute molecules whose sizes are greater than 1 nm, as is the case with some milk proteins. Table 2 shows the classification of the systems according to the particle size of the solute.

Table 2 Characteristics of the solutions according to the molecular size of the solute.

When the molecule has a higher range, they are called dispersions, and we may find the colloidal ones where the size of the particle or solute molecule is 1-100 nm, and the suspensions, when their size exceeds 100 nm.

The process of mixing two substances for the transport of APIs is one of the main physicochemical topics that a pharmacist must know. The correct application of this knowledge significantly shortens the development and manufacturing process of the final product formulation. That is why in this chapter, we will focus on the propriety of these mixtures so widely used by the pharmaceutical professional.

Forms of Expression of Solubility

Solubility is the maximum capacity of the solvent to contain a solute in a solution at a previously established temperature. This amount may be expressed in an exact or definite way with what is known as Descriptive Terms. These define approximate quantities of a solvent that may be capable of dissolving a single unit of solute. The descriptive terms may have some variation according to the bibliographical reference, so in order to standardize them, pharmacopeias have included them. Table 3 shows the descriptive terms defined by USP-NF 28.

Table 3 Descriptive Terms defined in USP-NF 28.

In order to make a more precise description of the solubility, variables related to the proportions of solute and solvent mass can be used. These magnitudes have the characteristic of not changing with temperature. However, when volumes are used (as it is the case of gases or liquids), they undergo changes with temperature or pressure, so the operating conditions in which the value was obtained must be strictly informed. This situation occurs when the proportion of the solution is given by one unit of the solute in one unit volume of the solvent or the solution. Changes also occur if the unit of the solute is reported in volume, as it is the case of liquid solutes. These magnitudes are modified by expansion, contraction, or compression, and therefore their solubility values are also modified. This also occurs to a greater extent in gases.

Adding Solids to the Solvent

Many times, it is necessary to give guidelines to the technical staff of a pharmaceutical office that performs the preparation of solutions. These guidelines are related to the preparation of solutions of approximate concentrations that are used for coloring reactions, preparation of solutions for preliminary tests or amounts of solid necessary to saturate a defined volume of solvent. However, the description of the incorporation of solids within a container presents difficulties for professionals or technicians, since some terms may be interpreted differently, depending on the countries or regions. For this, in Table 4, a series of terms are illustrated with the aim of defining a common criterion for the addition of a solid within a container.

Table 4 Descriptive Terms for the addition of solid in a container.

Expressions and Forms of Concentration

Physical units or chemical units may be used to express concentration. The physical units commonly used are the weight and the volume, while the chemical units are related to the molecular structure, such as the molecular weight or the number of charges of the species in the solution. The two nomenclatures to express concentration data are observed in Tables 5 and 6.

Table 5 Physical description of the concentration of solutions.

Table 6 Description in chemical units of the concentration of solutions.

The Molality and Molarity define the quantity of moles of solute contained in a quantity of solvent. In the case of Molality, it is in 1000 grams of solvent, and in Molarity, it is in 1000 ml of solution. The mole fraction of a solution is the relationship between the number of moles of solute with respect to the total that makes it up. Normality is the number of Equivalent Weights (EqW) or Equivalent (Eq) of solute in 1000 ml of solution. In order to know how much an Eq of a compound represents in mass terms, it is necessary to identify its chemical formula and in which reaction it may participate. First, it is important to know that, by definition, an equivalent weight of a substance is the total weight of that compound, which reacts completely with another. For example, if 1 mole of NaOH reacts with 1 mole of HCl in an aqueous solution, both components will react completely. Therefore, its equivalent weight would be equal to its Molecular Weight (MW) / coefficient of change or Number of Equivalents, which in this case is 1. According to this definition, to calculate EqW, the following Equation is used:

The number of equivalents, in case of NaOH when it participates in acid-base reactions, is the number of hydroxyls that this molecule has, and for HCl, it is the number of protons. Following the same analysis, the EqW of SO4H2 EqW is equal to its Molecular Weight/2. This EqW of sulfuric acid is the one that would react completely with an EqW of NaOH. In the case of an ion, the EqW of that species in solution is equal to its atomic weight / its valence. In the case of salts, it is more complex because it depends on the charge of the ions and their stoichiometric coefficients. Table 7 shows some examples.

