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3Physicochemical properties of drugs in solution

In this chapter we examine some of the important physicochemical properties of drugs in aqueoussolution which are of relevance to such liquid dosage forms as injections, solutions, and eyedrops. Some basic thermodynamic concepts will be introduced, particularly that of thermo-dynamic activity, an important parameter in determining drug potency. It is important thatparenteral solutions are formulated with osmotic pressure similar to that of blood serum; in thischapter we will see how to adjust the tonicity of the formulation to achieve this aim. Most drugsare at least partially ionised at physiological pH and many studies have suggested that thecharged group is essential for biological activity. We look at the influence of pH on the ionisationof several types of drug in solution and consider equations that allow the calculation of the pH ofsolutions of these drugs.

First, however, we recount the various ways to express the strength of a solution, since it is offundamental importance that we are able to interpret the various units used to denote solutionconcentration and to understand their interrelationships, not least in practice situations.

3.1 Concentration units 56

3.2 Thermodynamics – a brief

introduction 57

3.3 Activity and chemical potential 62

3.4 Osmotic properties of drug

solutions 69

3.5 Ionisation of drugs in solution 75

3.6 Diffusion of drugs in solution 89

Summary 90

References 91

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Physicochemical Principles of Pharmacy, 4th edition (ISBN: 0 85369 608 X) © Pharmaceutical Press 2006

3.1 Concentration units

A wide range of units is commonly used toexpress solution concentration, and confusionoften arises in the interconversion of oneset of units to another. Wherever possiblethroughout this book we have used the SIsystem of units. Although this is the currentlyrecommended system of units in GreatBritain, other more traditional systems are stillwidely used and these will be also described inthis section.

3.1.1 Weight concentration

Concentration is often expressed as a weightof solute in a unit volume of solution; forexample, g dm�03, or % w�v, which is thenumber of grams of solute in 100 cm�3 of solu-tion. This is not an exact method whenworking at a range of temperatures, sincethe volume of the solution is temperature-dependent and hence the weight concentra-tion also changes with temperature.

Whenever a hydrated compound is used,it is important to use the correct state ofhydration in the calculation of weightconcentration. Thus 10% w�v CaCl�2 (anhy-drous) is approximately equivalent to 20% w�vCaCl�2·6H�2��O and consequently the use of thevague statement ‘10% calcium chloride’ couldresult in gross error. The SI unit of weightconcentration is kg m�03 which is numericallyequal to g dm�03.

3.1.2 Molarity and molality

These two similar-sounding terms must not beconfused. The molarity of a solution is thenumber of moles (gram molecular weights) ofsolute in 1 litre (1 dm�3) of solution. The molal-ity is the number of moles of solute in 1 kg ofsolvent. Molality has the unit, mol kg�01, which

is an accepted SI unit. Molarity may be con-verted to SI units using the relationship1 mol litre � 01 # 1 mol dm � 03 # 10 � 3 mol m � 03.Interconversion between molarity and molal-ity requires a knowledge of the density of thesolution.

Of the two units, molality is preferable for aprecise expression of concentration because itdoes not depend on the solution temperatureas does molarity; also, the molality of a com-ponent in a solution remains unaltered by theaddition of a second solute, whereas themolarity of this component decreases becausethe total volume of solution increases follow-ing the addition of the second solute.

3.1.3 Milliequivalents

The unit milliequivalent (mEq) is commonlyused clinically in expressing the concentrationof an ion in solution. The term ‘equivalent’, orgram equivalent weight, is analogous to themole or gram molecular weight. Whenmonovalent ions are considered, these twoterms are identical. A 1 molar solution ofsodium bicarbonate, NaHCO�3, contains 1 molor 1 Eq of Na�! and 1 mol or 1 Eq of HCO perlitre (dm�03) of solution. With multivalentions, attention must be paid to the valency ofeach ion; for example, 10% w�v CaCl�2·2H�2��Ocontains 6.8 mmol or 13.6 mEq of Ca�2! in10 cm�3.

The Pharmaceutical Codex�1 gives a table ofmilliequivalents for various ions and also asimple formula for the calculation of milli-equivalents per litre (see Box 3.1).

In analytical chemistry a solution whichcontains 1 Eq dm�03 is referred to as a normalsolution. Unfortunately the term ‘normal’ isalso used to mean physiologically normal withreference to saline solution. In this usage, aphysiologically normal saline solution con-tains 0.9 g NaCl in 100 cm�3 aqueous solutionand not 1 equivalent (58.44 g) per litre.

�−3

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3.1.4 Mole fraction

The mole fraction of a component of a solu-tion is the number of moles of that com-ponent divided by the total number of molespresent in solution. In a two-component(binary) solution, the mole fraction of solvent,x�1, is given by x�1 # n�1�(n�1 ! n�2), where n�1 andn�2 are respectively the numbers of moles ofsolvent and of solute present in solution.Similarly, the mole fraction of solute, x�2, isgiven by x�2 # n�2�(n�1 ! n�2). The sum of themole fractions of all components is, of course,unity, i.e. for a binary solution, x�1 ! x�2 # 1.

EXAMPLE 3 .1 Units of concentration

Isotonic saline contains 0.9% w�v of sodiumchloride (mol. wt. # 58.5). Express the concen-tration of this solution as: (a) molarity; (b)molality; (c) mole fraction and (d) milliequiv-alents of Na�! per litre. Assume that the densityof isotonic saline is 1 g cm�03.

Answer(a) 0.9% w�v solution of sodium chloride

contains 9 g dm�03 # 0.154 mol dm�03.

(b) 9 g of sodium chloride are dissolved in991 g of water (assuming density #1 g dm�03).

Therefore 1000 g of water contains 9.08 g ofsodium chloride # 0.155 moles, i.e. molality #0.155 mol kg�01.

(c) Mole fraction of sodium chloride, x�1, isgiven by

(Note 991 g of water contains 991�18moles, i.e. n�2 # 55.06.)

(d) Since Na�! is monovalent, the number ofmilliequivalents of Na�! # number of milli-moles.

Therefore the solution contains 154 mEq dm�03

of Na�!.

3.2 Thermodynamics – a briefintroduction

The importance of thermodynamics in thepharmaceutical sciences is apparent when it isrealised that such processes as the partitioningof solutes between immiscible solvents, thesolubility of drugs, micellisation and drug�–�receptor interaction can all be treated inthermodynamic terms. This brief sectionmerely introduces some of the concepts ofthermodynamics which are referred to through-out the book. Readers requiring a greaterdepth of treatment should consult standardtexts on this subject.�2, 3

3.2.1 Energy

Energy is a fundamental property of a system.Some idea of its importance may be gained byconsidering its role in chemical reactions,where it determines what reactions may occur,how fast the reaction may proceed and inwhich direction the reaction will occur.Energy takes several forms: kinetic energy isthat which a body possesses as a result of its

x�1 =n�1

n�1 + n�2=

0.154

0.154 + 55.06= 2.79 × 10��−3

Thermodynamics – a brief introduction 57

The number of milliequivalents in 1 g of substance isgiven by

For example, CaCl�2·2H�2��O (mol. wt. # 147.0)

and

that is, each gram of CaCl�2·2H�2��O represents 13.6 mEqof calcium and 13.6 mEq of chloride

= 13.6 mEq

mEq Cl��− in 1 g CaCl�2 · 2H�2��O =1 × 1000 × 2

147.0

= 13.6 mEq

mEq Ca��2+ in 1 g CaCl�2 · 2H�2��O =2 × 1000 × 1

147.0

mEq =

valency x 1000 x no. of specifiedunits in 1 atom/molecule/ion

atomic, molecular or ionic weight

Box 3.1 Calculation of milliequivalents

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motion; potential energy is the energy whicha body has due to its position, whether gravi-tational potential energy or coulombic poten-tial energy which is associated with chargedparticles at a given distance apart. All forms ofenergy are related, but in converting betweenthe various types it is not possible to create ordestroy energy. This forms the basis of the lawof conservation of energy.

The internal energy U of a system is the sumof all the kinetic and potential energy contri-butions to the energy of all the atoms, ionsand molecules in that system. In thermo-dynamics we are concerned with change ininternal energy, ∆U, rather than the internalenergy itself. (Notice the use of ∆ to denote afinite change). We may change the internalenergy of a closed system (one that cannotexchange matter with its surroundings) inonly two ways: by transferring energy as work(w) or as heat (q). An expression for the changein internal energy is

(3.1)

If the system releases its energy to thesurroundings ∆U is negative, i.e. the totalinternal energy has been reduced. Where heatis absorbed (as in an endothermic process) theinternal energy will increase and consequentlyq is positive. Conversely, in a process whichreleases heat (an exothermic process) theinternal energy is decreased and q is negative.Similarly, when energy is supplied to thesystem as work, w is positive; and when thesystem loses energy by doing work, w isnegative.

It is frequently necessary to consider infini-tesimally small changes in a property; wedenote these by the use of d rather than ∆ .Thus for an infinitesimal change in internalenergy we write equation (3.1) as

(3.2)

We can see from this equation that it doesnot really matter whether energy is suppliedas heat or work or as a mixture of the two:the change in internal energy is the same.Equation (3.2) thus expresses the principle ofthe law of conservation of energy but is muchwider in its application since it involves

changes in heat energy, which were notencompassed in the conservation law.

