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3

Physicochemical properties of drugs in solution

3.1 Concentration units 56 3.5 Ionisation of drugs in solution 75 3.2 Thermodynamics – a brief 3.6 Diffusion of drugs in solution 89 introduction 57 Summary 90 3.3 Activity and chemical potential 62 References 91 3.4 Osmotic properties of drug solutions 69

In this chapter we examine some of the important physicochemical properties of drugs in aqueous solution which are of relevance to such liquid dosage forms as injections, solutions, and eye drops. Some basic thermodynamic concepts will be introduced, particularly that of thermo- dynamic activity, an important parameter in determining drug potency. It is important that parenteral solutions are formulated with osmotic pressure similar to that of blood serum; in this chapter we will see how to adjust the tonicity of the formulation to achieve this aim. Most drugs are at least partially ionised at physiological pH and many studies have suggested that the charged group is essential for biological activity. We look at the influence of pH on the ionisation of several types of drug in solution and consider equations that allow the calculation of the pH of solutions of these drugs. First, however, we recount the various ways to express the strength of a solution, since it is of fundamental importance that we are able to interpret the various units used to denote solution concentration and to understand their interrelationships, not least in practice situations.

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3.1 Concentration units is an accepted SI unit. Molarity may be con- verted to SI units using the relationship 1 mol litre 01 # 1 mol dm 03 # 10 3 mol m 03. A wide range of units is commonly used to Interconversion between molarity and molal- express solution concentration, and confusion ity requires a knowledge of the density of the often arises in the interconversion of one solution. set of units to another. Wherever possible Of the two units, molality is preferable for a throughout this book we have used the SI precise expression of concentration because it system of units. Although this is the currently does not depend on the solution temperature recommended system of units in Great as does molarity; also, the molality of a com- Britain, other more traditional systems are still ponent in a solution remains unaltered by the widely used and these will be also described in addition of a second solute, whereas the this section. molarity of this component decreases because the total volume of solution increases follow- ing the addition of the second solute. 3.1.1 Weight concentration

Concentration is often expressed as a weight of solute in a unit volume of solution; for 3.1.3 Milliequivalents example, g dm03, or % wv, which is the number of grams of solute in 100 cm3 of solu- The unit milliequivalent (mEq) is commonly tion. This is not an exact method when used clinically in expressing the concentration working at a range of temperatures, since of an ion in solution. The term ‘equivalent’, or the volume of the solution is temperature- gram equivalent weight, is analogous to the dependent and hence the weight concentra- mole or gram molecular weight. When tion also changes with temperature. monovalent ions are considered, these two Whenever a hydrated compound is used, terms are identical. A 1 molar solution of it is important to use the correct state of sodium bicarbonate, NaHCO3, contains 1 mol ! − or 1 Eq of Na and 1 mol or 1 Eq of HCO3 per hydration in the calculation of weight litre (dm03) of solution. With multivalent concentration. Thus 10% w v CaCl2 (anhy- drous) is approximately equivalent to 20% wv ions, attention must be paid to the valency of CaCl ·6H O and consequently the use of the each ion; for example, 10% w v CaCl2·2H2 O 2 2 2! vague statement ‘10% calcium chloride’ could contains 6.8 mmol or 13.6 mEq of Ca in 3 result in gross error. The SI unit of weight 10 cm . 1 concentration is kg m03 which is numerically The Pharmaceutical Codex gives a table of equal to g dm03. milliequivalents for various ions and also a simple formula for the calculation of milli- equivalents per litre (see Box 3.1). 3.1.2 Molarity and molality In analytical chemistry a solution which contains 1 Eq dm03 is referred to as a normal These two similar-sounding terms must not be solution. Unfortunately the term ‘normal’ is confused. The molarity of a solution is the also used to mean physiologically normal with number of moles (gram molecular weights) of reference to saline solution. In this usage, a solute in 1 litre (1 dm3) of solution. The molal- physiologically normal saline solution con- ity is the number of moles of solute in 1 kg of tains 0.9 g NaCl in 100 cm3 aqueous solution solvent. Molality has the unit, mol kg01, which and not 1 equivalent (58.44 g) per litre.

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Thermodynamics – a brief introduction 57

(b) 9 g of sodium chloride are dissolved in Box 3.1 Calculation of milliequivalents 991 g of water (assuming density # 1gdm03). The number of milliequivalents in 1 g of substance is Therefore 1000 g of water contains 9.08 g of given by sodium chloride # 0.155 moles, i.e. molality # valency x 1000 x no. of specified 0.155 mol kg01. units in 1 atom/molecule/ion mEq = (c) Mole fraction of sodium chloride, x , is atomic, molecular or ionic weight 1 given by For example, CaCl2·2H2 O (mol. wt. # 147.0) n1 0.154 − 2 × 1000 × 1 x = = = × 3 2+ = 1 2.79 10 mEq Ca in 1 g CaCl2 · 2H2 O n + n 0.154 + 55.06 147.0 1 2 = 13.6 mEq (Note 991 g of water contains 99118

and moles, i.e. n2 # 55.06.) ! × × (d) Since Na is monovalent, the number of − 1 1000 2 ! mEq Cl in 1 g CaCl · 2H O = milliequivalents of Na # number of milli- 2 2 147.0 moles. = 13.6 mEq 03 that is, each gram of CaCl ·2H O represents 13.6 mEq Therefore the solution contains 154 mEq dm 2 2 ! of calcium and 13.6 mEq of chloride of Na .

3.1.4 Mole fraction 3.2 Thermodynamics – a brief introduction The mole fraction of a component of a solu- tion is the number of moles of that com- The importance of thermodynamics in the ponent divided by the total number of moles pharmaceutical sciences is apparent when it is present in solution. In a two-component realised that such processes as the partitioning (binary) solution, the mole fraction of solvent, of solutes between immiscible solvents, the # ! x1, is given by x1 n1 (n1 n2), where n1 and solubility of drugs, micellisation and drug– n2 are respectively the numbers of moles of receptor interaction can all be treated in solvent and of solute present in solution. thermodynamic terms. This brief section Similarly, the mole fraction of solute, x2, is merely introduces some of the concepts of # ! given by x2 n2 (n1 n2). The sum of the thermodynamics which are referred to through- mole fractions of all components is, of course, out the book. Readers requiring a greater ! # unity, i.e. for a binary solution, x1 x2 1. depth of treatment should consult standard texts on this subject.2, 3

EXAMPLE 3.1 Units of concentration Isotonic saline contains 0.9% wv of sodium 3.2.1 Energy chloride (mol. wt. # 58.5). Express the concen- tration of this solution as: (a) molarity; (b) Energy is a fundamental property of a system. molality; (c) mole fraction and (d) milliequiv- Some idea of its importance may be gained by alents of Na! per litre. Assume that the density considering its role in chemical reactions, of isotonic saline is 1 g cm03. where it determines what reactions may occur, how fast the reaction may proceed and in Answer which direction the reaction will occur. (a) 0.9% wv solution of sodium chloride Energy takes several forms: kinetic energy is contains 9 g dm03 # 0.154 mol dm03. that which a body possesses as a result of its

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58 Chapter 3 • Physicochemical properties of drugs in solution

motion; potential energy is the energy which changes in heat energy, which were not a body has due to its position, whether gravi- encompassed in the conservation law. tational potential energy or coulombic poten- It follows from equation (3.2) that a system tial energy which is associated with charged which is completely isolated from its sur- particles at a given distance apart. All forms of roundings, such that it cannot exchange heat energy are related, but in converting between or interact mechanically to do work, cannot the various types it is not possible to create or experience any change in its internal energy. destroy energy. This forms the basis of the law In other words the internal energy of an isolated of conservation of energy. system is constant – this is the first law of The internal energy U of a system is the sum thermodynamics. of all the kinetic and potential energy contri- butions to the energy of all the atoms, ions and molecules in that system. In thermo- 3.2.2 Enthalpy dynamics we are concerned with change in internal energy, ∆U, rather than the internal Where a change occurs in a system at constant energy itself. (Notice the use of ∆ to denote a pressure as, for example, in a chemical reaction finite change). We may change the internal in an open vessel, then the increase in internal energy of a closed system (one that cannot energy is not equal to the energy supplied as exchange matter with its surroundings) in heat because some energy will have been lost only two ways: by transferring energy as work by the work done (against the atmosphere) (w) or as heat (q). An expression for the change during the expansion of the system. It is in internal energy is convenient, therefore, to consider the heat change in isolation from the accompanying ∆U = w + q (3.1) changes in work. For this reason we consider a If the system releases its energy to the property that is equal to the heat supplied at surroundings ∆U is negative, i.e. the total constant pressure: this property is called the internal energy has been reduced. Where heat enthalpy (H). We can define enthalpy by is absorbed (as in an endothermic process) the ∆H = q at constant pressure (3.3) internal energy will increase and consequently q is positive. Conversely, in a process which ∆H is positive when heat is supplied to a system releases heat (an exothermic process) the which is free to change its volume and negative internal energy is decreased and q is negative. when the system releases heat (as in an exo- Similarly, when energy is supplied to the thermic reaction). Enthalpy is related to the system as work, w is positive; and when the internal energy of a system by the relationship system loses energy by doing work, w is H = U + pV (3.4) negative. It is frequently necessary to consider infini- where p and V are respectively the pressure tesimally small changes in a property; we and volume of the system. denote these by the use of d rather than ∆. Enthalpy changes accompany such pro- Thus for an infinitesimal change in internal cesses as the dissolution of a solute, the forma- energy we write equation (3.1) as tion of micelles, chemical reaction, adsorption onto solids, vaporisation of a solvent, hydra- dU = dw + dq (3.2) tion of a solute, neutralisation of acids and We can see from this equation that it does bases, and the melting or freezing of solutes. not really matter whether energy is supplied as heat or work or as a mixture of the two: the change in internal energy is the same. 3.2.3 Entropy Equation (3.2) thus expresses the principle of the law of conservation of energy but is much The first law, as we have seen, deals with the wider in its application since it involves conservation of energy as the system changes

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from one state to another, but it does not For instance, the entropy of a perfect gas specify which particular changes will occur changes with its volume V according to the spontaneously. The reason why some changes relationship have a natural tendency to occur is not that the V system is moving to a lower-energy state but ∆S = nR ln f (3.7) that there are changes in the randomness of Vi the system. This can be seen by considering a where the subscripts f and i denote the final specific example: the diffusion of one gas into and initial states. Note that if Vf p Vi (i.e. if the another occurs without any external interven- gas expands into a larger volume) the loga- tion – i.e. it is spontaneous – and yet there are rithmic (ln) term will be positive and the equa- no differences in either the potential or kinetic tion predicts an increase of entropy. This is energies of the system in its equilibrium state expected since expansion of a gas is a spon- and in its initial state where the two gases are taneous process and will be accompanied by segregated. The driving force for such spontan- an increase in the disorder because the mol- eous processes is the tendency for an increase ecules are now moving in a greater volume. in the chaos of the system – the mixed system Similarly, increasing the temperature of a is more disordered than the original. system should increase the entropy because at A convenient measure of the randomness or higher temperature the molecular motion is disorder of a system is the entropy (S). When a more vigorous and hence the system more system becomes more chaotic, its entropy chaotic. The equation which relates entropy increases in line with the degree of increase in change to temperature change is disorder caused. This concept is encapsulated T in the second law of thermodynamics which ∆ = f S CV ln (3.8) states that the entropy of an isolated system Ti increases in a spontaneous change. where C is the molar heat capacity at con- The second law, then, involves entropy V ∆ stant volume. Inspection of equation (3.8) change, S, and this is defined as the heat ∆ shows that S will be positive when Tf p Ti , as absorbed in a reversible process, q , divided by rev predicted. the temperature (in kelvins) at which the The entropy of a substance will also change change occurred. when it undergoes a phase transition, since For a finite change this too leads to a change in the order. For example, when a crystalline solid melts, it q ∆S = rev (3.5) changes from an ordered lattice to a more T chaotic liquid (see Fig. 3.1) and consequently and for an infinitesimal change an increase in entropy is expected. The entropy change accompanying the melting of dq dS = rev (3.6) a solid is given by T ∆H ∆S = fus (3.9) By a ‘reversible process’ we mean one in T which the changes are carried out infini- ∆ tesimally slowly, so that the system is always where Hfus is the enthalpy of fusion (melting) in equilibrium with its surroundings. In this and T is the melting temperature. Similarly, case we infer that the temperature of the we may determine the entropy change when a surroundings is infinitesimally higher than liquid vaporises from that of the system, so that the heat changes ∆ Hvap are occurring at an infinitely slow rate, so that ∆S = (3.10) T the heat transfer is smooth and uniform. ∆ We can see the link between entropy and dis- where Hvap is the enthalpy of vaporisation order by considering some specific examples. and T now refers to the boiling point. Entropy