The number of equivalents is a very important number to know and identify correctly, since the calculation of equivalent mass units that participate in a reaction depends on it. If the species that make up a solution participate in the reactions of an electrochemical nature, it is necessary to know their equivalent weight to know the number of electrons that they exchange in the reaction. For example, when Fe⁰ is oxidized to Fe+2, its equivalent weight is its Atomic Weight/2. However, if Fe⁰ is oxidized to Fe+3, its equivalent weight is its Atomic Weight/3. In these reactions, it is necessary to know the number of electrons that are exchanged, since in these conditions, the equivalent masses of substances that are going to react, which, like acid-base reactions, exchange whole units of elements. In the case of electrochemical reactions, they are electrons. In more complex molecular structures where the species are organic, it is necessary to know the molecular weight of the substance that exchanges electrons and how much of them they can give for each mole. As we mentioned, to calculate the equivalent weight, it is necessary to know the reaction in which the species will participate. For example, in the salt NaH2PO4, and according to Table 6, its EqW = MW/1. However, if the solution containing this salt is used so that the H2PO4- ion reacts with OH- ions, its equivalent number would be 2. If it reacted with H+ ions added or generated in the solution, its equivalent number would be 1, since the mentioned ion admits the taking of a single proton.

Table 7 Number of Equivalent of a salt according to its stoichiometry.

One of the frequent problems that the Pharmaceutical professional finds, in any workspace such as a private office, a hospital or a laboratory, is the preparation of solutions for nutrition or solutions that require the replacement of electrolytes. For this, it is necessary to use the series of salts which is available at the pharmacy, which may not always be the most suitable for the required purpose. The bibliography normally reports the quantities available to the body in the internal environment for each of the ions, in milliEquivalent/L or any other equivalent unit. For example, blood plasma has an average Ca+2 levels of 9.5 mg/dL or 4.7 mEq/L (milliequivalent/L). To supplement these levels in the Pharmacy office, the Professional may have several available options, such as Ca3(PO4)2 or CaCl2. To calculate the necessary amounts of these salts to prepare a solution that has a Ca+2 concentration of 4.7 mEq/L, it is necessary to introduce another variable that defines the equivalents of a species that is provided per mole of salt.

For example, if CaCl2 were used, 2 equivalents of calcium would be provided per mole of the salt. This is because the EqW of Ca+2 is its MW/charge, that is, divided by two. Therefore, each mole of the salt provides one mole of Ca+2, and therefore, 2 equivalents of Ca. However, if calcium were incorporated with the following compound Ca3(PO4)2, 6 equivalents of Ca+2 would be provided per mole of salt. Then, knowing the necessary concentration of salt that adds to a solution 4.7mEq/L of Ca+2 in g/L requires the use of the following Equations. The first Equation (2) comes from the original Equation (1) which has been transformed, and from Equation (3), the necessary amount of salt is obtained, expressed in grams/L of solution, which contains the mEq required in the literature.

In line with the same example of Ca+2 in which its concentration is required to be 4.7 mEq/L, Equation (3) can be seen in the following example:

If CaCl2 is used, its MW=110 g/mol, so its EqW = 110/2 = 55 g/Eq. If the necessary Ca were 4.7 mEq /L, the calculation would then be: g/L = 55 g/Eq x 4.7x10-3 Eq/L = 0.26 g/L CaCl2.

If Ca3(PO4)2 MW = 310.2 g/mol were used, its EqW = 310.2/6 = 51.7 g/Eq. If the Eq/L of Ca were equal, 4.7mEq/L, the calculation would then remain g/L = 51.7 g/Eq x 4.7x10-3 Eq/L, which would correspond to 0.24 g of Ca3(PO4)2.