It follows from equation (3.2) that a systemwhich is completely isolated from its sur-roundings, such that it cannot exchange heator interact mechanically to do work, cannotexperience any change in its internal energy.In other words the internal energy of an isolatedsystem is constant – this is the first law ofthermodynamics.

3.2.2 Enthalpy

Where a change occurs in a system at constantpressure as, for example, in a chemical reactionin an open vessel, then the increase in internalenergy is not equal to the energy supplied asheat because some energy will have been lostby the work done (against the atmosphere)during the expansion of the system. It isconvenient, therefore, to consider the heatchange in isolation from the accompanyingchanges in work. For this reason we consider aproperty that is equal to the heat supplied atconstant pressure: this property is called theenthalpy (H). We can define enthalpy by

(3.3)

∆H is positive when heat is supplied to a systemwhich is free to change its volume and negativewhen the system releases heat (as in an exo-thermic reaction). Enthalpy is related to theinternal energy of a system by the relationship

(3.4)

where p and V are respectively the pressureand volume of the system.

Enthalpy changes accompany such pro-cesses as the dissolution of a solute, the forma-tion of micelles, chemical reaction, adsorptiononto solids, vaporisation of a solvent, hydra-tion of a solute, neutralisation of acids andbases, and the melting or freezing of solutes.

3.2.3 Entropy

The first law, as we have seen, deals with theconservation of energy as the system changes

H = U + pV

∆H = q at constant pressure

d�U = d�w + d�q

∆U = w + q

58 Chapter 3 • Physicochemical properties of drugs in solution

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from one state to another, but it does notspecify which particular changes will occurspontaneously. The reason why some changeshave a natural tendency to occur is not that thesystem is moving to a lower-energy state butthat there are changes in the randomness ofthe system. This can be seen by considering aspecific example: the diffusion of one gas intoanother occurs without any external interven-tion – i.e. it is spontaneous – and yet there areno differences in either the potential or kineticenergies of the system in its equilibrium stateand in its initial state where the two gases aresegregated. The driving force for such spontan-eous processes is the tendency for an increasein the chaos of the system – the mixed systemis more disordered than the original.

A convenient measure of the randomness ordisorder of a system is the entropy (S). When asystem becomes more chaotic, its entropyincreases in line with the degree of increase indisorder caused. This concept is encapsulatedin the second law of thermodynamics whichstates that the entropy of an isolated systemincreases in a spontaneous change.

The second law, then, involves entropychange, ∆S, and this is defined as the heatabsorbed in a reversible process, q�rev, divided bythe temperature (in kelvins) at which thechange occurred.

For a finite change

(3.5)

and for an infinitesimal change

(3.6)

By a ‘reversible process’ we mean one inwhich the changes are carried out infini-tesimally slowly, so that the system is alwaysin equilibrium with its surroundings. In thiscase we infer that the temperature of thesurroundings is infinitesimally higher thanthat of the system, so that the heat changesare occurring at an infinitely slow rate, so thatthe heat transfer is smooth and uniform.

We can see the link between entropy and dis-order by considering some specific examples.

For instance, the entropy of a perfect gaschanges with its volume V according to therelationship

(3.7)

where the subscripts f and i denote the finaland initial states. Note that if V�f p V�i (i.e. if thegas expands into a larger volume) the loga-rithmic (ln) term will be positive and the equa-tion predicts an increase of entropy. This isexpected since expansion of a gas is a spon-taneous process and will be accompanied byan increase in the disorder because the mol-ecules are now moving in a greater volume.

Similarly, increasing the temperature of asystem should increase the entropy because athigher temperature the molecular motion ismore vigorous and hence the system morechaotic. The equation which relates entropychange to temperature change is

(3.8)

where C�V is the molar heat capacity at con-stant volume. Inspection of equation (3.8)shows that ∆S will be positive when T�f p T�i, aspredicted.

The entropy of a substance will also changewhen it undergoes a phase transition, sincethis too leads to a change in the order. Forexample, when a crystalline solid melts, itchanges from an ordered lattice to a morechaotic liquid (see Fig. 3.1) and consequentlyan increase in entropy is expected. Theentropy change accompanying the melting ofa solid is given by

(3.9)

where ∆H�fus is the enthalpy of fusion (melting)and T is the melting temperature. Similarly,we may determine the entropy change when aliquid vaporises from

(3.10)

where ∆H�vap is the enthalpy of vaporisationand T now refers to the boiling point. Entropy

∆S =∆H�vap

T

∆S =∆H�fus

T

∆S = C�V ln T�f

T�i

∆S = nR ln V�f

V�i

d�S =d�q�rev

T

∆S =q�rev

T


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changes accompanying other phase changes,such as change of the polymorphic form ofcrystals (see section 1.2), may be calculated ina similar manner.

At absolute zero all the thermal motions ofthe atoms of the lattice of a crystal will haveceased and the solid will have no disorder andhence a zero entropy. This conclusion formsthe basis of the third law of thermodynamics,which states that the entropy of a perfectlycrystalline material is zero when T # 0.

3.2.4 Free energy

The free energy is derived from the entropyand is, in many ways, a more useful functionto use. The free energy which is referred towhen we are discussing processes at constantpressure is the Gibbs free energy (G). This isdefined by

(3.11)

The change in the free energy at constanttemperature arises from changes in enthalpyand entropy and is

(3.12)

Thus, at constant temperature and pressure,

(3.13)

from which we can see that the change in freeenergy is another way of expressing thechange in overall entropy of a process occur-ring at constant temperature and pressure.

In view of this relationship we can nowconsider changes in free energy which occurduring a spontaneous process. From equation(3.13) we can see that ∆G will decrease duringa spontaneous process at constant temper-ature and pressure. This decrease will occuruntil the system reaches an equilibrium statewhen ∆G becomes zero. This process can bethought of as a gradual using up of thesystem’s ability to perform work as equi-librium is approached. Free energy can there-fore be looked at in another way in that itrepresents the maximum amount of work,w�max (other than the work of expansion),which can be extracted from a system under-going a change at constant temperature andpressure; i.e.

(3.14)

This nonexpansion work can be extracted fromthe system as electrical work, as in the case ofa chemical reaction taking place in an electro-chemical cell, or the energy can be storedin biological molecules such as adenosinetriphosphate (ATP).

When the system has attained an equi-librium state it no longer has the ability toreverse itself. Consequently all spontaneousprocesses are irreversible. The fact that all spon-taneous processes taking place at constanttemperature and pressure are accompanied bya negative free energy change provides a usefulcriterion of the spontaneity of any givenprocess.

By applying these concepts to chemical equi-libria we can derive (see Box 3.2) the followingsimple relationship between free energy changeand the equilibrium constant of a reversiblereaction, K:

where the standard free energy G�� is the freeenergy of 1 mole of gas at a pressure of 1 bar.

A similar expression may be derived forreactions in solutions using the activities (see

∆G�� = −RT ln K

at constant temperature and pressure∆G = w�max

∆G = −T ∆S

∆G = ∆H − T ∆S

G = H − TS


Figure 3.1 Melting of a solid involves a change from anordered arrangement of molecules, represented by (a), to amore chaotic liquid, represented by (b). As a result, themelting process is accompanied by an increase in entropy.

(a)

(b)

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section 3.3.1) of the components rather thanthe partial pressures. ∆G�� values can readily becalculated from the tabulated data and henceequation (3.19) is important because it pro-vides a method of calculating the equilibriumconstants without resort to experimentation.

A useful expression for the temperaturedependence of the equilibrium constant is thevan’t Hoff equation (equation 3.23), which maybe derived as outlined in Box 3.3. A moregeneral form of this equation is

(3.24)

Plots of log K against 1�T should be linear witha slope of 0∆H��2.303R, from which ∆H�� maybe calculated.

Equations (3.19) and (3.24) are fundamentalequations which find many applications inthe broad area of the pharmaceutical sciences:for example, in the determination of equi-librium constants in chemical reactions andfor micelle formation; in the treatment ofstability data for some solid dosage forms(see section 4.4.3); and for investigations ofdrug�–�receptor binding.

log K =−∆H��

2.303RT+ constant


Consider the following reversible reaction taking placein the gaseous phase

According to the law of mass action, the equilibriumconstant, K, can be expressed as

(3.15)

where p, terms represent the partial pressures of thecomponents of the reaction at equilibrium.The relationship between the free energy of a perfect

gas and its partial pressure is given by

(3.16)

where G�� is the free energy of 1 mole of gas at apressure of 1 bar.Applying equation (3.16) to each component of the

reaction gives

As

so

(3.17)

∆G�� is the standard free energy change of the reaction,given by

As we have noted previously, the free energy changefor systems at equilibrium is zero, and hence equation(3.17) becomes

(3.18)

Substituting from equation (3.15) gives

(3.19)

Substituting equation (3.19) into equation (3.17) gives

(3.20)

Equation (3.20) gives the change in free energy when amoles of A at a partial pressure p�A and b moles of B ata partial pressure p�B react together to yield c moles of Cat a partial pressure p�C and d moles of D at a partialpressure p�D. For such a reaction to occur spontaneously,the free energy change must be negative, and henceequation (3.20) provides a means of predicting theease of reaction for selected partial pressures (or con-centrations) of the reactants.