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60 Chapter 3 • Physicochemical properties of drugs in solution

from which we can see that the change in free energy is another way of expressing the change in overall entropy of a process occur- ring at constant temperature and pressure. In view of this relationship we can now consider changes in free energy which occur (a) during a spontaneous process. From equation (3.13) we can see that ∆G will decrease during a spontaneous process at constant temper- ature and pressure. This decrease will occur until the system reaches an equilibrium state when ∆G becomes zero. This process can be thought of as a gradual using up of the (b) system’s ability to perform work as equi- librium is approached. Free energy can there- Figure 3.1 Melting of a solid involves a change from an fore be looked at in another way in that it ordered arrangement of molecules, represented by (a), to a more chaotic liquid, represented by (b). As a result, the represents the maximum amount of work, melting process is accompanied by an increase in entropy. wmax (other than the work of expansion), which can be extracted from a system under- changes accompanying other phase changes, going a change at constant temperature and such as change of the polymorphic form of pressure; i.e. crystals (see section 1.2), may be calculated in ∆ = G wmax (3.14) a similar manner. at constant temperature and pressure At absolute zero all the thermal motions of the atoms of the lattice of a crystal will have This nonexpansion work can be extracted from ceased and the solid will have no disorder and the system as electrical work, as in the case of hence a zero entropy. This conclusion forms a chemical reaction taking place in an electro- the basis of the third law of thermodynamics, chemical cell, or the energy can be stored which states that the entropy of a perfectly in biological molecules such as adenosine crystalline material is zero when T # 0. triphosphate (ATP). When the system has attained an equi- librium state it no longer has the ability to 3.2.4 Free energy reverse itself. Consequently all spontaneous processes are irreversible. The fact that all spon- The free energy is derived from the entropy taneous processes taking place at constant and is, in many ways, a more useful function temperature and pressure are accompanied by to use. The free energy which is referred to a negative free energy change provides a useful when we are discussing processes at constant criterion of the spontaneity of any given pressure is the Gibbs free energy (G). This is process. defined by By applying these concepts to chemical equi- libria we can derive (see Box 3.2) the following G = H − TS (3.11) simple relationship between free energy change and the equilibrium constant of a reversible The change in the free energy at constant reaction, K: temperature arises from changes in enthalpy and entropy and is ∆G =−RT ln K ∆G = ∆H − T ∆S (3.12) where the standard free energy G is the free energy of 1 mole of gas at a pressure of 1 bar. Thus, at constant temperature and pressure, A similar expression may be derived for ∆G =−T ∆S (3.13) reactions in solutions using the activities (see

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Box 3.2 Relationship between free energy change and the equilibrium constant

Consider the following reversible reaction taking place ∆G is the standard free energy change of the reaction, in the gaseous phase given by aA + bB e cC + d D ∆ = + − − G cGC dGD aGA bGB According to the law of mass action, the equilibrium As we have noted previously, the free energy change constant, K, can be expressed as for systems at equilibrium is zero, and hence equation ( )c ( )d (3.17) becomes = p’ C p’ D K (3.15) a b (p’ A) (p’ B) (p’ )c (p’ )d ∆G =−RT ln C D (3.18) where p, terms represent the partial pressures of the a b (p’ A) (p’ B) components of the reaction at equilibrium. Substituting from equation (3.15) gives The relationship between the free energy of a perfect gas and its partial pressure is given by ∆G =−RT ln K (3.19) ∆ = − = G G G RT ln p (3.16) Substituting equation (3.19) into equation (3.17) gives where G is the free energy of 1 mole of gas at a c d (p ) (p ) pressure of 1 bar. ∆G=−RT ln K + RT ln C D (3.20) a b Applying equation (3.16) to each component of the (pA) (pB) reaction gives Equation (3.20) gives the change in free energy when a = + moles of A at a partial pressure p and b moles of B at aGA a(GA RT ln pA) A = + a partial pressure pB react together to yield c moles of C bGB b(GB RT ln pB) at a partial pressure p and d moles of D at a partial C pressure p . For such a reaction to occur spontaneously, etc. D the free energy change must be negative, and hence As equation (3.20) provides a means of predicting the ∆ =∑ −∑ ease of reaction for selected partial pressures (or con- G Gprod Greact centrations) of the reactants. so (p )c (p )d ∆G= ∆G + RT ln C D (3.17) a b (pA) (p B)

section 3.3.1) of the components rather than Plots of log K against 1T should be linear with the partial pressures. ∆G values can readily be a slope of 0∆H2.303R, from which ∆H may calculated from the tabulated data and hence be calculated. equation (3.19) is important because it pro- Equations (3.19) and (3.24) are fundamental vides a method of calculating the equilibrium equations which find many applications in constants without resort to experimentation. the broad area of the pharmaceutical sciences: A useful expression for the temperature for example, in the determination of equi- dependence of the equilibrium constant is the librium constants in chemical reactions and van’t Hoff equation (equation 3.23), which may for micelle formation; in the treatment of be derived as outlined in Box 3.3. A more stability data for some solid dosage forms general form of this equation is (see section 4.4.3); and for investigations of −∆H drug–receptor binding. log K = + constant (3.24) 2.303RT

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62 Chapter 3 • Physicochemical properties of drugs in solution

activity and concentration. The ratio of the Box 3.3 Derivation of the van’t Hoff equation activity to the concentration is called the activity coefficient, γ; that is, From equation (3.19), γ = activity −∆G (3.25) = ln K concentration RT Depending on the units used to express Since concentration we can have either a molal ∆G = ∆H − T ∆S γ activity coefficient, m, a molar activity co- then at a temperature T γ 1 efficient, c, or, if mole fractions are used, a −∆ −∆ ∆ rational activity coefficient, γ . = G = H + S x ln K1 (3.21) RT1 RT1 R In order to be able to express the activity of a particular component numerically, it is and at temperature T2 −∆ −∆ ∆ necessary to define a reference state in which = G = H + S ln K2 (3.22) the activity is arbitrarily unity. The activity of RT RT R 2 2 a particular component is then the ratio of its If we assume that the standard enthalpy change ∆H value in a given solution to that in the refer- and the standard entropy change ∆S are independent ence state. For the solvent, the reference state of temperature, then subtracting equation (3.21) from is invariably taken to be the pure liquid and, if equation (3.22) gives this is at a pressure of 1 atmosphere and at a ∆ − =− H 1 − 1 definite temperature, it is also the standard ln K2 ln K1 R T2 T1 state. Since the mole fraction as well as the γ or activity is unity: x # 1. Several choices are available in defining the K ∆H (T − T ) log 2 = 2 1 (3.23) standard state of the solute. If the solute is a K 2.303R T T 1 1 2 liquid which is miscible with the solvent (as, Equation (3.23), which is often referred to as the van’t for example, in a benzene–toluene mixture), Hoff equation, is useful for the prediction of the equilib- then the standard state is again the pure rium constant K at a temperature T from its value K at 2 2 1 liquid. Several different standard states have another temperature T . 1 been used for solutions of solutes of limited solubility. In developing a relationship between drug activity and thermodynamic activity, the pure substance has been used as 3.3 Activity and chemical potential the standard state. The activity of the drug in solution was then taken to be the ratio of its 3.3.1 Activity and standard states concentration to its saturation solubility. The use of a pure substance as the standard state is The term activity is used in the description of of course of limited value since a different the departure of the behaviour of a solution state is used for each compound. A more feasi- from ideality. In any real solution, inter- ble approach is to use the infinitely dilute actions occur between the components which solution of the compound as the reference reduce the effective concentration of the solu- state. Since the activity equals the concentra- tion. The activity is a way of describing this tion in such solutions, however, it is not equal effective concentration. In an ideal solution or to unity as it should be for a standard state. in a real solution at infinite dilution, there are This difficulty is overcome by defining the no interactions between components and the standard state as a hypothetical solution of activity equals the concentration. Nonideality unit concentration possessing, at the same in real solutions at higher concentrations time, the properties of an infinitely dilute causes a divergence between the values of solution. Some workers4 have chosen to

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Activity and chemical potential 63

define the standard state in terms of an alkane using a theoretical method based on the solvent rather than water; one advantage of Debye–Hückel theory. In this theory each ion this solvent is the absence of specific solute– is considered to be surrounded by an ‘atmos- solvent interactions in the reference state phere’ in which there is a slight excess of ions which would be highly sensitive to molecular of opposite charge. The electrostatic energy structure. due to this effect is related to the chemical potential of the ion to give a limiting expres- sion for dilute solutions 3.3.2 Activity of ionised drugs − log γ& = z+z−A√I (3.36)

A large proportion of the drugs that are where z! and z0 are the valencies of the ions, A administered in aqueous solution are salts is a constant whose value is determined by the which, on dissociation, behave as electrolytes. dielectric constant of the solvent and the tem- Simple salts such as ephedrine hydrochloride perature (A # 0.509 in water at 298 K), and I is (C6 H5 CH(OH)CH(NHCH3)CH3 HCl) are 1 : 1 the total ionic strength defined by (or uni-univalent) electrolytes; that is, on 1 2 1 2 2 ... I = ∑(mz ) = (m z + m z + ) (3.37) dissociation each mole yields one cation, 2 2 1 1 2 2 ! C6 H5 CH(OH)CH(N H2 CH3)CH3, and one where the summation is continued over all the anion, Cl0. Other salts are more complex in different species in solution. It can readily be their ionisation behaviour; for example, shown from equation (3.37) that for a 1 : 1 ephedrine sulfate is a 1 : 2 electrolyte, each electrolyte the ionic strength is equal to its mole giving two moles of the cation and one molality; for a 1 : 2 electrolyte I # 3m; and for a 2− # mole of SO4 ions. 2:2 electrolyte, I 4m. The activity of each ion is the product of its The Debye–Hückel expression as given by activity coefficient and its concentration, that equation (3.36) is valid only in dilute solution is (I ` 0.02 mol kg01). At higher concentrations a modified expression has been proposed: a+ = γ+m+anda− = γ− m− −Az+z−√I The anion and cation may each have a log γ = (3.38) & + β√ different ionic activity in solution and it is not 1 ai I possible to determine individual ionic acti- where ai is the mean distance of approach of vities experimentally. It is therefore necessary the ions or the mean effective ionic diameter, to use combined terms, for example the com- and β is a constant whose value depends on bined activity term is the mean ionic activity, the solvent and temperature. As an approxi- a&. Similarly, we have the mean ion activity mation, the product a β may be taken to be γ i coefficient, &, and the mean ionic molality, m&. unity, thus simplifying the equation. Equation The relationship between the mean ionic (3.38) is valid for I less than 0.1 mol kg01 parameters is then

a& γ& = m& EXAMPLE 3.2 Calculation of mean ionic activ- ity coefficient More details of these combined terms are given in Box 3.4. Calculate: (a) the mean ionic activity coeffi- Values of the mean ion activity coefficient cient and the mean ionic activity of a may be determined experimentally using 0.002 mol kg01 aqueous solution of ephedrine several methods, including electromotive sulfate; (b) the mean ionic activity coefficient force measurement, solubility determinations of an aqueous solution containing 0.002 and colligative properties. It is possible, mol kg01 ephedrine sulfate and 0.01 mol kg01 γ however, to calculate & in very dilute solution sodium chloride. Both solutions are at 25°C.

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64 Chapter 3 • Physicochemical properties of drugs in solution

Answer The mean ionic activity may be calculated (a) Ephedrine sulfate is a 1 : 2 electrolyte and from equation (3.35): hence the ionic strength is given by equa- 2 13 a = 0.834 × 0.002 × (2 × 1) tion (3.37) as & = 0.00265 I = 1[(0.002 × 2 × 12) + (0.002 × 22)] 2 01 # = −1 (b) Ionic strength of 0.01 mol kg NaCl 0.006 mol kg 1 2 2 01 2 (0.01 " 1 ) ! (0.01 " 1 ) # 0.01 mol kg . From the Debye–Hückel equation (equa- Total ionic strength = 0.006 + 0.01 = 0.016 tion 3.36), −log γ& = 0.509 × 2 × √ 0.016 − log γ& = 0.509 × 1 × 2 × √0.006 log γ& =−0.1288 log γ& =−0.0789 γ& = 0.743 γ& = 0.834

Box 3.4 Mean ionic parameters

In general, we will consider a strong electrolyte which that is, mean ionic molality may be equated with the dissociates according to molality of the solution. z+ z− The activity of each ion is the product of its activity Cν+Aν− 2 ν+C + ν−A z! coefficient and its concentration where ν! is the number of cations, C , of valence z!, z0 a+ = γ+m+anda− = γ−m− and ν0 is the number of anions, A , of valence z0. The activity, a, of the electrolyte is then so that ν+ ν− ν a = a+ a− = a& (3.26) a+ a− γ+ = andγ− = where ν # ν! ! ν0. m+ m− In the simple case of a solution of the 1 : 1 electrolyte Expressed as the mean ionic parameters, we have sodium chloride, the activity will be γ = a& = + × − = 2 & (3.33) a aNa aCl a& m&

whereas for morphine sulfate, which is a 1 : 2 elec- Substituting for m& from equation (3.31) gives trolyte, a& 2 2− 3 γ = = + × = & (3.34) a amorph aSO4 a & ν+ ν− 1ν (m+ m− ) Similarly, we may also define a mean ion activity This equation applies in any solution, whether the ions γ coefficient, &, in terms of the individual ionic activity are added together, as a single salt, or separately as a γ γ coefficients ! and 0: mixture of salts. For a solution of a single salt of molality ν ν+ ν− γ& = γ+ γ− (3.27) m:

or m+ = ν+mandm− = ν−m ν+ ν− 1ν γ& = (γ+ γ− ) (3.28) Equation (3.34) reduces to

For a 1 : 1 electrolyte equation (3.28) reduces to a& γ& = (3.35) 12 νν+ νν− 1ν γ& = (γ+γ−) (3.29) m( + − ) ν # ν # Finally, we define a mean ionic molality, m&, as For example, for morphine sulfate, ! 2, 0 1, and ν ν+ ν− thus m& = m+ m− (3.30) a& a& or γ& = = 2 × 13 13 ν+ ν− 1ν (2 1) m 4 m m& = (m+ m − ) (3.31) For a 1 : 1 electrolyte, equation (3.31) reduces to 12 m+ = (m+m−) = m (3.32)