Equation (3) can be written as Equation (4) for a better operation in terms of mEq/L, which is in the units in which the literature commonly reports these concentrations of each species in the internal environment.

On the other hand, when salts are dissolved in an aqueous medium, we may have ions that are in different concentrations according to their charge and the stoichiometric coefficient of the salt that originates them. For example, a 1 M solution of NaCl generates one mole of Na+ ions in solution and one mole of Cl- ion. However, a 1 M solution of Na2CO3 generates two moles/L of Na+ ion and 2M and one mole/L of CO3-2 ion in solution.

The thermodynamics of solutions

Under saturation conditions, solubility is a dynamic equilibrium, in which small modifications of the conditions generate immediate changes in the state of equilibrium. This dynamic condition occurs at the interface between the solute and the solvent, when an almost imperceptible quantity (thermodynamic reversibility) of molecules leaves the surface into the solution and, in the same way, from within the solution, the reverse phenomenon occurs. This can be represented as follows (5) for any API, solid or liquid:

However, the solubilization process of the solute, that is, when equilibrium is not achieved, occurs if the process is spontaneous. This spontaneity of the solubilization process is governed by two thermodynamic variables: the enthalpy change (∆H) and the entropy change (∆S). According to the second law of thermodynamics, spontaneous processes require energy loss and entropy gain. The variable quantifies the spontaneity of a process in Gibbs free energy and is calculated from Equation (6).

Under equilibrium conditions, the system presents a Gibbs free energy variation (∆G) equal to 0. Under ideal conditions, the solution has ∆H = 0 and the engine of the dissolution process would be associated with an entropy gain, the entropy of the mixture can be calculated using the Equation (7):

where x2 and x1 are the molar fractions of the solute and solvent, respectively, and R is the constant of the gases.

However, for non-ideal solutions, ∆H contributes which is directly proportional to the product of the molar fractions of both components, the parameter delta (δ) being another contribution related to the interactions between the solute and the solvent, as shown in (8).

In the daily reality of the laboratory, the δ coefficient is the result of the interactions involved in the dissolution process, which is not only related to the interactions between the solute and the solvent, but which also involves other processes. These processes are described below in three fundamental stages:

1.- The first one involves the removal of a solute molecule within its structure. This process requires power consumption or requires the system to do work w1.

2.- The second stage requires that the solvent molecules generate the necessary space within, to locate the solute molecule detached in stage 1. This process also requires energy w2.

3.- The third stage begins after the solute molecule occupies the space generated and interacts with the solvent molecules. This stage releases energy –w3, and then the solvent begins to reduce the space created in stage 2, producing a final work in this stage equivalent to 2w3. By reducing the space generated, the solvent facilitates the solubilization process. Therefore, the total work is:

Knowing the Equation (9), Hildebrand and Scott [1], and then Hansen and Beerbower [2], reported solubility parameters in a series of solvents. Hansen was the one to propose that the solubility parameter is a summation of the solvent’s capacities to interact with the solute, expressed in the Equation (10):

where D refers to the nonpolar effects of solvents, P to the effects related to interactions of polar characters and H to the interactions by hydrogen bridging. However, Equation (10) does not consider the energy requirements necessary to separate a molecule from the solute from its structure.

Among the various expressions that have been proposed for the description of energies, linear free energy relations (LSER) is one of the most used. It was proposed by M. J. Kamlet and colaborators [3] and P. W. Carr [4], in which XYZ is a property linearly related to Gibbs free energy, as presented in (11).