∆G= −RT ln K + RT ln (��p �C)��c (��p �D)��d

(��p �A)��a (��p �B)��b

∆G�� = −RT ln K

∆G�� = −RT ln (��p’ ��C)��c (��p’ ��D)��d

(��p’ ��A)��a (��p’ ��B)��b

∆G�� = cG��C + dG��D − aG��A − bG��B

∆G= ∆G�� + RT ln (��pC )��c (��pD)��d

(��pA)��a (��p B)��b

∆G = ∑G�prod − ∑G�react

etc. � bG�B = b(G��B + RT ln p�B)

aG�A = a(G��A + RT ln p�A)

∆G = G − G�� = RT ln p

K =(��p’��C)��c (��p’��D)��d

(��p’��A)��a (��p’��B)��b

aA + bB�� e ��cC + d D

Box 3.2 Relationship between free energy change and the equilibrium constant

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3.3 Activity and chemical potential

3.3.1 Activity and standard states

The term activity is used in the description ofthe departure of the behaviour of a solutionfrom ideality. In any real solution, inter-actions occur between the components whichreduce the effective concentration of the solu-tion. The activity is a way of describing thiseffective concentration. In an ideal solution orin a real solution at infinite dilution, there areno interactions between components and theactivity equals the concentration. Nonidealityin real solutions at higher concentrationscauses a divergence between the values of

activity and concentration. The ratio of theactivity to the concentration is called theactivity coefficient, γ; that is,

(3.25)

Depending on the units used to expressconcentration we can have either a molalactivity coefficient, γ�m, a molar activity co-efficient, γ�c, or, if mole fractions are used, arational activity coefficient, γ�x.

In order to be able to express the activity ofa particular component numerically, it isnecessary to define a reference state in whichthe activity is arbitrarily unity. The activity ofa particular component is then the ratio of itsvalue in a given solution to that in the refer-ence state. For the solvent, the reference stateis invariably taken to be the pure liquid and, ifthis is at a pressure of 1 atmosphere and at adefinite temperature, it is also the standardstate. Since the mole fraction as well as theactivity is unity: γ�x # 1.

Several choices are available in defining thestandard state of the solute. If the solute is aliquid which is miscible with the solvent (as,for example, in a benzene�–�toluene mixture),then the standard state is again the pureliquid. Several different standard states havebeen used for solutions of solutes of limitedsolubility. In developing a relationshipbetween drug activity and thermodynamicactivity, the pure substance has been used asthe standard state. The activity of the drug insolution was then taken to be the ratio of itsconcentration to its saturation solubility. Theuse of a pure substance as the standard state isof course of limited value since a differentstate is used for each compound. A more feasi-ble approach is to use the infinitely dilutesolution of the compound as the referencestate. Since the activity equals the concentra-tion in such solutions, however, it is not equalto unity as it should be for a standard state.This difficulty is overcome by defining thestandard state as a hypothetical solution ofunit concentration possessing, at the sametime, the properties of an infinitely dilutesolution. Some workers�4 have chosen to

γ =activity

concentration


From equation (3.19),

Since

then at a temperature T�1

(3.21)

and at temperature T�2

(3.22)

If we assume that the standard enthalpy change ∆H��

and the standard entropy change ∆S�� are independentof temperature, then subtracting equation (3.21) fromequation (3.22) gives

or

(3.23)

Equation (3.23), which is often referred to as the van’tHoff equation, is useful for the prediction of the equilib-rium constant K�2 at a temperature T�2 from its value K�1 atanother temperature T�1.

log K�2

K�1=

∆H��

2.303R (T�2 − T�1)

T�1 T�2

ln K�2 − ln K�1 = −��∆H��

R � 1

T�2−

1T�1 �

ln K�2 =−∆G��

RT�2=

−∆H��

RT�2+

∆S��

R

ln K�1 =−∆G��

RT�1=

−∆H��

RT�1+

∆S��

R

∆G�� = ∆H�� − T ∆S��

−��∆G��

RT= ln K

Box 3.3 Derivation of the van’t Hoff equation

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define the standard state in terms of an alkanesolvent rather than water; one advantage ofthis solvent is the absence of specific solute�–�solvent interactions in the reference statewhich would be highly sensitive to molecularstructure.

3.3.2 Activity of ionised drugs

A large proportion of the drugs that areadministered in aqueous solution are saltswhich, on dissociation, behave as electrolytes.Simple salts such as ephedrine hydrochloride(C�6��H�5��CH(OH)CH(NHCH�3)CH�3��HCl) are 1 : 1(or uni-univalent) electrolytes; that is, ondissociation each mole yields one cation,C�6��H�5��CH(OH)CH(N�!��H�2��CH�3)CH�3, and oneanion, Cl�0. Other salts are more complex intheir ionisation behaviour; for example,ephedrine sulfate is a 1 : 2 electrolyte, eachmole giving two moles of the cation and onemole of ions.

The activity of each ion is the product of itsactivity coefficient and its concentration, thatis

The anion and cation may each have adifferent ionic activity in solution and it is notpossible to determine individual ionic acti-vities experimentally. It is therefore necessaryto use combined terms, for example the com-bined activity term is the mean ionic activity,a�&. Similarly, we have the mean ion activitycoefficient, γ�&, and the mean ionic molality, m�&.The relationship between the mean ionicparameters is then

More details of these combined terms aregiven in Box 3.4.

Values of the mean ion activity coefficientmay be determined experimentally usingseveral methods, including electromotiveforce measurement, solubility determinationsand colligative properties. It is possible,however, to calculate γ�& in very dilute solution

using a theoretical method based on theDebye�–�Hückel theory. In this theory each ionis considered to be surrounded by an ‘atmos-phere’ in which there is a slight excess of ionsof opposite charge. The electrostatic energydue to this effect is related to the chemicalpotential of the ion to give a limiting expres-sion for dilute solutions

(3.36)

where z�! and z�0 are the valencies of the ions, Ais a constant whose value is determined by thedielectric constant of the solvent and the tem-perature (A # 0.509 in water at 298 K), and I isthe total ionic strength defined by

(3.37)

where the summation is continued over all thedifferent species in solution. It can readily beshown from equation (3.37) that for a 1 : 1electrolyte the ionic strength is equal to itsmolality; for a 1 : 2 electrolyte I # 3m; and for a2 : 2 electrolyte, I # 4m.

The Debye�–�Hückel expression as given byequation (3.36) is valid only in dilute solution(I ` 0.02 mol kg�01). At higher concentrations amodified expression has been proposed:

(3.38)

where a�i is the mean distance of approach ofthe ions or the mean effective ionic diameter,and β is a constant whose value depends onthe solvent and temperature. As an approxi-mation, the product a�i β may be taken to beunity, thus simplifying the equation. Equation(3.38) is valid for I less than 0.1 mol kg�01

EXAMPLE 3 .2 Calculation of mean ionic activ-ity coefficient

Calculate: (a) the mean ionic activity coeffi-cient and the mean ionic activity of a0.002 mol kg�01 aqueous solution of ephedrinesulfate; (b) the mean ionic activity coefficientof an aqueous solution containing 0.002mol kg�01 ephedrine sulfate and 0.01 mol kg�01

sodium chloride. Both solutions are at 25°C.

log γ�& =−Az�+��z�−√I

1 + a�i��β√I

I = �12∑(mz��2) = �12(m�1��z�21 + m�2��z�22 + ...)

− log γ�& = z�+��z�−��A��√I

γ�& =a�&

m�&

a�+ = γ�+��m�+��and��a�− = γ�− m�−

SO�2−4

Activity and chemical potential 63

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Answer(a) Ephedrine sulfate is a 1 : 2 electrolyte and

hence the ionic strength is given by equa-tion (3.37) as

From the Debye�–�Hückel equation (equa-tion 3.36),

The mean ionic activity may be calculatedfrom equation (3.35):

(b) Ionic strength of 0.01 mol kg�01 NaCl #(0.01 " 1�2) ! (0.01 " 1�2) # 0.01 mol kg�01.

γ�& = 0.743

log γ�& = −0.1288

��−log γ�& = 0.509 × 2 × √ 0.016

0.006 + 0.01 = 0.016Total ionic strength =

�12

= 0.00265

a�& = 0.834 × 0.002 × (2��2 × 1)��1�3

γ�& = 0.834

log γ�& = −0.0789

− log γ�& = 0.509 × 1 × 2 × √0.006

= 0.006 mol��kg��−1

I = �12[(0.002 × 2 × 1��2) + (0.002 × 2��2)]


In general, we will consider a strong electrolyte whichdissociates according to

where ν�! is the number of cations, C�z!, of valence z!,and ν�0 is the number of anions, A�z0, of valence z�0.The activity, a, of the electrolyte is then

(3.26)

where ν # ν�! ! ν�0.In the simple case of a solution of the 1 : 1 electrolyte

sodium chloride, the activity will be

whereas for morphine sulfate, which is a 1 : 2 elec-trolyte,

Similarly, we may also define a mean ion activitycoefficient, γ�&, in terms of the individual ionic activitycoefficients γ�! and γ�0:

(3.27)

or

(3.28)

For a 1 : 1 electrolyte equation (3.28) reduces to

(3.29)

Finally, we define a mean ionic molality, m�&, as

(3.30)

or

(3.31)

For a 1 : 1 electrolyte, equation (3.31) reduces to

(3.32)

that is, mean ionic molality may be equated with themolality of the solution.The activity of each ion is the product of its activity

coefficient and its concentration

so that

Expressed as the mean ionic parameters, we have

(3.33)