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3.3.3 Solvent activity activity of the solute from measurements of vapour pressure. Although the phrase ‘activity of a solution’ usually refers to the activity of the solute in Water activity and bacterial growth the solution as in the preceding section, we also can refer to the activity of the solvent. When the aqueous solution in the environ-

Experimentally, solvent activity a1 may be ment of a microorganism is concentrated by determined as the ratio of the vapour pressure the addition of solutes such as sucrose, the

p1 of the solvent in a solution to that of the consequences for microbial growth result pure solvent p1 , that is mainly from the change in water activity aw. Every microorganism has a limiting aw below p1 a = = γ x (3.39) which it will not grow. The minimum a levels 1 1 1 w p1 for growth of human bacterial pathogens such γ as streptococci, Klebsiella, Escherichia coli, where 1 is the solvent activity coefficient and Corynebacterium, Clostridium perfringens and x1 is the mole fraction of solvent. 5 The relationship between the activities of other clostridia, and Pseudomonas is 0.91. the components of the solution is expressed Staphylococcus aureus can proliferate at an aw as by the Gibbs–Duhem equation low as 0.86. Figure 3.2 shows the influence of aw, adjusted by the addition of sucrose, on the x d ln a + x d ln a = 0 (3.40) 1 ( 1) 2 ( 2) growth rate of this microorganism at 35°C and which provides a way of determining the pH 7.0. The control medium, with a water

10

aw = 0.993

9

aw = 0.885

8

7

aw = 0.867 6

5 Log [number of cells/ml]

4 aw = 0.858

3

2

0 24 48 72 96 120 144 168 192 216 Time (h)

Figure 3.2 Staphylococcal growth at 35°C in medium alone (aw # 0.993) and in media with aw values lowered by additional sucrose. Reproduced from J. Chirife, G. Scarmato and C. Herszage, Lancet, i, 560 (1982) with permission.

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66 Chapter 3 • Physicochemical properties of drugs in solution

activity value of aw # 0.993, supported rapid 1000 growth of the test organism. Reduction of aw of the medium by addition of sucrose progres- sively increased generation times and lag Controls periods and lowered the peak cell counts. Complete growth inhibition was achieved at 100 an aw of 0.858 (195 g sucrose per 100 g water) with cell numbers declining slowly through- out the incubation period. The results reported in this study explain why the old remedy of treating infected 10 wounds with sugar, honey or molasses is successful. When the wound is filled with sugar, the sugar dissolves in the tissue water, 50% paste with serum creating an environment of low aw, which CFU/ml as % of initial inoculum inhibits bacterial growth. However, the differ- ence in water activity between the tissue and 1 75% paste with serum the concentrated sugar solution causes migra- tion of water out of the tissue, hence diluting

the sugar and raising aw. Further sugar must 100% paste then be added to the wound to maintain growth inhibition. Sugar may be applied as a 0.1 paste with a consistency appropriate to the 0 40 80 120 wound characteristics; thick sugar paste is suit- Time (min) able for cavities with wide openings, a thinner Figure 3.3 The effects of sucrose pastes diluted with serum paste with the consistency of thin honey on the colony-forming ability of P. mirabilis. being more suitable for instillation into Reproduced from reference 6 with permission. cavities with small openings. An in vitro study has been reported6 of the than 0.86 (the critical level for inhibition of efficacy of such pastes, and also of those pre- growth of S. aureus) for more than 3 hours pared using xylose as an alternative to sucrose, after packing of the wound, nevertheless clini- in inhibiting the growth of bacteria com- cal experience had shown that twice-daily monly present in infected wounds. Poly- dressing was adequate to remove infected ethylene glycol was added to the pastes as a slough from dirty wounds within a few days. lubricant and hydrogen peroxide was included in the formulation as a preservative. To simu- late the dilution that the pastes invariably 3.3.4 Chemical potential experience as a result of fluid being drawn into the wound, serum was added to the formula- Properties such as volume, enthalpy, free tions in varying amounts. Figure 3.3 illustrates energy and entropy, which depend on the the effects of these sucrose pastes on the quantity of substance, are called extensive colony-forming ability of Proteus mirabilis and properties. In contrast, properties such as tem- shows the reduction in efficiency of the pastes perature, density and refractive index, which as a result of dilution and the consequent are independent of the amount of material, increase of their water activity (see Fig. 3.4). It are referred to as intensive properties. The is clear that P. mirabilis was susceptible to the quantity denoting the rate of increase in the antibacterial activity of the pastes, even when magnitude of an extensive property with they were diluted by 50%. It was reported that increase in the number of moles of a substance

although aw may not be maintained at less added to the system at constant temperature

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Activity and chemical potential 67

1.0 Partial molar quantities are of importance in the consideration of open systems, that is those involving transference of matter as well

0.9 as energy. For an open system involving two components ∂G ∂G dG = dT + dP 0.8 ∂ ∂ T P P, n1, n2 T, n1, n2

w ∂ ∂ a G G + dn + dn (3.43) ∂ 1 ∂ 2 n1 n2 0.7 T, P, n2 T, P, n1 At constant temperature and pressure equa- tion (3.43) reduces to 0.6 = µ + µ dG 1 dn1 2 dn2 (3.44) ∴ =∫ = µ + µ G dG 1 n1 2 n2 (3.45) 0.5 The chemical potential therefore represents 0 20 40 60 80 100 the contribution per mole of each component Percentage paste in serum to the total free energy. It is the effective free (v/v) energy per mole of each component in the mixture and is always less than the free energy Figure 3.4 Effects on aw of adding xylose (solid symbols) and sucrose paste (open symbols) to serum. of the pure substance. Reproduced from reference 6 with permission. It can readily be shown (see Box 3.5) that the chemical potential of a component in a and pressure is termed a partial molar quantity. two-phase system (for example, oil and water), Such quantities are distinguished by a bar at equilibrium at a fixed temperature and pres- above the symbol for the particular property. sure, is identical in both phases. Because of the For example, need for equality of chemical potential at ∂V equilibrium, a substance in a system which is =VK (3.41) ∂ 2 not at equilibrium will have a tendency to n2 T, P, n1 diffuse spontaneously from a phase in which Note the use of the symbol ∂ to denote a partial it has a high chemical potential to another in change which, in this case, occurs under con- which it has a low chemical potential. In this ditions of constant temperature, pressure and respect the chemical potential resembles elec- number of moles of solvent (denoted by the trical potential; hence its name is an apt subscripts outside the brackets). description of its nature. In practical terms the partial molar volume, VK , represents the change in the total volume of Chemical potential of a component in solution a large amount of solution when one addi- tional mole of solute is added – it is the effective Where the component of the solution is a non- volume of 1 mole of solute in solution. electrolyte, its chemical potential in dilute solu- Of particular interest is the partial molar free tion at a molality m, can be calculated from energy, IG , which is also referred to as the µ = µ + RT ln m chemical potential, µ, and is defined for compo- 2 nent 2 in a binary system by where ∂ µ = µ + − G 2 RT ln M1 RT ln 1000 = IG = µ (3.42) ∂ 2 2 n2 T, P, n1 and M1 # molecular weight of the solvent.

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68 Chapter 3 • Physicochemical properties of drugs in solution

Box 3.5 Chemical potential in two-phase systems

Consider a system of two phases, a and b, in equi- A decrease of dn moles of component in phase a leads librium at constant temperature and pressure. If a small to an increase of exactly dn moles of this component in quantity of substance is transferred from phase a to phase b, that is phase b, then, because the overall free energy change =− dna dnb (3.48) is zero, we have Substitution of equation (3.48) into equation (3.47) + = dGa dG b 0 (3.46) leads to the result

where dGa and dGb are the free energy changes µ = µ a b (3.49) accompanying the transfer of material for each phase. In general, the chemical potential of a component is From equation (3.44), identical in all the phases of a system at equilibrium at a = µ = µ dGa a dnaanddG b b dnb fixed temperature and pressure. and thus µ + µ = a dna b dnb 0 (3.47)

Box 3.6 Chemical potential of a component in solution

Nonelectrolytes solution, is equal to the sum of the chemical potentials In dilute solutions of nonvolatile solutes, Raoult’s law of each of the component ions. Thus (see section 2.3.1) can usually be assumed to be µ+ = µ+ + RT ln a+ (3.54) obeyed and the chemical potential of the solute is given and by equation (3.50): µ = µ + µ = µ + − − RT ln a− (3.55) 2 2 RT ln x2 (3.50) and therefore It is usually more convenient to express solute concen- µ = µ + tration as molality, m, rather than mole fraction, using 2 2 RT ln a (3.56) where µ is the sum of the chemical potentials of the mM1 2 x = 2 1000 ions, each in their respective standard state, i.e. µ = ν µ + ν µ 2 + + − − where M1 # molecular weight of the solvent. Thus ν ν µ = µ + where ! and 0 are the number of cations and anions, 2 RT ln m (3.51) respectively, and a is the activity of the electrolyte as where given in section 3.3.2. µ = µ + − 2 RT ln M1 RT ln 1000 For example, for a 1 : 1 electrolyte, from equation At higher concentrations, the solution generally exhibits (3.26), 2 significant deviations from Raoult’s law and mole frac- a = a& tion must be replaced by activity: Therefore µ = µ + 2 2 RT ln a2 (3.52) µ = µ + 2 2 2RT ln a& or From equation (3.33), µ = µ + γ + 2 2 RT ln 2 RT ln x2 (3.53) a& = mγ& ∴ µ = µ + γ Electrolytes 2 2 2RT ln m & The chemical potential of a strong electrolyte, which may be assumed to be completely dissociated in

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Osmotic properties of drug solutions 69

In the case of strong electrolytes, the chemical is the van’t Hoff equation: potential is the sum of the chemical potentials ΠV = n RT (3.57) of the ions. For the simple case of a 1 : 1 elec- 2 trolyte, the chemical potential is given by On application of the van’t Hoff equation to the drug molecules in solution, consideration µ = µ + 2RT ln mγ 2 2 & must be made of any ionisation of the mol- The derivations of these equations are given in ecules, since osmotic pressure, being a colli- Box 3.6. gative property, will be dependent on the total number of particles in solution (including the free counterions). To allow for what was at the time considered to be anomalous behaviour of 3.4 Osmotic properties of drug electrolyte solutions, van’t Hoff introduced a solutions correction factor, i. The value of this factor approaches a number equal to that of the ν A nonvolatile solute added to a solvent affects number of ions, , into which each molecule not only the magnitude of the vapour pressure dissociates as the solution is progressively ν above the solvent but also the freezing point diluted. The ratio i is termed the practical φ and the boiling point to an extent that is osmotic coefficient, . proportional to the relative number of solute For nonideal solutions, the activity and molecules present, rather than to the weight osmotic pressure are related by the expression

concentration of the solute. Properties that are −νmM = 1 φ dependent on the number of molecules in ln a1 (3.58) 1000 solution in this way are referred to as colligative properties, and the most important of such where M1 is the molecular weight of the properties from a pharmaceutical viewpoint is solvent and m is the molality of the solution. the osmotic pressure. The relationship between the osmotic pressure and the osmotic coefficient is thus

RT νmM Π = 1 φ (3.59) 3.4.1 Osmotic pressure K V1 1000 K Whenever a solution is separated from a where V 1 is the partial molal volume of the solvent by a membrane that is permeable only solvent. to solvent molecules (referred to as a semi- permeable membrane), there is a passage of solvent across the membrane into the solution. 3.4.2 Osmolality and osmolarity This is the phenomenon of osmosis. If the solu- tion is totally confined by a semipermeable The experimentally derived osmotic pressure ξ membrane and immersed in the solvent, then is frequently expressed as the osmolality m, a pressure differential develops across the which is the mass of solute which, when dis- membrane, which is referred to as the osmotic solved in 1 kg of water, will exert an osmotic pressure. Solvent passes through the membrane pressure, Π,, equal to that exerted by 1 mole of because of the inequality of the chemical an ideal unionised substance dissolved in 1 kg potentials on either side of the membrane. of water. The unit of osmolality is the osmole Since the chemical potential of a solvent mol- (abbreviated as osmol), which is the amount ecule in solution is less than that in pure of substance that dissociates in solution to solvent, solvent will spontaneously enter the form one mole of osmotically active particles, solution until this inequality is removed. The thus 1 mole of glucose (not ionised) forms 1 equation which relates the osmotic pressure of osmole of solute, whereas 1 mole of NaCl the solution, Π, to the solution concentration forms 2 osmoles (1 mole of Na! and 1 mole of

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70 Chapter 3 • Physicochemical properties of drugs in solution