In Equation (11), the value XYZ0 is a constant that only depends on the solute, the energy for the formation of the cavity (energyfc) shows dependence on the solvent and the following term is the sum of all the forms of interaction of the solute with the solvent (solute-solvent energy) [5]. Considering the different forms of interaction that can occur, it is possible to express the above equation as follows:

In Equation (12), δH is the Hildebrand solubility parameter, which represents the cohesive forces of the solvent, the main variable to represent the energy necessary to separate two molecules of the solvent. π* is a solvatochromic parameter that describes a combination of properties related to the polarizability and polarity of the solvent. α and β are solvatochromic properties of the solvent, related to the donation capacity in the formation of hydrogen bonds (HBD), and in the case of α and β, to the acceptance capacity of the formation of hydrogen bonds (HBA), or the capacity to donate a pair of electrons in the formation of a coordinate bond. Table 8 shows the parameters of some solvents of interest. These parameters belong to the intrinsic properties of solvents.

Table 8 Solvatochromic parameters of solvent.

*Y. Marcus, Chem. Soc. Rev. 22 (1993) 409–416 and Hansen C.M., Beerbower A. Solubility parameters Kirk & Othmer: Encyclopedia of chemical technology, Suppl. 2ed. New York: John Wiley & Sons, 1971. 889-910.

These parameters of the solvents are determined through the energies of the absorption peaks of certain solutes when they are carefully selected in the solvents in question, after subtracting the effect of the solvents that do not experience HBD and/or HBA used in the test. For practical purposes, they were assigned numerical values so that they ideally describe the effect of the unique HBD and HBA properties of solvents, and their values are not affected by polarizability, polarity, or cohesion forces [6].

Later studies have revealed that the interactions that occur between a solute and the solvent are related to positive and negative factors. For some dissolution processes, some of the XYZ0, m, p, a and/or b coefficients may be insignificant or close to 0, showing that the property does not play a significant role in the dissolution process of the drug under study. On the contrary, if these coefficients have significant values with respect to the others, it is indicative that this property is gravitating in the solute dissolution process. These coefficients can present a positive sign when the existence of that property improves the solubility of the solute, and a negative sign when the presence of that property decreases it. This allows, through a multiparametric analysis, to identify which solvent parameters exert positive effects and which parameters exert negative effects with their corresponding values of solubility of a substance. In this way, it is possible to establish how solvents interact with each API, this being one of the best predictive elements that we may have. In Table 9, some parameters of drug interaction with the solvents studied are observed.

Table 9 Recently published API solubility parameters.

*(1) Filippa M.A. and Gasul E.I. Ibuprofen solubility in pure organic solvents and aqueous mixtures of cosolvents: Interactions and thermodynamic parameters relating to the solvation process. Fluid Phase Equilib. 2013, 354, 185-190. (2) Filippa M. and Gasul E. Experimental determination of Naproxen solubility in organic solvents and aqueous binary mixtures: Interactions and thermodynamic parameters relating to the solvation process. J. Mol. Liq. 2014, 198, 78–83. (3) Filippa M. Melo G. Gasull E. Ketoprofen Solubility in Organic Solvents and Aqueous Co-solvent Systems: Interactions and Thermodynamic Parameters of Solvation. J. Pharm. Chem. Biol. Sci. 2016, 3(4), 440-453. (4) Castro, G., Filippa, M., Peralta, C., Davin M., Almandoz, M., Gasull, E. Solubility and Preferential Solvation of Piroxicam in Neat Solvents and Binary Systems Z. Phys. Chem., 2018, 232(2), 257–280. (5) Castro, G.T., Filippa, M.A., Sancho, M.I., Gasull, E.I., Almandoz, M.C. Solvent effect on the solubility and absorption spectra of meloxicam: experimental and theoretical calculations. Phys. Chem. Liq., 2020, 58(3), 337–348. (6) Sun R., Wang Y., He H., Wan Y., Li L., Sha J., Jiang G., Li Y., Li T., Ren B. J. Mol. Liq. 2020, 319, 114139. (7) Zidan Cao, Ruke Zhang, Xiaoran Hu, Jiao Sha, Gaoliang Jiang, Yu Li Tao Li Baozeng Ren. J Chem Thermodynamics 2020, 151, 106239. (8) He H., Wan Y., Sun R., Zhang P., Jiang G., Sha J., Li Y., Li T., Ren B. Piperonylonitrile solubility in thirteen pure solvents: Determination, Correlation, Hansen solubility parameter, solvent effect, and thermodynamic analysis J. Chem. Thermodynamics. 2020 150, 106191. (9) Huang Z., Zun Y., Gong Y., Hu X., Sha J., Li Y., Li T., Ren B. Solid-liquid equilibrium solubility, thermodynamic properties, solvent effect of Ipriflavone in twelve pure solvents at various temperatures. J. Chem. Thermodynamics, 2020, 150, 106231. (10) Li R., Chen X., He G., Wu C., Gan Z., He Z., Zhao J., Han D. The dissolution behaviour and thermodynamic properties calculation of praziquantel in pure and mixed organic solvents J. Chem. Thermodynamics, 144, 2020, 106062. (11) Renjie X., Jian W. Measurement and Correlation of Solubility of Gatifloxacin in 12 Pure Solvents from 273.15 K to 318.15 K. J. Chem. Eng. Data, 2019, 64, 2, 676–681. (12) Hu X., Gong Y., Cao Z., Huang Z., Sha J., Li Y., Li T. Solubility, Hansen solubility parameter and thermodynamic properties of etodolac in twelve organic pure solvents at different temperatures. J. Mol. Liq. 2020, 316, 113779.