Substituting for m�&� from equation (3.31) gives

(3.34)

This equation applies in any solution, whether the ionsare added together, as a single salt, or separately as amixture of salts. For a solution of a single salt of molalitym:

Equation (3.34) reduces to

(3.35)

For example, for morphine sulfate, ν�! # 2, ν�0 # 1, andthus

γ�& =a�&

(2��2 × 1)��1�3��m=

a�&

4��1�3��m

γ�& =a�&

m(ν�ν++ ν�ν−

− )��1�ν

m�+ = ν�+��m��and��m�− = ν�−��m

γ�& =a�&

(m�ν++ m�ν−

− )��1�ν

γ�& =a�&

m�&

γ�+ =a�+

m�+��and��γ�− =

a�−

m�−

a�+ = γ�+��m�+��and��a�− = γ�−��m�−

m�+ = (m�+��m�−)��1�2 = m

m�& = (m�ν++ m −

��ν−)��1�ν

m�ν& = m�ν++ m�ν−

−

γ�& = (��γ�+��γ�−)��1�2

γ�& = (��γ�ν++ γ�ν−

− )��1�ν

γ�ν& = γ�ν++ γ�ν−

−

a = a�2morph��+ × a�SO�42− = a�3&

a = a�+Na × a�−Cl = a�&2

a = a�ν++ a�ν−

− = a�ν&

C�ν+ ��A�ν−�� 2 ��ν�+��C��z+ + ν�−��A��z−

Box 3.4 Mean ionic parameters

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3.3.3 Solvent activity

Although the phrase ‘activity of a solution’usually refers to the activity of the solute inthe solution as in the preceding section, wealso can refer to the activity of the solvent.Experimentally, solvent activity a�1 may bedetermined as the ratio of the vapour pressurep�1 of the solvent in a solution to that of thepure solvent , that is

(3.39)

where γ�1 is the solvent activity coefficient andx�1 is the mole fraction of solvent.

The relationship between the activities ofthe components of the solution is expressedby the Gibbs�–�Duhem equation

(3.40)

which provides a way of determining the

activity of the solute from measurements ofvapour pressure.

Water activity and bacterial growth

When the aqueous solution in the environ-ment of a microorganism is concentrated bythe addition of solutes such as sucrose, theconsequences for microbial growth resultmainly from the change in water activity a�w.Every microorganism has a limiting a�w belowwhich it will not grow. The minimum a�w levelsfor growth of human bacterial pathogens suchas streptococci, Klebsiella, Escherichia coli,Corynebacterium, Clostridium perfringens andother clostridia, and Pseudomonas is 0.91.�5

Staphylococcus aureus can proliferate at an a�w aslow as 0.86. Figure 3.2 shows the influence ofa�w, adjusted by the addition of sucrose, on thegrowth rate of this microorganism at 35°C andpH 7.0. The control medium, with a water

x�1 d(ln a�1) + x�2 d(ln a�2) = 0

a�1 =p�1

p��1= γ�1��x�1

p��1


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

0

2

3

4

5

6

7

8

9

aw = 0.993

aw = 0.885

aw = 0.867

aw = 0.858

10

24 48 72 96 120 144 168 192 216

Log

[num

ber

of c

ells/

ml]

Time (h)

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activity value of a�w # 0.993, supported rapidgrowth of the test organism. Reduction of a�wof the medium by addition of sucrose progres-sively increased generation times and lagperiods and lowered the peak cell counts.Complete growth inhibition was achieved atan a�w 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 explainwhy the old remedy of treating infectedwounds with sugar, honey or molasses issuccessful. When the wound is filled withsugar, the sugar dissolves in the tissue water,creating an environment of low a�w, whichinhibits bacterial growth. However, the differ-ence in water activity between the tissue andthe concentrated sugar solution causes migra-tion of water out of the tissue, hence dilutingthe sugar and raising a�w. Further sugar mustthen be added to the wound to maintaingrowth inhibition. Sugar may be applied as apaste with a consistency appropriate to thewound characteristics; thick sugar paste is suit-able for cavities with wide openings, a thinnerpaste with the consistency of thin honeybeing more suitable for instillation intocavities with small openings.

An in vitro study has been reported�6 of theefficacy of such pastes, and also of those pre-pared using xylose as an alternative to sucrose,in inhibiting the growth of bacteria com-monly present in infected wounds. Poly-ethylene glycol was added to the pastes as alubricant and hydrogen peroxide was includedin the formulation as a preservative. To simu-late the dilution that the pastes invariablyexperience as a result of fluid being drawn intothe wound, serum was added to the formula-tions in varying amounts. Figure 3.3 illustratesthe effects of these sucrose pastes on thecolony-forming ability of Proteus mirabilis andshows the reduction in efficiency of the pastesas a result of dilution and the consequentincrease of their water activity (see Fig. 3.4). Itis clear that P. mirabilis was susceptible to theantibacterial activity of the pastes, even whenthey were diluted by 50%. It was reported thatalthough a�w may not be maintained at less

than 0.86 (the critical level for inhibition ofgrowth of S. aureus) for more than 3 hoursafter packing of the wound, nevertheless clini-cal experience had shown that twice-dailydressing was adequate to remove infectedslough from dirty wounds within a few days.

3.3.4 Chemical potential

Properties such as volume, enthalpy, freeenergy and entropy, which depend on thequantity of substance, are called extensiveproperties. In contrast, properties such as tem-perature, density and refractive index, whichare independent of the amount of material,are referred to as intensive properties. Thequantity denoting the rate of increase in themagnitude of an extensive property withincrease in the number of moles of a substanceadded to the system at constant temperature


Figure 3.3 The effects of sucrose pastes diluted with serumon the colony-forming ability of P. mirabilis.Reproduced from reference 6 with permission.

0

0.1

1

10

100

1000

40 80 120

Time (min)

CFU

/ml a

s %

of i

nitia

l ino

culu

m100% paste

75% paste with serum

50% pastewith serum

Controls

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and pressure is termed a partial molar quantity.Such quantities are distinguished by a barabove the symbol for the particular property.For example,

(3.41)

Note the use of the symbol ∂ to denote a partialchange which, in this case, occurs under con-ditions of constant temperature, pressure andnumber of moles of solvent (denoted by thesubscripts outside the brackets).

In practical terms the partial molar volume,, represents the change in the total volume of

a large amount of solution when one addi-tional mole of solute is added – it is the effectivevolume of 1 mole of solute in solution.

Of particular interest is the partial molar freeenergy, , which is also referred to as thechemical potential, µ, and is defined for compo-nent 2 in a binary system by

(3.42)

Partial molar quantities are of importance inthe consideration of open systems, that isthose involving transference of matter as wellas energy. For an open system involving twocomponents

(3.43)

At constant temperature and pressure equa-tion (3.43) reduces to

(3.44)

(3.45)

The chemical potential therefore representsthe contribution per mole of each componentto the total free energy. It is the effective freeenergy per mole of each component in themixture and is always less than the free energyof the pure substance.

It can readily be shown (see Box 3.5) thatthe chemical potential of a component in atwo-phase system (for example, oil and water),at equilibrium at a fixed temperature and pres-sure, is identical in both phases. Because of theneed for equality of chemical potential atequilibrium, a substance in a system which isnot at equilibrium will have a tendency todiffuse spontaneously from a phase in whichit has a high chemical potential to another inwhich it has a low chemical potential. In thisrespect the chemical potential resembles elec-trical potential; hence its name is an aptdescription of its nature.

Chemical potential of a component in solution

Where the component of the solution is a non-electrolyte, its chemical potential in dilute solu-tion at a molality m, can be calculated from

where

and M�1 # molecular weight of the solvent.

µ�� = µ��2 + RT ln M�1 − RT ln 1000

µ�2 = µ�� + RT ln m

∴ G = ∫ d�G = µ�1��n�1 + µ�2��n�2

d�G = µ�1 d�n�1 + µ�2 d�n�2

+ � ∂G

∂n�1 �T,��P,��n��2

��d�n�1 + � ∂G

∂n�2 �T,��P,��n��1

��d�n�2

d�G�� = ��∂G

∂T �P,��n��1,��n��2

��d�T + �∂G

∂P �T,��n��1,��n��2

��d�P

� ∂G

∂n�2 �T, P, n��1

= IG�2 = µ�2

IG

KV

� ∂V

∂n�2 �T,��P,��n��1

= KV�2


Figure 3.4 Effects on a�w of adding xylose (solid symbols)and sucrose paste (open symbols) to serum.Reproduced from reference 6 with permission.

0 20 40 60 80 100

Percentage paste in serum(v/v)

0.5

0.6

0.7

0.8

0.9

1.0

a w

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NonelectrolytesIn dilute solutions of nonvolatile solutes, Raoult’s law(see section 2.3.1) can usually be assumed to beobeyed and the chemical potential of the solute is givenby equation (3.50):

(3.50)

It is usually more convenient to express solute concen-tration as molality, m, rather than mole fraction, using

where M�1 # molecular weight of the solvent. Thus

(3.51)

where

At higher concentrations, the solution generally exhibitssignificant deviations from Raoult’s law and mole frac-tion must be replaced by activity:

(3.52)

or

(3.53)

ElectrolytesThe chemical potential of a strong electrolyte, whichmay be assumed to be completely dissociated in

solution, is equal to the sum of the chemical potentialsof each of the component ions. Thus

(3.54)

and

(3.55)

and therefore

(3.56)

where is the sum of the chemical potentials of theions, each in their respective standard state, i.e.

where ν�! and ν�0 are the number of cations and anions,respectively, and a is the activity of the electrolyte asgiven in section 3.3.2.