Cl0). In practical terms, this means that a 1 Table 3.1 Tonicities (osmolalities) of intravenous molal solution of NaCl will have (approxi- fluids mately) twice the osmolality (osmotic pres- sure) as a 1 molal solution of glucose. Solution Tonicity ξ ΠΠ 01 According to the definition, m # ,. The (mosmol kg ) value of Π, may be obtained from equation (3.59) by noting that for an ideal unionised Vamin 9 0700 Vamin 9 Glucose 1350 substance ν # φ # 1, and since m is also unity, Vamin14 1145 equation (3.59) becomes Vamin 14 Electrolyte-free 0810 Vamin 18 Electrolyte-free 1130 RT M Π’ = 1 Vaminolact 0510 K V1 1000 Vitrimix KV 1130 Thus Intralipid 10% Novum 0300 Intralipid 20% 0350 ξ = ν φ m m (3.60) Intralipid 30% 0310 Intrafusin 22 1400 Hyperamine 30 1450 Gelofusine 00279a EXAMPLE 3.3 Calculation of osmolality Hyperamine 30 1450 a A 0.90% ww solution of sodium chloride Lipofundin MCT LCT 10% 00345 Lipofundin MCTLCT 20% 00380a # (mol. wt. 58.5) has an osmotic coefficient of Nutriflex 32 01400a 0.928. Calculate the osmolality of the solution. Nutriflex 48 02300a Nutriflex 70 02100a Answer Sodium Bicarbonate Intravenous Infusion BP Osmolality is given by equation (3.60) as 8.4% wv 02000a 4.2% wv 01000a ξ = ν φ m m a 03 so Osmolarity (mosmol dm ). 9.0 ξ = × × = −1 m 2 0.928 286 mosmol kg logical membranes, notably the red blood cell 58.5 membrane, behave in a manner similar to that of semipermeable membranes. Consequently, Pharmaceutical labelling regulations some- when red blood cells are immersed in a times require a statement of the osmolarity; solution of greater osmotic pressure than that for example, the USP 27 requires that sodium of their contents, they shrink as water passes chloride injection should be labelled in this out of the cells in an attempt to reduce the way. Osmolarity is defined as the mass of solute chemical potential gradient across the cell which, when dissolved in 1 litre of solution, membrane. Conversely, on placing the cells in will exert an osmotic pressure equal to that an aqueous environment of lower osmotic exerted by 1 mole of an ideal unionised sub- pressure, the cells swell as water enters and stance dissolved in 1 litre of solution. The rela- eventually lysis may occur. It is an important tionship between osmolality and osmolarity consideration, therefore, to ensure that the has been discussed by Streng et al.7 effective osmotic pressure of a solution for Table 3.1 lists the osmolalities of commonly injection is approximately the same as that of used intravenous fluids. blood serum. This effective osmotic pressure, which is termed the tonicity, is not always identical to the osmolality because it is con- 3.4.3 Clinical relevance of osmotic effects cerned only with those solutes in solution that can exert an effect on the passage of water Osmotic effects are particularly important through the biological membrane. Solutions from a physiological viewpoint since bio- that have the same tonicity as blood serum are

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Osmotic properties of drug solutions 71

said to be isotonic with blood. Solutions with a NEC has been reported8 among premature higher tonicity are hypertonic and those with a infants fed undiluted calcium lactate than lower tonicity are termed hypotonic solutions. among those fed no supplemental calcium or Similarly, in order to avoid discomfort on calcium lactate eluted with water or formula. administration of solutions to the delicate White and Harkavy9 have discussed a similar membranes of the body, such as the eyes, case of the development of NEC following these solutions are made isotonic with the medication with calcium glubionate elixir. relevant tissues. These authors have measured osmolalities of The osmotic pressures of many of the several medications by freezing point depres- products of Table 3.1 are in excess of that of sion and compared these with the osmolalities 0 plasma (291 mosmol dm 3). It is generally of analogous intravenous (i.v.) preparations recommended that any fluid with an osmotic (see Table 3.2). Except in the case of digoxin, 0 pressure above 550 mosmol dm 3 should not the osmolalities of the i.v. preparations were be infused rapidly as this would increase the very much lower than those of the corres- incidence of venous damage. The rapid ponding oral preparations despite the fact that infusion of marginally hypertonic solutions the i.v. preparations contained at least as 0 (in the range 300–500 mosmol dm 3) would much drug per millilitre as did the oral forms. appear to be clinically practicable; the higher This striking difference may be attributed to the osmotic pressure of the solution within the additives, such as ethyl alcohol, sorbitol this range, the slower should be its rate of and propylene glycol, which make a large infusion to avoid damage. Patients with cen- contribution to the osmolalities of the oral trally inserted lines are not normally affected preparations. The vehicle for the i.v. digoxin by limits on tonicity as infusion is normally consists of 40% propylene glycol and 10% slow and dilution is rapid. ethyl alcohol with calculated osmolalities of Certain oral medications commonly used in 5260 and 2174 mosmol kg01 respectively, thus the intensive care of premature infants have explaining the unusually high osmolality of very high osmolalities. The high tonicity of this i.v. preparation. These authors have rec- enteral feedings has been implicated as a cause ommended that extreme caution should be of necrotising enterocolitis (NEC). A higher exercised in the administration of these oral frequency of gastrointestinal illness including preparations and perhaps any medication in a

Table 3.2 Measured and calculated osmolalities of drugsa

Drug (route) Concentration Mean measured Calculated available of drug osmolality milliosmoles in (mosmol kg01)1kg of drug preparationb

Theophylline elixir (oral) 80 mg15 cm3 p3000 4980 Aminophylline (i.v.) 25 mg cm03 00116 0200 Calcium glubionate (oral) 115 mg5cm3 p3000 2270 Calcium gluceptate (i.v.) 90 mg5cm3 00507 0950 Digoxin elixir 25 mg dm03 p3000 4420 Digoxin (i.v.) 100 mg dm03 p3000 9620 Dexamethasone elixir (oral) 0.5 mg5cm3 p3000 3980 Dexamethasone sodium phosphate (i.v.) 4 mg cm03 00284 0312

a Reproduced from reference 9. b This would be the osmolality of the drug if the activity coefficient were equal to 1 in the full-strength preparation. The osmolalities of serial dilutions of the drug were plotted against the concentrations of the solution, and a least-squares regression line was drawn. The value for the osmolality of the full-strength solution was then estimated from the line. This is the ‘calculated available milliosmoles’.

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72 Chapter 3 • Physicochemical properties of drugs in solution

syrup or elixir form when the infant is at risk osmolarity within the physiological range from necrotising enterocolitis. In some cases (209–305 mosmol dm03) have only a small the osmolality of the elixir is so high that even effect on the liquidgas partition coefficient, mixing with infant formula does not reduce changes in the serum osmolarity and the the osmolality to a tolerable level. For example, concentration of serum constituents at the when a clinically appropriate dose of dexa- extremes of the physiological range may sig- methasone elixir was mixed in volumes of nificantly decrease the liquidgas partition formula appropriate for a single feeding for a coefficient. For example, the bloodgas parti- 1500 g infant, the osmolalities of the mixes tion coefficient of isoflurane decreases signifi- increased by at least 300% compared to cantly after an infusion of mannitol. This may formula alone (see Table 3.3). be attributed to both a transient increase in the osmolarity of the blood and a more pro- Volatile anaesthetics longed decrease in the concentration of serum The aqueous solubilities of several volatile constituents caused by the influx of water due anaesthetics can be related to the osmolarity to the osmotic gradient. of the solution.10 The inverse relationship between solubility (expressed as the liquidgas Rehydration solutions partition coefficient) of those anaesthetics and An interesting application of the osmotic the osmolarity is shown in Table 3.4. effect has been in the design of rehydration These findings have practical applications solutions. During the day the body moves for the clinician. Although changes in serum many litres of fluid from the blood into the

Table 3.3 Osmolalities of drug-infant formula mixturesa

Drug (dose) Volume of drug (cm3) Mean measured osmolality ! volume of formula (cm3) (mosmol kg01)

Infant formula – 0292 Theophylline elixir, 1 mg kg01 00.3 ! 15 0392 00.3 ! 30 0339 Calcium glubionate syrup, 0.5 mmol kg01 00.5 ! 15 0378 00.5 ! 30 0330 Digoxin elixir, 5 µgkg01 0.15 ! 15 0347 0.15 ! 30 0322 Dexamethasone elixir, 0.25 mg kg01 03.8 ! 15 1149 03.8 ! 30 0791

a Reproduced from reference 9.

Table 3.4 Liquidgas partition coefficients of anaesthetics in four aqueous solutions at 37°Ca

Solution Osmolarity Partition coefficient (mosmol dm03) Isoflurane Enflurane Halothane Methoxyflurane

Distilled H2 O 0000 0.626 & 0.050 0.754 & 0.060 0.859 & 0.02 4.33 & 0.50 Normal saline 0308 0.590 & 0.010 0.713 & 0.010 0.825 & 0.02 4.22 & 0.30 Isotonic heparin 0308 0.593 & 0.010 0.715 & 0.0100 – 4.08 & 0.22 (1000 U cm03) Mannitol (20%) 1098 0.476 & 0.023 0.575 & 0.024 0.747 & 0.03 3.38 & 0.14

a Reproduced from reference 10.

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Osmotic properties of drug solutions 73

intestine and back again. The inflow of water enhance the co-transport system. If this is into the intestine, which aids the breakdown done, however, the osmolarity of the glucose of food, is an osmotic effect arising from the will become greater than that of normal secretion of Cl0 ions by the crypt cells of the blood, and water would now flow from the intestinal lining (see section 9.2.2) into the blood to the intestine and so exacerbate the intestine. Nutrients from the food are taken up problem. An alternative is to substitute by the villus cells in the lining of the small starches for simple glucose in the ORT. When intestine. The villus cells also absorb Na! ions, these polymer molecules are broken down in which they pump out into the extracellular the intestinal lumen they release many hun- spaces, from where they return to the circu- dreds of glucose molecules, which are imme- lation. As a consequence of this flow of Na!, diately taken up by the co-transport system water and other ions follow by osmotic flow and removed from the lumen. The effect is and hence are also transferred to the blood. therefore as if a high concentration of glucose This normal functioning is disrupted by were administered, but because osmotic pres- diarrhoea-causing microorganisms which sure is a colligative property (dependent on the either increase the Cl0-secreting activity of the number of molecules rather than the mass of crypt cells or impair the absorption of Na! by substance), there is no associated problem of a the villus cells, or both. Consequently, the high osmolarity when starches are used. The fluid that is normally returned to the blood process is summarised in Fig. 3.5. A similar across the intestinal wall is lost in watery effect is achieved by the addition of proteins, stool. If untreated, diarrhoea can eventually since there is also a co-transport mechanism lead to a severe decline in the volume of the whereby amino acids (released on breakdown blood, the circulation may become danger- of the proteins in the intestine) and Na! ions ously slow, and death may result. are simultaneously taken up by the villus cells. This process of increasing water uptake from Oral rehydration therapy the intestine has an added appeal since the Treatment of dehydration by oral rehydration source of the starch and protein can be cereals, therapy (ORT) is based on the discovery that beans and rice, which are likely to be available the diarrhoea-causing organisms do not usually in the parts of the world where problems interfere with the carrier systems which bring arising from diarrhoea are most prevalent. sodium and glucose simultaneously into the Food-based ORT offers additional advantages: villus cells from the intestinal cavity. This ‘co- it can be made at home from low-cost ingre- transport’ system only operates when both dients and can be cooked, which kills the sodium and glucose are present. The principle pathogens in water. behind ORT is that if glucose is mixed into an electrolyte solution it activates the co-transport system, causing electrolyte and then water to 3.4.4 Preparation of isotonic solution pass through the intestinal wall and to enter the blood, so minimising the dehydration. Since osmotic pressure is not a readily measur- ORT requires administration to the patient able quantity, it is usual to make use of the of small volumes of fluid throughout the day relationship between the colligative properties (to prevent vomiting); it does not reduce the and to calculate the osmotic pressure from a duration or severity of the diarrhoea, it simply more easily measured property such as the replaces lost fluid and electrolytes. Let us freezing point depression. In so doing, how- examine, using the principles of the osmotic ever, it is important to realise that the red effect, two possible methods by which the blood cell membrane is not a perfect semi- process of fluid uptake from the intestine permeable membrane and allows through might be speeded up. It might seem reason- small molecules such as urea and ammonium able to suggest that more glucose should be chloride. Therefore, although the quantity of added to the formulation in an attempt to each substance required for an isotonic

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74 Chapter 3 • Physicochemical properties of drugs in solution

ORT solution in Dehydrated blood Standard ORT lumen of intestine Osmotic flow of (osmolarity equals the normal osmolarity of blood) water and ions Solute Effect: Co-transport of glucose and sodium induces a bloodward osmotic flow of water, which drags along additional ions. ORT exactly replaces water, sodium and other ions lost from the blood but does not reduce the extent or duration of diarrhoea. Glucose-induced Solute sodium transport Cation

Undesirable If extra glucose is added osmotic flow (high osmolarity)

Effect: Solution is unacceptable because osmotic flow yields a net loss of water and ions from the blood – an osmotic penalty. Dehydration and risk of death increase.

Glucose-induced sodium transport

Food-based ORT Starch Osmotic flow (low osmolarity)

Effect: Each polymer has the same osmotic effect as a single glucose or amino acid molecule but markedly enhances nutrient-induced sodium transport when the polymer is broken apart at the villus cell surface. (Rapid uptake at the surface avoids an osmotic penalty.) Water and ions are Glucose-induced returned to the blood quickly, and or amino acid- less of both are lost in the stool. induced sodium The extent and duration of diarrhoea Protein are reduced. transport Figure 3.5 How osmosis affects the performance of solutions used in oral rehydration therapy (ORT).