Effect of Temperature on the Modification of Equilibrium

When the solubilization process reaches equilibrium, the effect of temperature in the process can be evaluated.

For this, the van’t Hoff Equation defines the relationship between the equilibrium constant and the absolute temperature. Considering the equilibrium described in (5), the Equation is (13):

integrating this Equation resulting in Equation (14)

and ordering Equation (14)

Equation (15) has a linear form, and, experimentally, the modification of the equilibrium constant versus the absolute temperature is studied, and two different situations can be observed depending on the positive or negative sign than the enthalpy variation of the process, as can be seen in Fig. (1).

Fig. (1))

Possible graphic profiles in the analysis of the equilibrium constant with temperature.

Dissolution Rate

As we have previously discussed, solubility is a dynamic phenomenon, as well as an equilibrium condition when high concentration levels are achieved, such as in saturated solutions. Under these conditions, it is necessary to know that the solute molecules that leave the solid do so at the same speed as the solute molecules that are deposited on the solid. This speed is directly related to the flow of molecules that circulate in both directions. Measuring this flow with respect to time will give us an idea of the real process of the flow of molecules that transpose the interface between the solute and the solvent. To explain this in mathematical terms, it is necessary to introduce Fick's first law. It establishes that the speed of the flow of a material through any surface is observed by the following Equation (16):

where J is the flow of material that passes in a unit area in a unit of time, M is the amount in moles, A is the area and t is time. It is also known that the flow of materials within a system is mainly due to a difference in concentration, or what is normally known as a concentration gradient. In the same way that gases spontaneously balance their pressures, from places of higher pressure to places of lower pressure, the dissolved species in a system migrate from places of higher concentration to places of lower concentration, as shown in Equation (17):

where dC is the concentration differential, dx is the unit of separation of concentrations and D a constant of proportionality or specific flow.

The stagnant layer theory proposed by Nernst [7] in 1904, states that when a particle of a soluble solid comes into contact with the solvent, a very thin film of a saturated solution (C1) of the solid accompanies the particle in every moment (see Fig. (2)). Taking this into account, and integrating Equations (16) and (17), the flow of material that enters the solution per unit of time would be Equation (18):

Fig. (2))

Description of the particle in the solution with the accompanying stagnant layer. Within the stagnant layer zone, the concentration is close to C1 or saturation concentration.

where dC is (C2-C1), being C2 the concentration of the solid in the solution. This concentration differential is the engine of the dissolution process, and h describes the thickness of the layer that accompanies the solid. Therefore, when both concentrations are equal, the system is in equilibrium and the solution is saturated and stable.