For example, for a 1 : 1 electrolyte, from equation(3.26),

Therefore

From equation (3.33),

∴��µ�2 = µ��2 + 2RT ln mγ�&

a�& = mγ�&

µ�2 = µ��2 + 2RT ln a�&

a = a�2&

µ��2 = ν�+��µ��+ + ν�−��µ��−

µ��2

µ�2 = µ��2 + RT ln a�

µ�− = µ��− + RT ln a�−

µ�+ = µ��+ + RT ln a�+

µ�2 = µ��2 + RT ln γ�2 + RT ln x�2

µ�2 = µ��2 + RT ln a�2

µ�� = µ��2 + RT ln M�1 − RT ln 1000

µ�2 = µ�� + RT ln m

x�2 =mM�1

1000

µ�2 = µ��2 + RT ln x�2

Box 3.6 Chemical potential of a component in solution

Consider a system of two phases, a and b, in equi-librium at constant temperature and pressure. If a smallquantity of substance is transferred from phase a tophase b, then, because the overall free energy changeis zero, we have

(3.46)

where dG�a and dG�b are the free energy changesaccompanying the transfer of material for each phase.From equation (3.44),

and thus

(3.47)

A decrease of dn moles of component in phase a leadsto an increase of exactly dn moles of this component inphase b, that is

(3.48)

Substitution of equation (3.48) into equation (3.47)leads to the result

(3.49)

In general, the chemical potential of a component isidentical in all the phases of a system at equilibrium at afixed temperature and pressure.

µ�a = µ�b

d�n�a = −d�n�b

µ�a d�n�a + µ�b d�n�b = 0

d�G�a = µ�a d�n�a��and��d�G� b = µ�b d�n�b

d�G�a + d�G� b = 0

Box 3.5 Chemical potential in two-phase systems

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In the case of strong electrolytes, the chemicalpotential is the sum of the chemical potentialsof the ions. For the simple case of a 1 : 1 elec-trolyte, the chemical potential is given by

The derivations of these equations are given inBox 3.6.

3.4 Osmotic properties of drugsolutions

A nonvolatile solute added to a solvent affectsnot only the magnitude of the vapour pressureabove the solvent but also the freezing pointand the boiling point to an extent that isproportional to the relative number of solutemolecules present, rather than to the weightconcentration of the solute. Properties that aredependent on the number of molecules insolution in this way are referred to as colligativeproperties, and the most important of suchproperties from a pharmaceutical viewpoint isthe osmotic pressure.

3.4.1 Osmotic pressure

Whenever a solution is separated from asolvent by a membrane that is permeable onlyto solvent molecules (referred to as a semi-permeable membrane), there is a passage ofsolvent across the membrane into the solution.This is the phenomenon of osmosis. If the solu-tion is totally confined by a semipermeablemembrane and immersed in the solvent, thena pressure differential develops across themembrane, which is referred to as the osmoticpressure. Solvent passes through the membranebecause of the inequality of the chemicalpotentials on either side of the membrane.Since the chemical potential of a solvent mol-ecule in solution is less than that in puresolvent, solvent will spontaneously enter thesolution until this inequality is removed. Theequation which relates the osmotic pressure ofthe solution, Π, to the solution concentration

is the van’t Hoff equation:

(3.57)

On application of the van’t Hoff equation tothe drug molecules in solution, considerationmust be made of any ionisation of the mol-ecules, since osmotic pressure, being a colli-gative property, will be dependent on the totalnumber of particles in solution (including thefree counterions). To allow for what was at thetime considered to be anomalous behaviour ofelectrolyte solutions, van’t Hoff introduced acorrection factor, i. The value of this factorapproaches a number equal to that of thenumber of ions, ν, into which each moleculedissociates as the solution is progressivelydiluted. The ratio i�ν is termed the practicalosmotic coefficient, φ.

For nonideal solutions, the activity andosmotic pressure are related by the expression

(3.58)

where M�1 is the molecular weight of thesolvent and m is the molality of the solution.The relationship between the osmotic pressureand the osmotic coefficient is thus

(3.59)

where �1��is the partial molal volume of thesolvent.

3.4.2 Osmolality and osmolarity

The experimentally derived osmotic pressureis frequently expressed as the osmolality ξ�m,which is the mass of solute which, when dis-solved in 1 kg of water, will exert an osmoticpressure, Π ,, equal to that exerted by 1 mole ofan ideal unionised substance dissolved in 1 kgof water. The unit of osmolality is the osmole(abbreviated as osmol), which is the amountof substance that dissociates in solution toform one mole of osmotically active particles,thus 1 mole of glucose (not ionised) forms 1osmole of solute, whereas 1 mole of NaClforms 2 osmoles (1 mole of Na�! and 1 mole of

KV

Π = �RT

KV�1 � νmM�1

1000 φ

ln a�1 =−νmM�1

1000 φ

ΠV = n�2��RT

µ�2 = µ��2 + 2RT ln mγ�&

Osmotic properties of drug solutions 69

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Cl�0). In practical terms, this means that a 1molal solution of NaCl will have (approxi-mately) twice the osmolality (osmotic pres-sure) as a 1 molal solution of glucose.

According to the definition, ξ�m # Π�Π,. Thevalue of Π, may be obtained from equation(3.59) by noting that for an ideal unionisedsubstance ν # φ # 1, and since m is also unity,equation (3.59) becomes

Thus

(3.60)

EXAMPLE 3.3 Calculation of osmolality

A 0.90% w�w solution of sodium chloride(mol. wt. # 58.5) has an osmotic coefficient of0.928. Calculate the osmolality of the solution.

AnswerOsmolality is given by equation (3.60) as

so

Pharmaceutical labelling regulations some-times require a statement of the osmolarity;for example, the USP 27 requires that sodiumchloride injection should be labelled in thisway. Osmolarity is defined as the mass of solutewhich, when dissolved in 1 litre of solution,will exert an osmotic pressure equal to thatexerted by 1 mole of an ideal unionised sub-stance dissolved in 1 litre of solution. The rela-tionship between osmolality and osmolarityhas been discussed by Streng et al.�7

Table 3.1 lists the osmolalities of commonlyused intravenous fluids.

3.4.3 Clinical relevance of osmotic effects

Osmotic effects are particularly importantfrom a physiological viewpoint since bio-

logical membranes, notably the red blood cellmembrane, behave in a manner similar to thatof semipermeable membranes. Consequently,when red blood cells are immersed in asolution of greater osmotic pressure than thatof their contents, they shrink as water passesout of the cells in an attempt to reduce thechemical potential gradient across the cellmembrane. Conversely, on placing the cells inan aqueous environment of lower osmoticpressure, the cells swell as water enters andeventually lysis may occur. It is an importantconsideration, therefore, to ensure that theeffective osmotic pressure of a solution forinjection is approximately the same as that ofblood serum. This effective osmotic pressure,which is termed the tonicity, is not alwaysidentical to the osmolality because it is con-cerned only with those solutes in solution thatcan exert an effect on the passage of waterthrough the biological membrane. Solutionsthat have the same tonicity as blood serum are

ξ�m = 2 ×9.0

58.5× 0.928 = 286 mosmol kg��−1

ξ�m = νmφ

ξ�m = νmφ

Π’ = �RT

KV�1 � M�1

1000


Table 3.1 Tonicities (osmolalities) of intravenousfluids

Solution Tonicity(mosmol kg�01)

Vamin 9 0700Vamin 9 Glucose 1350Vamin14 1145Vamin 14 Electrolyte-free 0810Vamin 18 Electrolyte-free 1130Vaminolact 0510Vitrimix KV 1130Intralipid 10% Novum 0300Intralipid 20% 0350Intralipid 30% 0310Intrafusin 22 1400Hyperamine 30 1450Gelofusine 00279�a

Hyperamine 30 1450Lipofundin MCT�LCT 10% 00345�a

Lipofundin MCT�LCT 20% 00380�a

Nutriflex 32 01400�a



Sodium Bicarbonate Intravenous Infusion BP8.4% w�v 02000�a

4.2% w�v 01000�a

�a Osmolarity (mosmol dm�03).

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said to be isotonic with blood. Solutions with ahigher tonicity are hypertonic and those with alower tonicity are termed hypotonic solutions.Similarly, in order to avoid discomfort onadministration of solutions to the delicatemembranes of the body, such as the eyes,these solutions are made isotonic with therelevant tissues.

The osmotic pressures of many of theproducts of Table 3.1 are in excess of that ofplasma (291 mosmol dm�03). It is generallyrecommended that any fluid with an osmoticpressure above 550 mosmol dm�03 should notbe infused rapidly as this would increase theincidence of venous damage. The rapidinfusion of marginally hypertonic solutions(in the range 300�–�500 mosmol dm�03) wouldappear to be clinically practicable; the higherthe osmotic pressure of the solution withinthis range, the slower should be its rate ofinfusion to avoid damage. Patients with cen-trally inserted lines are not normally affectedby limits on tonicity as infusion is normallyslow and dilution is rapid.