∆ solution may be calculated from freezing depression, Tf , of 0.52°C. One has therefore point depression values, these solutions may to adjust the freezing point of the drug cause cell lysis when administered. solution to this value to give an isotonic solu- It has been shown that a solution which is tion. Freezing point depressions for a series of isotonic with blood has a freezing point compounds are given in reference texts1, 11

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Ionisation of drugs in solution 75

and it is a simple matter to calculate the con- Therefore, the weight of sodium chloride to be centration required for isotonicity from these added to 50 cm3 of solution is 0.395 g. values. For example, a 1% NaCl solution has a freezing point depression of 0.576°C. The percentage concentration of NaCl required to make isotonic saline solution is therefore 3.5 Ionisation of drugs in solution (0.520.576) " 1.0 # 0.90% wv. With a solution of a drug, it is not of course Many drugs are either weak organic acids (for possible to alter the drug concentration in this example, acetylsalicylic acid [aspirin]) or weak manner, and an adjusting substance must be organic bases (for example, procaine), or their added to achieve isotonicity. The quantity of salts (for example, ephedrine hydrochloride). adjusting substance can be calculated as shown The degree to which these drugs are ionised in in Box 3.7 solution is highly dependent on the pH. The exceptions to this general statement are the nonelectrolytes, such as the steroids, and the Box 3.7 Preparation of isotonic solutions quaternary ammonium compounds, which are completely ionised at all pH values and in 3 If the drug concentration is x g per 100 cm solution, this respect behave as strong electrolytes. The then extent of ionisation of a drug has an impor- ∆ = × ∆ = Tf for drug solution x ( Tf of 1% drug solution) a tant effect on its absorption, distribution and Similarly, if w is the weight in grams of adjusting elimination and there are many examples of substance to be added to 100 cm3 of drug solution the alteration of pH to change these prop- to achieve isotonicity, then erties. The pH of urine may be adjusted (for ∆ example by administration of ammonium Tf for adjusting solution = × ∆ chloride or sodium bicarbonate) in cases of w ( Tf of 1% adjusting substance) overdosing with amfetamines, barbiturates, = w × b narcotics and salicylates, to ensure that these For an isotonic solution, drugs are completely ionised and hence a + (w × b) = 0.52 readily excreted. Conversely, the pH of the 0.52 − a ∴w = (3.61) urine may be altered to prevent ionisation of a b drug in cases where reabsorption is required for therapeutic reasons. Sulphonamide crystal- luria may also be avoided by making the urine EXAMPLE 3.4 Isotonic solutions alkaline. An understanding of the relationship between pH and drug ionisation is of use in Calculate the amount of sodium chloride the prediction of the causes of precipitation in which should be added to 50 cm3 of a admixtures, in the calculation of the solubility 0.5% wv solution of lidocaine hydrochloride of drugs and in the attainment of optimum to make a solution isotonic with blood serum. bioavailability by maintaining a certain ratio of ionised to unionised drug. Table 3.5 shows Answer the nominal pH values of some body fluids From reference lists, the values of b for sodium and sites, which are useful in the prediction of chloride and lidocaine hydrochloride are the percentage ionisation of drugs in vivo. 0.576°C and 0.130°C, respectively. From equation (3.61) we have a = 0.5 × 0.130 = 0.065 3.5.1 Dissociation of weakly acidic and basic drugs and their salts Therefore, 0.52 − 0.065 According to the Lowry–Brønsted theory of w = = 0.790 g 0.576 acids and bases, an acid is a substance which Physicochemical Principles of Pharmacy, 4th edition (ISBN: 0 85369 608 X) © Pharmaceutical Press 2006 09_1439PP_CH03 17/10/05 10:47 pm Page 76

76 Chapter 3 • Physicochemical properties of drugs in solution

acetylsalicylate ion acts as a base, because it Table 3.5 Nominal pH values of some body fluids a accepts a proton to yield an acid. An acid and and sites base represented by such an equilibrium is Site Nominal pH said to be a conjugate acid–base pair. Scheme 3.1 is not a realistic expression, Aqueous humour 7.21 however, since protons are too reactive to Blood, arterial 7.40 exist independently and are rapidly taken up Blood, venous 7.39 by the solvent. The proton-accepting entity, Blood, maternal umbilical 7.25 by the Lowry–Brønsted definition, is a base, Cerebrospinal fluid 7.35 Duodenum 5.50 and the product formed when the proton has Faecesb 7.15 been accepted by the solvent is an acid. Thus a Ileum, distal 8.00 second acid–base equilibrium occurs when the Intestine, microsurface 5.30 solvent accepts the proton, and this may be Lacrimal fluid (tears) 7.40 represented by Milk, breast 7.00 c + + e + Muscle, skeletal 6.00 H2 O H H3 O Nasal secretions 6.00 Prostatic fluid 6.45 The overall equation on summing these equa- Saliva 6.40 tions is shown in Scheme 3.2, or, in general, Semen 7.20 + e − + + Stomach 1.50 HA H2 O A H3 O Sweat 5.40 Urine, female 5.80 By similar reasoning, the dissociation of Urine, male 5.70 benzocaine, a weak base, may be represented Vaginal secretions, premenopause 4.50 by the equilibrium Vaginal secretions, postmenopause 7.00 NH C H COOC H ! H O 2 6 5 2 5 2 a Reproduced from D. W. Newton and R. B. Kluza, Drug Intell. Clin. base 1 acid 2 Pharm., 12, 547 (1978). + b Value for normal soft, formed stools, hard stools tend to be more e 0 NH3 C6 H5 COOC2 H5 ! OH alkaline, whereas watery, unformed stools are acidic. acid 1 base 2 c Studies conducted intracellularly in the rat. or, in general, will donate a proton and a base is a substance + − B + H O e BH + OH which will accept a proton. Thus the disso- 2 ciation of acetylsalicylic acid, a weak acid, Comparison of the two general equations could be represented as in Scheme 3.1. In this shows that H2 O can act as either an acid or a equilibrium, acetylsalicylic acid acts as an base. Such solvents are called amphiprotic acid, because it donates a proton, and the solvents.

OOC CH3 OOC CH3 ϩ ϩ H

Ϫ COOH COO Scheme 3.1

OOC CH3 OOC CH3 ϩ ϩ H2O ϩ H3O Ϫ COOH COO Acid 1 Base 2 Base 1 Acid 2 Scheme 3.2

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Ionisation of drugs in solution 77

Salts of weak acids or bases are essentially The pKa and pKb values provide a convenient completely ionised in solution. For example, means of comparing the strengths of weak

ephedrine hydrochloride (salt of the weak base acids and bases. The lower the pKa, the ephedrine, and the strong acid HCl) exists in stronger the acid; the lower the pKb, the aqueous solution in the form of the conjugate stronger is the base. The pKa values of a series acid of the weak base, C6 H5 CH(OH)CH(CH3) of drugs are given in Table 3.6. pKa and pKb ! 0 N H2 CH3, together with its Cl counterions. values of conjugate acid–base pairs are linked In a similar manner, when sodium salicylate (salt of the weak acid salicylic acid, and the strong base NaOH) is dissolved in water, it ionises almost entirely into the conjugate base Box 3.8 The degree of ionisation of weak acids and 0 ! bases of salicylic acid, HOC6 H5 COO , and Na ions. The conjugate acids and bases formed in this way are, of course, subject to acid–base Weak acids equilibria described by the general equations Taking logarithms of the expression for the dissociation above. constant of a weak acid (equation 3.63) − + [A ] −log K =−log [H O ] − log a 3 [HA] 3.5.2 The effect of pH on the ionisation of which can be rearranged to weakly acidic or basic drugs and their [A−] salts pH = pK + log (3.70) a [HA] If the ionisation of a weak acid is represented Equation (3.70) may itself be rearranged to facilitate the as described above, we may express an equi- direct determination of the molar percentage ionisation librium constant as follows: as follows: − [ ] = [ ] ( − ) + × − HA A antilog pKa pH aH O aA = 3 Ka (3.62) Therefore, a HA − = [A ] × Assuming the activity coefficients approach percentage ionisation − 100 [HA] + [A ] unity in dilute solution, the activities may be 100 replaced by concentrations: percentage ionisation = (3.71) 1 + antilog (pK − pH) + − a [H O ] [A ] = 3 Ka (3.63) [HA] Weak bases An analogous series of equations for the percentage Ka is variously referred to as the ionisation constant, dissociation constant, or acidity con- ionisation of a weak base may be derived as follows. stant for the weak acid. The negative logarithm Taking logarithms of equation (3.65) and rearranging gives of K is referred to as pK , just as the negative a a + logarithm of the hydrogen ion concentration − [BH ] −log K =−log [OH ] − log is called the pH. Thus b [B] =− Therefore, pKa log Ka (3.64) [BH+] pH = pK − pK − log (3.72) Similarly, the dissociation constant or basicity w b [B] constant for a weak base is Rearranging to facilitate calculation of the percentage − + a − × a + [OH ] [BH ] ionisation leads to = OH BH Kb Q (3.65) a [B] 100 B percentage ionisation = 1 + antilog(pH − pK + pK ) and w b (3.73) =− pKb log Kb (3.66)

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78 Chapter 3 • Physicochemical properties of drugs in solution

a Table 3.6 pKa values of some medicinal compounds

Compound pKa Compound pKa

Acid Base Acid Base

Acebutolol – 9.2 Diamorphine – 7.6 Acetazolamide – 7.2, 9.0 Diazepam – 3.4 Acetylsalicylic acid 3.5 – Diclofenac 4.0 – Aciclovir – 2.3, 9.3 Diethylpropion – 8.7 Adrenaline 9.9 8.5 Diltiazem – 8.0 Adriamycin – 8.2 Diphenhydramine – 9.1 Allopurinol 9.4 (10.2)b – Disopyramide –– Alphaprodine – 8.7 Dithranol – 9.4 Alprenolol – 9.6 Doxepin – 8.0 Amikacin – 8.1 Doxorubicin – 8.2, 10.2 p-Aminobenzoic acid 4.9 2.4 Doxycycline 7.7 3.4, 9.3 Aminophylline – 5.0 Enalapril – 5.5 Amitriptyline – 9.4 Enoxacin 6.3 8.6 Amiodarone – 6.6 Ergometrine – 6.8 Amoxicillin 2.4, 7.4, 9.6 – Ergotamine – 6.4 Ampicillin 2.5 7.2 Erythromycin – 8.8 Apomorphine 8.9 7.0 Famotidine – 6.8 Atenolol – 9.6 Fenoprofen 4.5 Ascorbic acid 4.2, 11.6 – Flucloxacillin 2.7 – Atropine – 9.9 Flufenamic acid 3.9 – Azapropazone `1.8 6.6 Flumequine 6.5 – Azathioprine 8.2 – Fluorouracil 8.0, 13.0 – Benzylpenicillin 2.8 – Fluphenazine – 3.9, 8.1 Benzocaine – 2.8 Flurazepam 8.2 1.9 Bupivacaine – 8.1 Flurbiprofen 4.3 – Captopril 3.5 – Furosemide 3.9 – Cefadroxil 7.6 2.7 Glibenclamide 5.3 – Cefalexin 7.1 2.3 Guanethidine – 11.9 Cefaclor 7.2 2.7 Guanoxan – 12.3 Celiprolol – 9.7 Haloperidol – 8.3 Cetirizine 2.9 2.2, 8.0 Hexobarbital 8.3 – Chlorambucil 4.5 (4.9)b 2.5 Hydralazine – 0.5, 7.1 Chloramphenicol – 5.5 Ibuprofen 4.4 – Chlorcyclizine – 8.2 Imipramine – 9.5 Chlordiazepoxide – 4.8 Indometacin 4.5 – Chloroquine – 8.1, 9.9 Isoniazid 2.0, 3.9 – Chlorothiazide 6.5 9.5 Ketoprofen 4.0 – Chlorphenamine – 9.0 Labetalol 7.4 9.4 Chlorpromazine – 9.3 Levodopa 2.3, 9.7, 13.4 – Chlorpropamide – 4.9 Lidocaine – 7.94 (26°C), Chlorprothixene – 8.8 7.55 (36°C) Cimetidine – 6.8 Lincomycin – 7.5 Cinchocaine – 8.3 Maprotiline – 10.2 Clindamycin – 7.5 Meclofenamic acid 4.0 – Cocaine – 8.5 Metoprolol – 9.7 Codeine – 8.2 Methadone – 8.3 Cyclopentolate – 7.9 Methotrexate 3.8, 4.8 5.6 Daunomycin – 8.2 Metronidazole – 2.5 Desipramine – 10.2 Minocycline 7.8 2.8, 5.0, 9.5 Dextromethorphan – 8.3 continued

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Ionisation of drugs in solution 79

Table 3.6 (continued)

Compound pKa Compound pKa

Acid Base Acid Base

Minoxidil – 4.6 Prazocin – 6.5 Morphine 8.0 (phenol) 9.6 (amine) Procaine – 8.8 Nadolol – 9.7 Prochlorperazine – 3.7,8.1 Nafcillin 2.7 – Promazine – 9.4 Nalidixic acid 6.4 – Promethazine – 9.1 Nalorphine – 7.8 Propranolol – 9.5 Naloxone – 7.9 Quinidine – 4.2, 8.3 Naltrexone 9.5 8.1 Quinine – 4.2, 8.8 Naproxen 4.2 – Ranitidine – 2.7, 8.2 Nitrofurantoin – 7.2 Sotalol 8.3 9.8 Nitrazepam 10.8 3.2 (3.4)b Sulfadiazine 6.5 2.0 Norfloxacin 6.2 8.6 Sulfaguanidine 12.1 2.8 Nortriptyline – 9.7 Sulfamerazine 7.1 2.3 Novobiocin 4.3, 9.1 – Sulfathiazole 7.1 2.4 Ofloxacin 6.1 8.3 Tamoxifen – 8.9 Oxolinic acid 6.6 – Temazepam – 1.6 Oxprenolol – 9.5 Tenoxicam 1.1 5.3 Oxycodone – 8.9 Terfenadine – 9.5 Oxytetracycline 7.3 3.3, 9.1 Tetracaine – 8.4 Pentazocine – 8.8 Tetracycline 7.7 3.3, 9.5 Pethidine – 8.7 Theophylline 8.6 3.5 Phenazocine – 8.5 Thiopental 7.5 – Phenytoin 8.3 – Timolol – 9.2 (8.8)b Physostigmine – 2.0, 8.1 Tolbutamide 5.3 – Pilocarpine – 1.6, 7.1 Triflupromazine – 9.2 Pindolol – 8.8 Trimethoprin – 7.2 Piperazine – 5.6, 9.8 Valproate 5.0 – Piroxicam 2.3 – Verapamil – 8.8 Polymyxin B – 8.9 Warfarin 5.1 –

a For a more complete list see: D. W. Newton and R. B. Kluza, Drug Intell. Clin. Pharm., 12, 547 (1978); G. C. Raymond and J. L. Born, Drug Intell. Clin. Pharm., 20, 683 (1986); D. B. Jack, Handbook of Clinical Pharmacokinetic Data, Macmillan, London, 1992; The Pharmaceutical Codex 12th edn, Pharmaceutical Press, London, 1994. b Values in parentheses represent alternative values from the literature.