This theory also makes it possible to describe that when the solution is stirred, the flow speed is increased, since the stirring reduces the thickness of the stagnant layer and makes it easier for the species on the surface of the solute to pass into the solution. Noyes and Whitney [8] studied the dissolution rate of some drugs in rotating rings, thus keeping the surface constant, and established a fundamental Equation that explains their experience:

where dC/dt represents the change in concentration over time, this being the dissolution rate of a solid in a constant volume of solvent, driven by the concentration gradient; and K is a constant of proportionality, called the dissolution constant. Equation (19) describes the dissolution process as a first-order Equation, so if we represent the log of concentration with respect to time, we will obtain a straight line and the value of K from the slope. This condition is commonly referred to as non-sink. According to the operating conditions, when the volume of the liquid where the particle is located is very large, it could be said that C1 is insignificant compared to C2, so that Equation (19) would be simplified as presented in Equation (20):

In Equation (20), it is observed that dC/dt is referenced to two constants and can be represented mathematically as a zero-order reaction, so the graphical representation of concentration with respect to time is linear. This condition is called sink and it occurs on many occasions when a particle of a solid dissolves within the lumen of the gastrointestinal tract and its epithelium carries out an immediate transport effect to other areas in a short period of time. An analysis of the constant K allows us to know other variables that modify the dissolution rate. Brunner [9] included the diffusion coefficient (D), the thickness of the staked layer (h) and the volume of the diffusion medium (v) in the Equation of Fick’s first law, where k2 is the specific velocity of dissolution in Equation (21):

Colligative Properties of Solutions

The colligative properties of the solutions are properties that depend on Raoult's law. When a non-electrolyte and non-volatile solute is added to a volatile solvent, it undergoes a decrease in vapor pressure. This decrease is directly proportional to the number of solute particles present in the solution and does not depend on their nature. At a previously defined temperature, the vapor pressure of a solvent is directly proportional to its activity and knowing that the activity is equal to the coefficient multiplied by the concentration of the solvent a = γ x1, in the pure state, the x1 as the coefficient has a unit value. In Equation (22), according to Raoult’s law, we can say that

where P and Pº are the pressures of the solvents in solution and that of the pure solvent respectively, under the same operating conditions, and a is the solvent activity. Taking this Equation into account, the activity of the solvent in the solution state is a ratio of the vapor pressures of the solvent in the solution to the solvent in the pure state. Now we have a single solute x1 +x2 = 1. By convention, we will use number 2 to identify the solvent and number 1 for the solute. Replacing the Equation mentioned in Equation (22) we obtain:

and knowing that the pressure difference is equal to P-P0=-∆P, then

Equation (23) calculates the change in the vapor pressure of the solvent in solution that also has consequences on the change in the boiling temperature of the solution with respect to the solvent in its pure state, and this change is directly proportional to the change in vapor pressure, as can see in Equation (24)

considering the previous Equations

and in diluted conditions x2 =m PM2/1000, replacing this in Equation (25)

or the Equation (26) can be written as la Equation (27):

and according to the Clapeyron Equation, written in Equation (28):

and considering the original Equation of Raoult’s law

In Equation (29), Ke is called the molal ebulloscopic constant of the solvent, or in other words, it is the increase in the boiling temperature of the solution when the solute concentration is 1 molal.

The same criterion is applied for the decrease in the freezing temperature that the solutions present with respect to the solvent in its pure state.

In Equation (30), Kc is the molal cryoscopic decrease constant of the solvent or decrease in the freezing temperature of the solution when it has a concentration of 1 molal.

The phenomenon of Osmotic Pressure (π) is another colligative property. This phenomenon occurs due to the loss of activity of the solvent when it is in a solution, with respect to its pure state, and is observed as the passage of solvent through a selective solvent or semi-permeable membrane, from an area of greater activity, towards one of less activity, as can be seen in Fig. (3).

Fig.