Certain oral medications commonly used inthe intensive care of premature infants havevery high osmolalities. The high tonicity ofenteral feedings has been implicated as a causeof necrotising enterocolitis (NEC). A higherfrequency of gastrointestinal illness including

NEC has been reported�8 among prematureinfants fed undiluted calcium lactate thanamong those fed no supplemental calcium orcalcium lactate eluted with water or formula.White and Harkavy�9 have discussed a similarcase of the development of NEC followingmedication with calcium glubionate elixir.These authors have measured osmolalities ofseveral medications by freezing point depres-sion and compared these with the osmolalitiesof analogous intravenous (i.v.) preparations(see Table 3.2). Except in the case of digoxin,the osmolalities of the i.v. preparations werevery much lower than those of the corres-ponding oral preparations despite the fact thatthe i.v. preparations contained at least asmuch drug per millilitre as did the oral forms.This striking difference may be attributed tothe additives, such as ethyl alcohol, sorbitoland propylene glycol, which make a largecontribution to the osmolalities of the oralpreparations. The vehicle for the i.v. digoxinconsists of 40% propylene glycol and 10%ethyl alcohol with calculated osmolalities of5260 and 2174 mosmol kg�01 respectively, thusexplaining the unusually high osmolality ofthis i.v. preparation. These authors have rec-ommended that extreme caution should beexercised in the administration of these oralpreparations and perhaps any medication in a


Table 3.2 Measured and calculated osmolalities of drugs�a

Drug (route) Concentration Mean measured Calculated availableof drug osmolality milliosmoles in

(mosmol kg�01) 1 kg of drugpreparation�b

Theophylline elixir (oral) 80 mg�15 cm�3 p3000 4980Aminophylline (i.v.) 25 mg cm�03 00116 0200Calcium glubionate (oral) 115 mg�5 cm�3 p3000 2270Calcium gluceptate (i.v.) 90 mg�5 cm�3 00507 0950Digoxin elixir 25 mg dm�03 p3000 4420Digoxin (i.v.) 100 mg dm�03 p3000 9620Dexamethasone elixir (oral) 0.5 mg�5 cm�3 p3000 3980Dexamethasone sodium phosphate (i.v.) 4 mg cm�03 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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syrup or elixir form when the infant is at riskfrom necrotising enterocolitis. In some casesthe osmolality of the elixir is so high that evenmixing with infant formula does not reducethe osmolality to a tolerable level. For example,when a clinically appropriate dose of dexa-methasone elixir was mixed in volumes offormula appropriate for a single feeding for a1500 g infant, the osmolalities of the mixesincreased by at least 300% compared toformula alone (see Table 3.3).

Volatile anaestheticsThe aqueous solubilities of several volatileanaesthetics can be related to the osmolarityof the solution.�10 The inverse relationshipbetween solubility (expressed as the liquid�gaspartition coefficient) of those anaesthetics andthe osmolarity is shown in Table 3.4.

These findings have practical applicationsfor the clinician. Although changes in serum

osmolarity within the physiological range(209�–�305 mosmol dm�03) have only a smalleffect on the liquid�gas partition coefficient,changes in the serum osmolarity and theconcentration of serum constituents at theextremes of the physiological range may sig-nificantly decrease the liquid�gas partitioncoefficient. For example, the blood�gas parti-tion coefficient of isoflurane decreases signifi-cantly after an infusion of mannitol. This maybe attributed to both a transient increase inthe osmolarity of the blood and a more pro-longed decrease in the concentration of serumconstituents caused by the influx of water dueto the osmotic gradient.

Rehydration solutionsAn interesting application of the osmoticeffect has been in the design of rehydrationsolutions. During the day the body movesmany litres of fluid from the blood into the


Table 3.3 Osmolalities of drug-infant formula mixtures�a

Drug (dose) Volume of drug (cm�3) Mean measured osmolality! volume of formula (cm�3) (mosmol kg�01)

Infant formula �–� 0292Theophylline elixir, 1 mg kg�01 00.3 ! 15 0392

00.3 ! 30 0339Calcium glubionate syrup, 0.5 mmol kg�01 00.5 ! 15 0378

00.5 ! 30 0330Digoxin elixir, 5 µg kg�01 0.15 ! 15 0347

0.15 ! 30 0322Dexamethasone elixir, 0.25 mg kg�01 03.8 ! 15 1149

03.8 ! 30 0791

�a Reproduced from reference 9.

Table 3.4 Liquid�gas partition coefficients of anaesthetics in four aqueous solutions at 37°C�a

Solution Osmolarity Partition coefficient(mosmol dm�03)

Isoflurane Enflurane Halothane Methoxyflurane

Distilled H�2��O 0000 0.626 & 0.050 0.754 & 0.060 0.859 & 0.02 4.33 & 0.50Normal saline 0308 0.590 & 0.010 0.713 & 0.010 0.825 & 0.02 4.22 & 0.30Isotonic heparin 0308 0.593 & 0.010 0.715 & 0.0100 �–� 4.08 & 0.22

(1000 U cm�03)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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intestine and back again. The inflow of waterinto the intestine, which aids the breakdownof food, is an osmotic effect arising from thesecretion of Cl�0 ions by the crypt cells of theintestinal lining (see section 9.2.2) into theintestine. Nutrients from the food are taken upby the villus cells in the lining of the smallintestine. The villus cells also absorb Na�! ions,which they pump out into the extracellularspaces, from where they return to the circu-lation. As a consequence of this flow of Na�!,water and other ions follow by osmotic flowand hence are also transferred to the blood.This normal functioning is disrupted bydiarrhoea-causing microorganisms whicheither increase the Cl�0-secreting activity of thecrypt cells or impair the absorption of Na�! bythe villus cells, or both. Consequently, thefluid that is normally returned to the bloodacross the intestinal wall is lost in waterystool. If untreated, diarrhoea can eventuallylead to a severe decline in the volume of theblood, the circulation may become danger-ously slow, and death may result.

Oral rehydration therapyTreatment of dehydration by oral rehydrationtherapy (ORT) is based on the discovery thatthe diarrhoea-causing organisms do not usuallyinterfere with the carrier systems which bringsodium and glucose simultaneously into thevillus cells from the intestinal cavity. This ‘co-transport’ system only operates when bothsodium and glucose are present. The principlebehind ORT is that if glucose is mixed into anelectrolyte solution it activates the co-transportsystem, causing electrolyte and then water topass through the intestinal wall and to enterthe blood, so minimising the dehydration.

ORT requires administration to the patientof small volumes of fluid throughout the day(to prevent vomiting); it does not reduce theduration or severity of the diarrhoea, it simplyreplaces lost fluid and electrolytes. Let usexamine, using the principles of the osmoticeffect, two possible methods by which theprocess of fluid uptake from the intestinemight be speeded up. It might seem reason-able to suggest that more glucose should beadded to the formulation in an attempt to

enhance the co-transport system. If this isdone, however, the osmolarity of the glucosewill become greater than that of normalblood, and water would now flow from theblood to the intestine and so exacerbate theproblem. An alternative is to substitutestarches for simple glucose in the ORT. Whenthese polymer molecules are broken down inthe intestinal lumen they release many hun-dreds of glucose molecules, which are imme-diately taken up by the co-transport systemand removed from the lumen. The effect istherefore as if a high concentration of glucosewere administered, but because osmotic pres-sure is a colligative property (dependent on thenumber of molecules rather than the mass ofsubstance), there is no associated problem of ahigh osmolarity when starches are used. Theprocess is summarised in Fig. 3.5. A similareffect is achieved by the addition of proteins,since there is also a co-transport mechanismwhereby amino acids (released on breakdownof the proteins in the intestine) and Na�! ionsare simultaneously taken up by the villus cells.

This process of increasing water uptake fromthe intestine has an added appeal since thesource of the starch and protein can be cereals,beans and rice, which are likely to be availablein the parts of the world where problemsarising from diarrhoea are most prevalent.Food-based ORT offers additional advantages:it can be made at home from low-cost ingre-dients and can be cooked, which kills thepathogens in water.

3.4.4 Preparation of isotonic solution

Since osmotic pressure is not a readily measur-able quantity, it is usual to make use of therelationship between the colligative propertiesand to calculate the osmotic pressure from amore easily measured property such as thefreezing point depression. In so doing, how-ever, it is important to realise that the redblood cell membrane is not a perfect semi-permeable membrane and allows throughsmall molecules such as urea and ammoniumchloride. Therefore, although the quantity ofeach substance required for an isotonic


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solution may be calculated from freezingpoint depression values, these solutions maycause cell lysis when administered.

It has been shown that a solution which isisotonic with blood has a freezing point

depression, ∆T�f, of 0.52°C. One has thereforeto adjust the freezing point of the drugsolution to this value to give an isotonic solu-tion. Freezing point depressions for a series ofcompounds are given in reference texts�1, 11


Figure 3.5 How osmosis affects the performance of solutions used in oral rehydration therapy (ORT).

ORT solution inlumen of intestine

Solute

SoluteGlucose-inducedsodium transport

Dehydratedblood

Osmotic flow ofwater and ions

Standard ORT(osmolarity equals the normalosmolarity of blood)

Effect: Co-transport ofglucose and sodium induces abloodward osmotic flow ofwater, which drags alongadditional ions. ORT exactlyreplaces water, sodium andother ions lost from the bloodbut does not reduce the extentor duration of diarrhoea.

If extra glucose is added(high osmolarity)

Effect: Solution isunacceptable becauseosmotic flow yields a netloss of water and ionsfrom the blood – an osmoticpenalty. Dehydration andrisk of death increase.