+ − + × − by the expression aH O aOH [H3 O ] [OH ] K = 3 Q (3.68) 2 2 + = aH O [H2 O] pKa pKb pKw (3.67) 2 The concentration of molecular water is con- where pKw is the negative logarithm of the dis- sidered to be virtually constant for dilute sociation constant for water, Kw. Kw is derived aqueous solutions. Therefore from consideration of the equilibrium + − K = [H O ][OH ] (3.69) + e + + − w 3 H2 O H2 O H3 O OH where the dissociation constant for water (the where one molecule of water is behaving as ionic product) now incorporates the term for the weak acid or base and the other is behav- molecular water and has the values given in ing as the solvent. Then Table 3.7.

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80 Chapter 3 • Physicochemical properties of drugs in solution

Answer Table 3.7 Ionic product for water From equation (3.73):

14 At pH 4.5: Temperature (°C) Kw " 10 pKw percentage ionisation 0 0.1139 14.94 100 10 0.2920 14.53 = 20 0.6809 14.17 1 + antilog(4.5 − 14.0 + 5.6) 25 1.0080 14.00 100 = 30 1.4690 13.83 1.000126 40 2.9190 13.54 = 50 5.4740 13.26 99.99% 60 9.6140 13.02 Thus the percentage existing as cocaine base # 70 15.10000 12.82 0.01%. 80 23.40000 12.63 At pH 8.0: When the pH of an aqueous solution of the percentage ionisation weakly acidic or basic drug approaches the pKa 100 = or pKb, there is a very pronounced change in 1 + antilog(8.0 − 14.0 + 5.6) the ionisation of that drug. An expression that 100 enables predictions of the pH dependence of = 1.398 the degree of ionisation to be made can be = derived as shown in Box 3.8. The influence of 71.53% pH on the percentage ionisation may be Thus the percentage existing as cocaine base # determined for drugs of known pKa using 28.47% Table 3.8. If we carry out calculations such as those of Example 3.5 over the whole pH range for both EXAMPLE 3.5 Calculation of percentage acidic and basic drugs, we arrive at the graphs ionisation shown in Fig. 3.6. Notice from this figure that

Calculate the percentage of cocaine existing ● The basic drug is virtually completely as the free base in a solution of cocaine ionised at pH values up to 2 units below its

hydrochloride at pH 4.5, and at pH 8.0. The pKa, and virtually completely unionised at pKb of cocaine is 5.6. pH values greater than 2 units above its pKa.

100 Basic Acidic

50 Ionised (%)

Acidic Basic 0 Ϫ ϩ pKa 2pKa pKa 2

pH

Figure 3.6 Percentage ionisation of weakly acidic and weakly basic drugs as a function of pH.

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Ionisation of drugs in solution 81

Table 3.8 Percentage ionisation of anionic and cationic compounds as a function of pH

At pH above pKa At pH below pKa

pH–pKa If anionic If cationic pKa –pH If anionic If cationic

6.0 99.999 90 00.000 099 9 0.1 44.27 55.73 5.0 99.999 00 00.000 999 9 0.2 38.68 61.32 4.0 99.990 0 00.009 999 0 0.3 33.39 66.61 –– 0– 0.4 28.47 71.53 –– 0– 0.5 24.03 75.97 3.5 99.968 00.031 6 ––– 3.4 99.960 00.039 8 ––– 3.3 99.950 00.050 1 0.6 20.07 79.93 3.2 99.937 00.063 0 0.7 16.63 83.37 3.1 99.921 00.079 4 0.8 13.70 86.30 –– 0– 0.9 11.19 88.81 –– 0– 1.0 09.09 90.91 3.0 99.90 00.099 9 – 0–– 2.9 99.87 00.125 7 – 0–– 2.8 99.84 00.158 2 1.1 07.36 92.64 2.7 99.80 00.199 1 1.2 05.93 94.07 2.6 99.75 00.250 5 1.3 04.77 95.23 –– 0– 1.4 03.83 96.17 –– 0– 1.5 03.07 96.93 2.5 99.68 00.315 2 – 0–– 2.4 99.60 00.396 6 – 0–– 2.3 99.50 00.498 7 1.6 02.450 97.55 2.2 99.37 00.627 0 1.7 01.956 98.04 2.1 99.21 00.787 9 1.8 01.560 98.44 –– 0– 1.9 01.243 98.76 –– 0– 2.0 00.990 99.01 2.0 99.01 00.990 – 0–– 1.9 98.76 01.243 – 0–– 1.8 98.44 01.560 2.1 00.787 9 99.21 1.7 98.04 01.956 2.2 00.627 0 99.37 1.6 97.55 02.450 2.3 00.498 7 99.50 –– 0– 2.4 00.396 6 99.60 –– 0– 2.5 00.315 2 99.68 1.5 96.93 03.07 – 0–– 1.4 96.17 03.83 – 0–– 1.3 95.23 04.77 2.6 00.250 5 99.75 1.2 94.07 05.93 2.7 00.199 1 99.80 1.1 92.64 07.36 2.8 00.158 2 99.84 2.9 00.125 7 99.87 1.0 90.91 09.09 3.0 00.099 9 99.90 0.9 88.81 11.19 – 0–– 0.8 86.30 13.70 3.1 00.079 4 99.921 0.7 83.37 16.63 3.2 00.063 0 99.937 0.6 79.93 20.07 3.3 00.050 1 99.950 ––– 3.4 00.039 8 99.960 0.5 75.97 24.03 3.5 00.031 6 99.968 0.4 71.53 28.47 – 0–– 0.3 66.61 33.39 4.0 00.009 999 0 99.990 0 0.2 61.32 38.68 5.0 00.000 999 9 99.999 00 0.1 55.73 44.27 6.0 00.000 099 9 99.999 90 0 50.00 50.00 – 0––

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82 Chapter 3 • Physicochemical properties of drugs in solution

● The acidic drug is virtually completely 1.2 unionised at pH values up to 2 units below Cation Neutral Anion 1.0 its pKa and virtually completely ionised at pH values greater than 2 units above its pKa. 0.8 ● Both acidic and basic drugs are exactly 50% 0.6 ionised at their pKa values. 0.4 Mole fraction

3.5.3 Ionisation of amphoteric drugs 0.2

Ampholytes (amphoteric electrolytes) can 0.0 function as either weak acids or weak bases in 135791113 pH aqueous solution and have pKa values corres- ponding to the ionisation of each group. They Figure 3.7 Distribution of ionic species for the ordinary may be conveniently divided into two cate- ampholyte m-aminophenol. gories – ordinary ampholytes and zwitterionic Redrawn from A. Pagliara, P.-A. Carrupt, G. Caron, P. Gaillard and B. Testa, ampholytes – depending on the relative acidity Chem. Rev., 97, 3385 (1997). of the two ionisable groups.12, 13 groups is much smaller because of overlapping of the two equilibria. Ordinary ampholytes

In this category of ampholytes, the pKa of the Zwitterionic ampholytes acidic acidic group, pKa , is higher than that of the basic group, pKbasic, and consequently the first This group of ampholytes is characterised by the a acidic ` basic group that loses its proton as the pH is relation pKa pKa . The most common increased is the basic group. Table 3.6 includes examples of zwitterionic ampholytes are the several examples of this type of ampholyte. amino acids, peptides and proteins. There are We will consider, as a simple example, the ion- essentially two types of zwitterionic electrolyte isation of m-aminophenol (I), which has depending on the difference between the acidic basic ∆ acidic = basic = pKa and pKa values, pKa. pKa 9.8 and pKa 4.4. ∆ The steps of the ionisation on increasing pH Large pKa. The simplest type to consider is are shown in the following equilibria: that of compounds having two widely sepa- rated pK values, for example glycine. The pK + e e a a NH3C6 H4 OH NH2 C6 H4 OH values of the carboxylate and amino groups on − NH2 C6 H5 O glycine are 2.34 and 9.6, respectively and the This compound can exist as a cation, as an changes in ionisation as the pH is increased are unionised form, or as an anion depending on described by the following equilibria: the pH of the solution, but because the differ- + e − + HOOCCH2 NH3 OOCCH2 NH3 ence between pKacidic and pKbasic is p2, there e − a a OOCCH2 NH2 will be no simultaneous ionisation of the two groups and the distribution of the species will Over the pH range 3–9, glycine exists in solu- 0 + be as shown in Fig. 3.7. The ionisation pattern tion predominantly in the form OOCCH2NH3 will become more complex, however, with Such a structure, having both positive and neg- ative charges on the same molecule, is referred drugs in which the difference in pKa of the two to as a zwitterion and can react both as an acid,

OH − + + e OOCCH2 NH3 H2 O − + + OOCCH2 NH2 H3 O or as a base, − + NH2 + e OOCCH2 NH3 H2 O e + + − Structure I m-Aminophenol HOOCCH2 NH3 OH Physicochemical Principles of Pharmacy, 4th edition (ISBN: 0 85369 608 X) © Pharmaceutical Press 2006 09_1439PP_CH03 17/10/05 10:47 pm Page 83

Ionisation of drugs in solution 83

This compound can exist as a cation, as a zwit- 1.2 terion, or as an anion depending on the pH of Cation Anion 1.0 the solution. The two pKa values of glycine are p Zwitterion 2pH units apart and hence the distribution 0.8 of the ionic species will be similar to that shown in Fig. 3.7. 0.6 At a particular pH, known as the isoelectric Mole fraction 0.4 Neutral pH or isoelectric point, pHi, the effective net charge on the molecule is zero. pHi can be 0.2 calculated from 0.0 acidic + basic 4 6 8 10 12 pKa pKa pH = i 2 pH Figure 3.8 Distribution of ionic species for the zwitterionic ∆ Small pKa. In cases where the two pKa ampholyte labetalol. values are <<2pH units apart there is overlap of Redrawn from A. Pagliara, P.-A. Carrupt, G. Caron, P. Gaillard and B. Testa, the ionisation of the acidic and basic groups, Chem. Rev., 97, 3385 (1997). with the result that the zwitterionic electrolyte can exist in four different electrical states – the Examples of zwitterionic drugs with both ∆ cation, the unionised form, the zwitterion, large and small pKa values are given in Box 3. and the anion (Scheme 3.3). 9; others can be noted in Table 3.6. In Scheme 3.3, (ABH)& is the zwitterion and, although possessing both positive and negative charges, is essentially neutral. The 3.5.4 Ionisation of polyprotic drugs unionised form, (ABH), is of course also neutral and can be regarded as a tautomer of the In the examples we have considered so far, the zwitterion. Although only two dissociation acidic drugs have donated a single proton. acidic basic There are several acids, for example citric, constants Ka and Ka (macrodissociation constants) can be determined experimentally, phosphoric and tartaric acids, that are capable each of these is composed of individual of donating more than one proton; these com- microdissociation constants because of simul- pounds are referred to as polyprotic or polybasic taneous ionisation of the two groups (see acids. Similarly, a polyprotic base is one section 3.5.5). These microdissociation con- capable of accepting two or more protons. stants represent equilibria between the cation Many examples of both types of polyprotic and zwitterion, the anion and the zwitterion, drugs can be found in Table 3.6, including the the cation and the unionised form, and the polybasic acids amoxicillin and fluorourocil, and the polyacidic bases pilocarpine, doxoru- anion and the unionised form. At pHi, the unionised form and the zwitterion always bicin and aciclovir. Each stage of the dissocia- coexist, but the ratio of the concentrations of tion may be represented by an equilibrium each will vary depending on the relative expression and hence each stage has a distinct magnitude of the microdissociation constants. pKa or pKb value. The dissociation of phos- The distribution of the ionic species for phoric acid, for example, occurs in three acidic = basic = stages; thus: labetalol which has pKa 7.4 and Ka 9.4 + is shown in Fig. 3.8. H3 PO4 H2 O e − + + = × −3 H2 PO4 H3 O K1 7.5 10 − + H2 PO4 H2 O acidic basic − + − K Ϯ K e 2 + = × 8 ϩ a (ABH) a Ϫ HPO4 H3 O K2 6.2 10 (ABH2) (AB) (ABH) 2− + HPO4 H2 O e 3− + + = × −13 Scheme 3.3 PO4 H3 O K3 2.1 10

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84 Chapter 3 • Physicochemical properties of drugs in solution

Box 3.9 Chemical structures and pKa values of some zwitterionic drugs

⌬ Zwitterions with a large pKa acidic ഛ pK a 1.8 Cl

H basic2 pK ϭ 8.00 O a

O N O N N N HO O N N

acidic pK ϭ 2.93 pKbasic1 ϭ 2.19 basic ϭ a a pK a 6.55 Azapropazone Cetirizine

⌬ Zwitterions with a small pK a basic ϭ pK a 9.38 H OH O

N NH2

OH Labetalol pK acidic ϭ 1.07 a acidic ϭ pKa 7.44 H O O

S N N

N H S CH3 basic ϭ O O pK a 5.34 Tenoxicam (Reproduced from reference 12.)