Food-based ORT(low osmolarity)

Effect: Each polymer has thesame osmotic effect as asingle glucose or amino acidmolecule but markedlyenhances nutrient-inducedsodium transport when thepolymer is broken apart at thevillus cell surface. (Rapid uptakeat the surface avoids an osmoticpenalty.) Water and ions arereturned to the blood quickly, andless of both are lost in the stool.The extent and duration of diarrhoeaare reduced.

Undesirableosmotic flow

Glucose-inducedsodium transport

Glucose-inducedor amino acid-induced sodiumtransport

Osmotic flow

Protein

Starch

Cation

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and it is a simple matter to calculate the con-centration required for isotonicity from thesevalues. For example, a 1% NaCl solution has afreezing point depression of 0.576°C. Thepercentage concentration of NaCl required tomake isotonic saline solution is therefore(0.52�0.576) " 1.0 # 0.90% w�v.

With a solution of a drug, it is not of coursepossible to alter the drug concentration in thismanner, and an adjusting substance must beadded to achieve isotonicity. The quantity ofadjusting substance can be calculated as shownin Box 3.7

EXAMPLE 3 .4 Isotonic solutions

Calculate the amount of sodium chloridewhich should be added to 50 cm�3 of a0.5% w�v solution of lidocaine hydrochlorideto make a solution isotonic with blood serum.

AnswerFrom reference lists, the values of b for sodiumchloride and lidocaine hydrochloride are0.576°C and 0.130°C, respectively.

From equation (3.61) we have

Therefore,

Therefore, the weight of sodium chloride to beadded to 50 cm�3 of solution is 0.395 g.

3.5 Ionisation of drugs in solution

Many drugs are either weak organic acids (forexample, acetylsalicylic acid [aspirin]) or weakorganic bases (for example, procaine), or theirsalts (for example, ephedrine hydrochloride).The degree to which these drugs are ionised insolution is highly dependent on the pH. Theexceptions to this general statement are thenonelectrolytes, such as the steroids, and thequaternary ammonium compounds, whichare completely ionised at all pH values and inthis respect behave as strong electrolytes. Theextent of ionisation of a drug has an impor-tant effect on its absorption, distribution andelimination and there are many examples ofthe alteration of pH to change these prop-erties. The pH of urine may be adjusted (forexample by administration of ammoniumchloride or sodium bicarbonate) in cases ofoverdosing with amfetamines, barbiturates,narcotics and salicylates, to ensure that thesedrugs are completely ionised and hencereadily excreted. Conversely, the pH of theurine may be altered to prevent ionisation of adrug in cases where reabsorption is requiredfor therapeutic reasons. Sulphonamide crystal-luria may also be avoided by making the urinealkaline. An understanding of the relationshipbetween pH and drug ionisation is of use inthe prediction of the causes of precipitation inadmixtures, in the calculation of the solubilityof drugs and in the attainment of optimumbioavailability by maintaining a certain ratioof ionised to unionised drug. Table 3.5 showsthe nominal pH values of some body fluidsand sites, which are useful in the prediction ofthe percentage ionisation of drugs in vivo.

3.5.1 Dissociation of weakly acidic and basicdrugs and their salts

According to the Lowry�–�Brønsted theory ofacids and bases, an acid is a substance whichw =

0.52 − 0.065

0.576= 0.790 g

a = 0.5 × 0.130 = 0.065

Ionisation of drugs in solution 75

If the drug concentration is x g per 100 cm�3 solution,then

Similarly, if w is the weight in grams of adjustingsubstance to be added to 100 cm�3 of drug solutionto achieve isotonicity, then

For an isotonic solution,

(3.61) ∴��w =0.52 − a

b

a + (w × b) = 0.52

�� = w × b

�� = w × (∆T�f of 1% adjusting substance)

∆T�f for adjusting solution

= x × (∆T�f of 1% drug solution) = a∆T�f for drug solution

Box 3.7 Preparation of isotonic solutions

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will donate a proton and a base is a substancewhich will accept a proton. Thus the disso-ciation of acetylsalicylic acid, a weak acid,could be represented as in Scheme 3.1. In thisequilibrium, acetylsalicylic acid acts as anacid, because it donates a proton, and the

acetylsalicylate ion acts as a base, because itaccepts a proton to yield an acid. An acid andbase represented by such an equilibrium issaid to be a conjugate acid�–�base pair.

Scheme 3.1 is not a realistic expression,however, since protons are too reactive toexist independently and are rapidly taken upby the solvent. The proton-accepting entity,by the Lowry�–�Brønsted definition, is a base,and the product formed when the proton hasbeen accepted by the solvent is an acid. Thus asecond acid�–�base equilibrium occurs when thesolvent accepts the proton, and this may berepresented by

The overall equation on summing these equa-tions is shown in Scheme 3.2, or, in general,

By similar reasoning, the dissociation ofbenzocaine, a weak base, may be representedby the equilibrium

NH�2��C�6��H�5��COOC�2��H�5 ! H�2��Obase 1 acid 2

e NH C�6��H�5��COOC�2��H�5 ! OH�0

acid 1 base 2

or, in general,

Comparison of the two general equationsshows that H�2��O can act as either an acid or abase. Such solvents are called amphiproticsolvents.

B + H�2��O�� e ��BH��+ + OH��−

�+3

HA + H�2��O�� e ��A��− + H�3��O��+

H�2��O + H��+�� e ��H�3��O��+


Scheme 3.1

COOH

OOC

� H�

CH3

COO�

OOC CH3

Scheme 3.2

� H3O�

COO�

OOC CH3

�

COOH

OOC CH3

H2O

Acid 1 Base 2 Base 1 Acid 2

Table 3.5 Nominal pH values of some body fluidsand sites�a

Site Nominal pH

Aqueous humour 7.21Blood, arterial 7.40Blood, venous 7.39Blood, maternal umbilical 7.25Cerebrospinal fluid 7.35Duodenum 5.50Faeces�b 7.15Ileum, distal 8.00Intestine, microsurface 5.30Lacrimal fluid (tears) 7.40Milk, breast 7.00Muscle, skeletal�c 6.00Nasal secretions 6.00Prostatic fluid 6.45Saliva 6.40Semen 7.20Stomach 1.50Sweat 5.40Urine, female 5.80Urine, male 5.70Vaginal secretions, premenopause 4.50Vaginal secretions, postmenopause 7.00

�a Reproduced from D. W. Newton and R. B. Kluza, Drug Intell. Clin.

Pharm., 12, 547 (1978).�b Value for normal soft, formed stools, hard stools tend to be more

alkaline, whereas watery, unformed stools are acidic.�c Studies conducted intracellularly in the rat.

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Salts of weak acids or bases are essentiallycompletely ionised in solution. For example,ephedrine hydrochloride (salt of the weak baseephedrine, and the strong acid HCl) exists inaqueous solution in the form of the conjugateacid of the weak base, C�6��H�5��CH(OH)CH(CH�3)N�!��H�2��CH�3, together with its Cl�0 counterions.In a similar manner, when sodium salicylate(salt of the weak acid salicylic acid, and thestrong base NaOH) is dissolved in water, itionises almost entirely into the conjugate baseof salicylic acid, HOC�6��H�5��COO�0, and Na�! ions.

The conjugate acids and bases formed inthis way are, of course, subject to acid�–�baseequilibria described by the general equationsabove.

3.5.2 The effect of pH on the ionisation ofweakly acidic or basic drugs and theirsalts

If the ionisation of a weak acid is representedas described above, we may express an equi-librium constant as follows:

(3.62)

Assuming the activity coefficients approachunity in dilute solution, the activities may bereplaced by concentrations:

(3.63)

K�a is variously referred to as the ionisationconstant, dissociation constant, or acidity con-stant for the weak acid. The negative logarithmof K�a is referred to as pK�a, just as the negativelogarithm of the hydrogen ion concentrationis called the pH. Thus

(3.64)

Similarly, the dissociation constant or basicityconstant for a weak base is

(3.65)

and

(3.66)

The pK�a and pK�b values provide a convenientmeans of comparing the strengths of weakacids and bases. The lower the pK�a, thestronger the acid; the lower the pK�b, thestronger is the base. The pK�a values of a seriesof drugs are given in Table 3.6. pK�a and pK�bvalues of conjugate acid�–�base pairs are linked

pK�b = −log K�b

K�b =a�OH�− × a�BH�+

a�B�� Q ��

[OH��−] [BH��+][B]

pK�a = −log K�a

K�a =[H�3��O��+] [A��−]

[HA]

K�a =a�H��3O

�+ × a�A�−

a�HA


Weak acidsTaking logarithms of the expression for the dissociationconstant of a weak acid (equation 3.63)

which can be rearranged to

(3.70)

Equation (3.70) may itself be rearranged to facilitate thedirect determination of the molar percentage ionisationas follows:

Therefore,

(3.71)

Weak basesAn analogous series of equations for the percentageionisation of a weak base may be derived as follows.Taking logarithms of equation (3.65) and rearranginggives

Therefore,

(3.72)

Rearranging to facilitate calculation of the percentageionisation leads to

(3.73)

=100

1 + antilog(��pH − pK� w + pK� b)percentage ionisation

pH = pK� w − pK� b − log [BH��+]

[B]

−log K� b = −log [OH��−] − log [BH��+]