3.5.5 Microdissociation constants groups. Experience suggests that the first pKa value is probably associated with the ionisa- The experimentally determined dissociation tion of the amino group and the second with constants for the various stages of dissociation that of the phenolic group; but it is not pos- of polyprotic and zwitterionic drugs are sible to assign the values of these groups referred to as macroscopic values. However, it unequivocally and for a more complete is not always easy to assign macroscopic disso- picture of the dissociation it is necessary to ciation constants to the ionisation of specific take into account all possible ways in which groups of the molecule, particularly when the the molecule may be ionised and all the pos-

pKa values are close together, as discussed in sible species present in solution. We may section 3.5.4. The diprotic drug morphine has represent the most highly protonated form of ! ! ! macroscopic pKa values of 8.3 and 9.5, arising morphine, HMOH, as O, where the ‘ ’ from ionisation of amino and phenolic refers to the protonated amino group and the

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ϩϪ shifts in pK values between native and de- k a 1 k12 natured states or between bound and free ϩO OϪ forms of a complex can cause changes in binding constants or stability of the protein k2 k21 OO due to pH effects. For example, the acid denat- Scheme 3.4 uration of proteins may be a consequence of anomalous shifts of the pKa values of a small 19 O refers to the uncharged phenolic group. number of amino acids in the native protein. Dissociation of the amino proton only pro- Several possible causes of such shifts have duces an uncharged form MOH, represented been proposed. They may arise from interac- by OO, while dissociation of the phenolic tions among ionisable groups on the protein proton gives a zwitterion !HMO0, represented molecule; for example, an acidic group will 0 by ‘!0’. The completely dissociated form MO have its pKa lowered by interactions with basic is represented as O0. The entire dissociation groups. Other suggested cases of shifts of pKa scheme is given in Scheme 3.4. include hydrogen-bonding interactions with nonionisable groups and the degree of expo- The constants K1, K2, K12 and K21 are termed microdissociation constants and are defined sure to the bulk solvent; for example, an acidic by group will have its pKa increased if it is fully or partially removed from solvent, but the effect +− + + [ ][H3 O ] [OO][H3 O ] may be reversed by strong hydrogen-bonding k = k = 1 [+O] 2 [+O] interactions with other groups. + + − − The calculation of pKa values of protein [O ][H3 O ] [O ][H3 O ] k = k = molecules thus requires a detailed consider- 12 [+−] 21 [OO] ation of the environment of each ionisable The micro- and macrodissociation constants group, and is consequently highly complex. are related by the following expressions: An additional complication is that a protein N = + with N ionisable residues has 2 possible K1 k1 k2 (3.74) ionisation states; the extent of the problem is 1 1 1 apparent when it is realised that a moderately = + (3.75) sized protein may contain as many as 50 K k k 2 12 21 ionisable groups. and

K K = k k = k k (3.76) 1 2 1 12 2 21 3.5.7 Calculation of the pH of drug solutions Various methods have been proposed whereby the microdissociation constants for the mor- We have considered above the effect on the phine system may be evaluated.14 Other drugs ionisation of a drug of buffering the solution for which microdissociation constants have at given pH values. When these weakly acidic been derived include the tetracyclines,15 doxo- or basic drugs are dissolved in water they will, rubicin,16 cephalosporin,17 dopamine18 and of course, develop a pH value in their own the group of drugs shown in Box 3.9.13 right. In this section we examine how the pH of drug solutions of known concentration can be simply calculated from a knowledge of the

3.5.6 pKa values of proteins pKa of the drug. We will consider the way in which one of these expressions may be

The pKa values of ionisable groups in proteins derived from the expression for the ionisation and other macromolecules can be significantly in solution; the derivation of the other expres- different from those of the corresponding sions follows a similar route and you may wish groups when they are isolated in solution. The to derive these for yourselves.

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86 Chapter 3 • Physicochemical properties of drugs in solution

Weakly acidic drugs Weakly basic drugs We saw above that the dissociation of these We can show by a similar derivation to that types of drugs may be represented by equation above that the pH of aqueous solutions of (3.63). We can now express the concentra- weakly basic drugs will be given by tions of each of the species in terms of the pH = 1pK + 1pK + 1log c (3.79) degree of dissociation, α, which is a number 2 w 2 a 2 with a value between 0 (no dissociation) and 1 (complete dissociation): EXAMPLE 3.7 Calculation of the pH of a e 0 ! HA ! H2 O A ! H3 O weakly basic drug (1 0 α)c αc αc Calculate the pH of a saturated solution of where c is the initial concentration of the # 03 codeine monohydrate (mol. wt. 317.4) given weakly acidic drug in mol dm . that its pK # 8.2 and its solubility at room α a Because the drugs are weak acids, will be temperature is 1 g in 120 cm3 water. very small and hence the term (1 0 α) can be approximated to 1. We may therefore write Answer α2 2 Codeine is a weakly basic drug and hence its c 2 K = Q α c (3.77) a (1 − α)c pH will be given by = 1 + 1 + 1 Therefore, pH 2pKw 2pKa 2log c 3 03 12 where c # 1g in 120 cm # 8.33 g dm # K α2 = a 0.02633 mol dm03. c ∴pH = 7.0 + 4.1 − 0.790 = 10.31 To introduce pH into the discussion we note that α = + = + c [H3 O ] [H ] ∴ + = 12 [H ] (Ka c) Drug salts + 1 1 ∴−log[H ] =− log K − log c 2 a 2 Because of the limited solubility of many weak = 1 − 1 pH 2pKa 2log c (3.78) acids and weak bases in water, drugs of these We now have an expression which enables us types are commonly used as their salts; for to calculate the pH of any concentration of example, sodium salicylate is the salt of a weak acid (salicylic acid) and a strong base (sodium the weakly acidic drug provided that its pKa value is known. hydroxide). The pH of a solution of this type of salt is given by = 1 + 1 + 1 pH 2pKw 2pKa 2log c (3.80) EXAMPLE 3.6 Calculation of the pH of a weak acid Alternatively, a salt may be formed between a weak base and a strong acid; for example, 03 Calculate the pH of a 50 mg cm solution of ephedrine hydrochloride is the salt of ascorbic acid (mol. wt # 176.1; pKa # 4.17). ephedrine and hydrochloric acid. Solutions of such drugs have a pH given by Answer = 1 − 1 For a weakly acidic drug, pH 2pKa 2log c (3.81) = 1 − 1 Finally, a salt may be produced by the combi- pH 2pKa 2log c − − nation of a weak base and a weak acid, as in c = 50 mgcm 3 = 50 gdm 3 the case of codeine phosphate. Solutions of = −3 0.2839 mol dm such drugs have a pH given by ∴ = + = pH 2.09 0.273 2.36 = 1 + 1 − 1 pH 2pKw 2pKa 2pKb (3.82)

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Notice that this equation does not include a buffering action of a weak acid HA and its concentration term and hence the pH is inde- ionised salt (for example, NaA) is that the A0 pendent of concentration for such drugs. ions from the salt combine with the added H! ions, removing them from solution as undissociated weak acid: EXAMPLE 3.8 Calculation of the pH of drug salts − + + e + A H3 O H2 O HA Calculate the pH of the following solutions Added OH0 ions are removed by combination with the weak acid to form undissociated (a) 5% oxycodone hydrochloride (pKa # 8.9, mol. wt. # 405.9). water molecules: − − (b) 600 mg of benzylpenicillin sodium (pKa # + e + HA OH H2 O A 2.76, mol. wt. # 356.4) in 2 cm3 of water for injection. The buffering action of a mixture of a weak ! (c) 100 mg cm03 chlorphenamine maleate base and its salt arises from the removal of H (mol. wt. # 390.8) in water for injection ions by the base B to form the salt and removal of OH0 ions by the salt to form undissociated (pKb chlorphenamine # 5.0, pKa maleic acid # 1.9). water: + + e + + B H3 O H2 O BH Answer + − BH + OH e H O + B (a) Oxycodone hydrochloride is the salt of a 2 weak base and a strong acid, hence The concentration of buffer components 1 1 required to maintain a solution at the required pH = pK − log c 2 a 2 pH may be calculated using equation (3.70). where c # 50 g dm03 # 0.1232 mol dm03. Since the acid is weak and therefore only very slightly ionised, the term [HA] in this equation ∴pH = 4.45 + 0.45 = 4.90 may be equated with the total acid concen- (b) Benzylpenicillin sodium is the salt of a tration. Similarly, the free A0 ions in solution weak acid and a strong base, hence may be considered to originate entirely from 0 1 1 1 the salt and the term [A ] may be replaced by pH = pK + pK + log c 2 w 2 a 2 the salt concentration. where c # 600 mg in 2 cm3 # 0.842 mol Equation (3.70) now becomes dm03. [salt] pH = pK + log (3.83) ∴pH = 7.00 + 1.38 − 0.037 = 8.34 a [acid] (c) Chlorphenamine maleate is the salt of a By similar reasoning, equation (3.72) may be weak acid and a weak base, hence modified to facilitate the calculation of the pH = 1 + 1 − 1 of a solution of a weak base and its salt, giving pH 2pKw 2pKa 2pKb ∴pH = 7.00 + 0.95 − 2.5 = 5.45 [base] pH = pK − pK + log (3.84) w b [salt] Equations (3.83) and (3.84) are often referred 3.5.8 Preparation of buffer solutions to as the Henderson–Hasselbalch equations.

A mixture of a weak acid and its salt (that is, a conjugate base), or a weak base and its con- EXAMPLE 3.9 Buffer solutions jugate acid, has the ability to reduce the large changes in pH which would otherwise result Calculate the amount of sodium acetate to be 0 from the addition of small amounts of acid or added to 100 cm3 of a 0.1 mol dm 3 acetic alkali to the solution. The reason for the acid solution to prepare a buffer of pH 5.20.

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88 Chapter 3 • Physicochemical properties of drugs in solution

Answer acid) needed to change the pH of 1 dm3 of The pKa of acetic acid is 4.76. Substitution in solution by an amount d(pH). If the addition equation (3.83) gives of 1 mole of alkali to 1 dm3 of buffer solution [salt] produces a pH change of 1 unit, the buffer 5.20 = 4.76 + log capacity is unity. [acid] Equation (3.70) may be rewritten in the form The molar ratio of [salt][acid] is 2.754. Since 1 c 100 cm3 of 0.1 mol dm03 acetic acid contains pH = pK + ln (3.86) a − 0.01 mol, we would require 0.027 54 mol of 2.303 c0 c sodium acetate (2.258 g), ignoring dilution where c0 is the total initial buffer concentra- effects. tion and c is the amount of alkali added. Rearrangement and subsequent differentia- tion yield Equations (3.83) and (3.84) are also useful in c calculating the change in pH which results = 0 (3.87) c + − − from the addition of a specific amount of acid 1 exp[ 2.303(pH pKa)] or alkali to a given buffer solution, as seen from Therefore, the following calculation in Example 3.10. dc 2.303 c exp[2.303 (pH − pK )] β = = 0 a + − 2 d(pH) 1 exp[2.303 exp(pH pKa)] EXAMPLE 3.10 Calculation of the pH change (3.88)

in buffer solutions + 2.303 c K [H O ] β = 0 a 3 (3.89) Calculate the change in pH following the + 2 ([H O ] + K ) addition of 10 cm3 of 0.1 mol dm03 NaOH to 3 a the buffer solution described in Example 3.9. EXAMPLE 3.11 Calculation of buffer capacity Answer The added 10 cm3 of 0.1 mol dm03 NaOH (equi- Calculate the buffer capacity of the acetic valent to 0.002 mol) combines with 0.001 mol acid–acetate buffer of Example 3.9 at pH 4.0. of acetic acid to produce 0.001 mol of sodium acetate. Reapplying equation (3.83) using the Answer revised salt and acid concentrations gives The total amount of buffer components in 100 cm3 of solution # 0.01 ! 0.027 54 # + = + (0.02754 0.001) = 0.037 54 moles. Therefore, pH 4.76 log − 5.26 (0.01 0.001) = −3 c0 0.3754 mol dm The pH of the buffer has been increased by pK of acetic acid # 4.76. Therefore, only 0.06 units following the addition of the a = × −5 alkali. Ka 1.75 10 The pH of the solution # 4.0. Therefore, + = −4 [H3 O ] 10 Buffer capacity Substituting in equation (3.89), The effectiveness of a buffer in reducing × × −5 × −4 changes in pH is expressed as the buffer capac- β = 2.303 0.3754 1.75 x 10 10 − − ity, β. The buffer capacity is defined by the ratio (10 4 + 1.75 × 10 5)2 = 0.1096 dc β = (3.85) d(pH) The buffer capacity of the acetic acid–acetate buffer is 0.1096 mol dm03 per pH unit. where dc is the number of moles of alkali (or

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Figure 3.9 shows the variation of buffer irritability of the various ionic species that are capacity with pH for the acetic acid–acetate commonly used as buffer components. The buffer used in the numerical examples above pH of blood is maintained at about 7.4 by 03 (c0 # 0.3754 mol dm ) as calculated from primary buffer components in the plasma (car- equation (3.89). It should be noted that β is at bonic acid–carbonate and the acid–sodium

a maximum when pH # pKa (that is, at pH salts of phosphoric acid) and secondary buffer 4.76). When selecting a weak acid for the components (oxyhaemoglobin–haemoglobin preparation of a buffer solution, therefore, the and acid–potassium salts of phosphoric acid)

chosen acid should have a pKa as close as pos- in the erythrocytes. Values of 0.025 and 03 sible to the pH required. Substituting pH # pKa 0.039 mol dm per pH unit have been quoted into equation (3.89) gives the useful result that for the buffer capacity of whole blood. β the maximum buffer capacity is max # 0.576c0, Parenteral solutions are not normally buffered, where c0 is the total buffer concentration. or alternatively are buffered at a very low Buffer solutions are widely used in phar- capacity, since the buffers of the blood are macy to adjust the pH of aqueous solutions to usually capable of bringing them within a that required for maximum stability or that tolerable pH range. needed for optimum physiological effect. Solutions for application to delicate tissues, Universal buffers particularly the eye, should also be formulated at a pH not too far removed from that of the We have seen from Fig. 3.9 that the buffer appropriate tissue fluid, as otherwise irritation capacity is at a maximum at a pH equal to the

may be caused on administration. The pH of pKa of the weak acid used in the formulation tears lies between 7 and 8, with an average of the buffer system and decreases appreciably value of 7.4. Fortunately, the buffer capacity of as the pH extends more than one unit either tears is high and, provided that the solutions side of this value. If, instead of a single weak to be administered have a low buffer capacity, monobasic acid, a suitable mixture of poly- a reasonably wide range of pH may be toler- basic and monobasic acids is used, it is pos- ated, although there is a difference in the sible to produce a buffer which is effective over a wide pH range. Such solutions are referred to as universal buffers. A typical 0.22 example is a mixture of citric acid (pKa1 # 3.06, 0.20 pKa2 # 4.78, pKa3 # 5.40), Na2 HPO4 (pKa of − conjugate acid H PO # 7.2), diethylbarbituric 0.18 2 4 acid (pKa1 # 7.43) and boric acid (pKa1 # 9.24). 0.16 Because of the wide range of pK values b a 0.14 involved, each associated with a maximum

0.12 buffer capacity, this buffer is effective over a correspondingly wide pH range (pH 2.4–12). 0.10 Buffer capacity, Buffer capacity, 0.08 0.06 3.6 Diffusion of drugs in solution 0.04

0.02 pKa Diffusion is the process by which a concentra- tion difference is reduced by a spontaneous 2.0 3.0 4.0 5.0 6.0 7.0 flow of matter. Consider the simplest case of a pH solution containing a single solute. The solute Figure 3.9 Buffer capacity of acetic acid–acetate buffer will spontaneously diffuse from a region of (initial concentration # 0.3754 mol dm03) as a function of high concentration to one of low concentra- pH. tion. Strictly speaking, the driving force for

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90 Chapter 3 • Physicochemical properties of drugs in solution

diffusion is the gradient of chemical potential, but it is more usual to think of the diffusion of Table 3.9 Effect of molecular weight on diffusion solutes in terms of the gradient of their con- coefficient (25°C) in aqueous media centration. Imagine the solution to be divided Compound Molecular D into volume elements. Although no indi- weight (10010 m2 s01) vidual solute particle in a particular volume element shows a preference for motion in any p-Hydroxybenzoatesa particular direction, a definite fraction of the Methyl hydroxybenzoate 152.2 8.44 molecules in this element may be considered Ethyl hydroxybenzoate 166.2 7.48 n-Propyl hydroxybenzoate 180.2 6.81 to be moving in, say, the x direction. In an Isopropyl hydroxybenzoate 180.2 6.94 adjacent volume element, the same fraction n-Butyl hydroxybenzoate 194.2 6.31 may be moving in the reverse direction. If the Isobutyl hydroxybenzoate 194.2 6.40 concentration in the first volume element is n-Amyl hydroxybenzoate 208.2 5.70 greater than that in the second, the overall Proteinsb effect is that more particles are leaving the first Cytochrome c (horse) 12 400 1.28 element for the second and hence there is a Lysozyme (chicken) 14 400 0.95 net flow of solute in the x direction, the Trypsin (bovine) 24 000 1.10 direction of decreasing concentration. The Albumin (bovine) 66 000 0.46

expression which relates the flow of material a From reference 20. to the concentration gradient (dc dx) is b From reference 21; see also Chapter 11 for further values of referred to as Fick’s first law: therapeutic peptides and proteins. dc J =−D (3.90) dx coefficients of more complex molecules such where J is the flux of a component across a as proteins will also be affected by the shape of plane of unit area and D is the diffusion the molecule, more asymmetric molecules coefficient (or diffusivity). The negative sign having a greater resistance to flow. indicates that the flux is in the direction of The diffusional properties of a drug have decreasing concentration. J is in mol m02 s01, c relevance in pharmaceutical systems in a con- is in mol m03 and x is in metres; therefore, the sideration of such processes as the dissolution units of D are m2 s01. of the drug and transport through artificial The relationship between the radius, a, of (e.g. polymer) or biological membranes. Diffu- the diffusing molecule and its diffusion coeffi- sion in tissues such as the skin or in tumours is cient (assuming spherical particles or mol- a process which relies on the same criteria as ecules) is given by the Stokes–Einstein equation discussed above, even though the diffusion as takes place in complex media. RT D = (3.91) πη 6 aNA Summary Table 3.9 shows diffusion coefficients of some p-hydroxybenzoate (paraben) preservatives ● We have looked at the meaning of some of and typical proteins in aqueous solution. the terms commonly used in thermo- Although the trend for a decrease of D with dynamics and how these are interrelated increase of molecular size (as predicted by in the three laws of thermodynamics. In equation 3.91) is clearly seen from the data of particular, we have seen that: this table, it is also clear that other factors, such as branching of the p-hydroxybenzoate ● Entropy is a measure of the disorder or molecules (as with the isoalkyl derivatives) chaos of a system and that processes, and the shape of the protein molecules, also such as the melting of a crystal, which affect the diffusion coefficients. The diffusion result in an increased disorder are

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References 91

accompanied by an increase in entropy. freezing point depressions of the drug and Since spontaneous processes always the adjusting substance. produce more disordered systems, it ● The strengths of weakly acidic or basic

follows that the entropy change for these drugs may be expressed by their pKa and processes is always positive. pKb values; the lower the pKa the stronger is ● During a spontaneous change at con- the acid; the lower the pKb the stronger is stant temperature and pressure, there is a the base. Acidic drugs are completely decrease in the free energy until the unionised at pH values up to 2 units below

system reaches an equilibrium state, their pKa and completely ionised at pH when the free energy change becomes values greater than 2 units above their pKa. zero. The free energy is therefore a Conversely, basic drugs are completely measure of the useful work that a system ionised at pH values up to 2 units below

can do; when the system has reached their pKa and completely unionised when equilibrium it has used up its free energy the pH is more than 2 units above their pKa. and consequently no longer has the Both types of drug are exactly 50% ionised

ability to do any further work; thus all at their pKa values. Some drugs can donate spontaneous processes are irreversible. or accept more than one proton and so may

● There is a simple relationship between free have several pKa values; other drugs can energy change and the equilibrium con- behave as both acids and bases, i.e. they are stant for a reaction from which an equa- amphoteric drugs. The pH of aqueous solu- tion has been derived for the change of tions of each of these types of drug and

equilibrium constant with temperature. their salts can be calculated from their pKa ● If there are no interactions between the and the concentration of the drug. components of a solution (ideal solu- ● A solution of a weak acid and its salt (conju- tion) then the activity equals the concen- gate base) or a weak base and its conjugate tration; in real solutions the ratio of the acid acts as a buffer solution. The quantities activity to the concentration is called the of buffer components required to prepare activity coefficient. buffers solutions of known pH can be cal- ● The chemical potential is the effective free culated from the Henderson–Hasselbalch energy per mole of each component of a equation. The buffering capacity of a buffer

solution. In general, the chemical poten- solution is maximum at the pKa of the weak tial of a component is identical in all acid component of the buffer. Universal phases of a system at equilibrium at a buffers are mixtures of polybasic and fixed temperature and pressure. monobasic acids that are effective over a wide range of pH. ● Parenteral solutions should be of approxi- ● The drug molecules in solution will spon- mately the same tonicity as blood serum; taneously diffuse from a region of high the amount of adjusting substance which chemical potential to one of low chemical must be added to a formulation to achieve potential; the rate of diffusion may be isotonicity can be calculated using the calculated from Fick’s law.

References

1. The Pharmaceutical Codex, 12th edn, The Pharma- 4. J. H. Rytting, S. S. Davis and T. Higuchi. Suggested ceutical Press, London, 1994 thermodynamic standard state for comparing drug 2. P. W. Atkins. Physical Chemistry, 7th edn, Oxford molecules in structure–activity studies. J. Pharm. University Press, 2001 Sci., 61, 816–8 (1972) 3. D. J. Shaw and H. E. Avery. Physical Chemistry, 5. J. H. B. Christian. In Water Activity: Influences on Macmillan Education Limited, London, 1994 Food Quality (ed. L. B. Rockland and G. F. Stewart), Academic Press, New York, 1981, p. 825

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92 Chapter 3 • Physicochemical properties of drugs in solution

6. U. Ambrose, K. Middleton and D. Seal. In vitro 14. P. J. Niebergall, R. L. Schnaare and E. T. Sugita. studies of water activity and bacterial growth Spectral determination of microdissociation con- inhibition of sucrose–polyethylene glycol 400– stants. J. Pharm. Sci., 61, 232 (1972) hydrogen peroxide and xylose–polyethylene 15. L. J. Leeson, J. E. Krueger and R. A. Nash. Structural glycol 400– hydrogen peroxide pastes used to treat assignment of the second and third acidity con- infected wounds. Antimicrob. Agents Chemother., stants of tetracycline antibiotics. Tetrahedron Lett., 35, 1799–803 (1991) 18, 1155–60 (1963) 7. W. H. Streng, H. E. Huber and J. T. Carstensen. 16. R. J. Sturgeon and S. G. Schulman. Electronic Relationship between osmolality and osmolarity. J. absorption spectra and protolytic equilibriums of Pharm. Sci., 67, 384–6 (1978) doxorubicin: direct spectrophotometric deter- 8. D. M. Willis, J. Chabot, I. C. Radde and G. W. mination of microconstants. J. Pharm. Sci., 66, Chance. Unsuspected hyperosmolality of oral solu- 958–61 (1977) tions contributing to necrotizing enterocolitis in 17. W. H. Streng, H. E. Huber, J. L. DeYoung and M. A. very-low-birth-weight infants. Pediatrics, 60, 535–8 Zoglio. Ionization constants of cephalosporin (1977) zwitterionic compounds. J. Pharm. Sci., 65, 1034 9. K. C. White and K. L. Harkavy. Hypertonic formula (1976); 66, 1357 (1977) resulting from added oral medications. Am. J. Dis. 18. T. Ishimitsu, S. Hirose and H. Sakurai. Microscopic Child., 136, 931–3 (1982) acid dissociation constants of 3,4-dihydrox- 10. J. Lerman, M. M. Willis, G. A. Gregory and E. I. yphenethylamine (dopamine). Chem. Pharm. Bull., Eger. Osmolarity determines the solubility of 26, 74–8 (1978) anesthetics in aqueous solutions at 37°C. 19. B. Honig and A. Nicholls. Classical electrostatics in Anesthesiology, 59, 554–8 (1983) biology and chemistry. Science, 268, 1144–9 (1995) 11. The Merck Index, 12th edn, Merck, Rahway, NJ, 20. T. Seki, M. Okamoto, O. Hosoya and K. Juni. 1996 Measurement of diffusion coefficients of parabens 12. A. Pagliara, P.-A. Carrupt, G. Caron, et al. by the chromatographic broadening method. J. Lipophilicity profiles of ampholytes. Chem. Rev., Pharm. Sci. Technol., Jpn. 60, 114–7 (2000) 97, 3385–400 (1997) 21. N. Baden and M. Terazima. A novel method for 13. G. Bouchard, A. Pagliara, P.-A. Carrupt, et al. measurement of diffusion coefficients of proteins Theoretical and experimental exploration of the and DNA in solution. Chem. Phys. Lett., 393, lipophilicity of zwitterionic drugs in 1,2-dichloro- 539–45 (2004) ethanewater system. Pharm. Res., 19, 1150–9 (2002)

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