[B]

=100

1 + antilog (pK�a − pH)percentage ionisation

=[A��−]

[HA] + [A��−]× 100percentage ionisation

[HA] = [A��−] antilog(pK�a − pH)

pH = pK�a + log [A��−]

[HA]

−log K�a = −log [H�3��O��+] − log [A��−]

[HA]

Box 3.8 The degree of ionisation of weak acids andbases

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Table 3.6 pK�a values of some medicinal compounds�a

Compound pK�a

Acid Base

Acebutolol �–� 9.2Acetazolamide �–� 7.2, 9.0Acetylsalicylic acid 3.5 �–�Aciclovir �–� 2.3, 9.3Adrenaline 9.9 8.5Adriamycin �–� 8.2Allopurinol 9.4 (10.2)�b –Alphaprodine �–� 8.7Alprenolol �–� 9.6Amikacin �–� 8.1p-Aminobenzoic acid 4.9 2.4Aminophylline �–� 5.0Amitriptyline �–� 9.4Amiodarone �–� 6.6Amoxicillin 2.4, 7.4, 9.6 �–�Ampicillin 2.5 7.2Apomorphine 8.9 7.0Atenolol �–� 9.6Ascorbic acid 4.2, 11.6 �–�Atropine �–� 9.9Azapropazone `1.8 6.6Azathioprine 8.2 �–�Benzylpenicillin 2.8 �–�Benzocaine �–� 2.8Bupivacaine �–� 8.1Captopril 3.5 �–�Cefadroxil 7.6 2.7Cefalexin 7.1 2.3Cefaclor 7.2 2.7Celiprolol �–� 9.7Cetirizine 2.9 2.2, 8.0Chlorambucil 4.5 (4.9)�b 2.5Chloramphenicol �–� 5.5Chlorcyclizine �–� 8.2Chlordiazepoxide �–� 4.8Chloroquine �–� 8.1, 9.9Chlorothiazide 6.5 9.5Chlorphenamine �–� 9.0Chlorpromazine �–� 9.3Chlorpropamide �–� 4.9Chlorprothixene �–� 8.8Cimetidine �–� 6.8Cinchocaine �–� 8.3Clindamycin �–� 7.5Cocaine �–� 8.5Codeine �–� 8.2Cyclopentolate �–� 7.9Daunomycin �–� 8.2Desipramine �–� 10.2Dextromethorphan �–� 8.3

Compound pK�a

Acid Base

Diamorphine �–� 7.6Diazepam �–� 3.4Diclofenac 4.0 �–�Diethylpropion �–� 8.7Diltiazem �–� 8.0Diphenhydramine �–� 9.1Disopyramide �–� �–�Dithranol �–� 9.4Doxepin �–� 8.0Doxorubicin �–� 8.2, 10.2Doxycycline 7.7 3.4, 9.3Enalapril �–� 5.5Enoxacin 6.3 8.6Ergometrine �–� 6.8Ergotamine �–� 6.4Erythromycin �–� 8.8Famotidine �–� 6.8Fenoprofen 4.5Flucloxacillin 2.7 �–�Flufenamic acid 3.9 �–�Flumequine 6.5 �–�Fluorouracil 8.0, 13.0 �–�Fluphenazine �–� 3.9, 8.1Flurazepam 8.2 1.9Flurbiprofen 4.3 �–�Furosemide 3.9 �–�Glibenclamide 5.3 �–�Guanethidine �–� 11.9Guanoxan �–� 12.3Haloperidol �–� 8.3Hexobarbital 8.3 �–�Hydralazine �–� 0.5, 7.1Ibuprofen 4.4 �–�Imipramine �–� 9.5Indometacin 4.5 �–�Isoniazid 2.0, 3.9 �–�Ketoprofen 4.0 �–�Labetalol 7.4 9.4Levodopa 2.3, 9.7, 13.4 �–�Lidocaine �–� 7.94 (26°C),

7.55 (36°C)Lincomycin �–� 7.5Maprotiline �–� 10.2Meclofenamic acid 4.0 �–�Metoprolol �–� 9.7Methadone �–� 8.3Methotrexate 3.8, 4.8 5.6Metronidazole �–� 2.5Minocycline 7.8 2.8, 5.0, 9.5

continued

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by the expression

(3.67)

where pK�w is the negative logarithm of the dis-sociation constant for water, K�w. K�w is derivedfrom consideration of the equilibrium

where one molecule of water is behaving asthe weak acid or base and the other is behav-ing as the solvent. Then

(3.68)

The concentration of molecular water is con-sidered to be virtually constant for diluteaqueous solutions. Therefore

(3.69)

where the dissociation constant for water (theionic product) now incorporates the term formolecular water and has the values given inTable 3.7.

K�w = [H�3��O��+][OH��−]

K =a�H��3O

�+ × a�OH�−

a�2H�2��O

�� Q ��[H�3��O��+] [OH��−]

[H�2��O]��2

H�2��O + H�2��O�� e ��H�3��O��+ + OH��−

pK�a + pK�b = pK�w


�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.

Table 3.6 (continued)

Compound pK�a

Acid Base

Minoxidil �–� 4.6Morphine 8.0 (phenol) 9.6 (amine)Nadolol �–� 9.7Nafcillin 2.7 �–�Nalidixic acid 6.4 �–�Nalorphine �–� 7.8Naloxone �–� 7.9Naltrexone 9.5 8.1Naproxen 4.2 �–�Nitrofurantoin �–� 7.2Nitrazepam 10.8 3.2 (3.4)�b

Norfloxacin 6.2 8.6Nortriptyline �–� 9.7Novobiocin 4.3, 9.1 �–�Ofloxacin 6.1 8.3Oxolinic acid 6.6 �–�Oxprenolol �–� 9.5Oxycodone �–� 8.9Oxytetracycline 7.3 3.3, 9.1Pentazocine �–� 8.8Pethidine �–� 8.7Phenazocine �–� 8.5Phenytoin 8.3 �–�Physostigmine �–� 2.0, 8.1Pilocarpine �–� 1.6, 7.1Pindolol �–� 8.8Piperazine �–� 5.6, 9.8Piroxicam 2.3 �–�Polymyxin B �–� 8.9

Compound pK�a

Acid Base

Prazocin �–� 6.5Procaine �–� 8.8Prochlorperazine �–� 3.7,8.1Promazine �–� 9.4Promethazine �–� 9.1Propranolol �–� 9.5Quinidine �–� 4.2, 8.3Quinine �–� 4.2, 8.8Ranitidine �–� 2.7, 8.2Sotalol 8.3 9.8Sulfadiazine 6.5 2.0Sulfaguanidine 12.1 2.8Sulfamerazine 7.1 2.3Sulfathiazole 7.1 2.4Tamoxifen �–� 8.9Temazepam �–� 1.6Tenoxicam 1.1 5.3Terfenadine �–� 9.5Tetracaine �–� 8.4Tetracycline 7.7 3.3, 9.5Theophylline 8.6 3.5Thiopental 7.5 �–�Timolol �–� 9.2 (8.8)b

Tolbutamide 5.3 �–�Triflupromazine �–� 9.2Trimethoprin �–� 7.2Valproate 5.0 �–�Verapamil �–� 8.8Warfarin 5.1 �–�

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When the pH of an aqueous solution of theweakly acidic or basic drug approaches the pK�aor pK�b, there is a very pronounced change inthe ionisation of that drug. An expression thatenables predictions of the pH dependence ofthe degree of ionisation to be made can bederived as shown in Box 3.8. The influence ofpH on the percentage ionisation may bedetermined for drugs of known pK�a usingTable 3.8.

EXAMPLE 3 .5 Calculation of percentageionisation

Calculate the percentage of cocaine existingas the free base in a solution of cocainehydrochloride at pH 4.5, and at pH 8.0. ThepK�b of cocaine is 5.6.

AnswerFrom equation (3.73):At pH 4.5:

Thus the percentage existing as cocaine base #0.01%.

At pH 8.0:

Thus the percentage existing as cocaine base #28.47%

If we carry out calculations such as those ofExample 3.5 over the whole pH range for bothacidic and basic drugs, we arrive at the graphsshown in Fig. 3.6. Notice from this figure that

● The basic drug is virtually completelyionised at pH values up to 2 units below itspK�a, and virtually completely unionised atpH values greater than 2 units above its pK�a.

�� = 71.53%

�� =100

1.398

�� =100

1 + antilog(8.0 − 14.0 + 5.6)

percentage ionisation

�� = 99.99%

�� =100

1.000126

�� =100

1 + antilog(4.5 − 14.0 + 5.6)

percentage ionisation


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

0

50

100

Ioni

sed

(%)

Basic

Basic

Acidic

Acidic

pH

pKa � 2 pKa pKa � 2

Table 3.7 Ionic product for water

Temperature (°C) K�w " 10�14 pK�w

0 0.1139 14.9410 0.2920 14.5320 0.6809 14.1725 1.0080 14.0030 1.4690 13.8340 2.9190 13.5450 5.4740 13.2660 9.6140 13.0270 15.10000 12.8280 23.40000 12.63

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

At pH above pK�a At pH below pK�a

pH – pK�a If anionic If cationic pKa� – pH If anionic If cationic

6.0 99.999 90 00.000 099 9 0.1 44.27 55.735.0 99.999 00 00.000 999 9 0.2 38.68 61.324.0 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 75.973.5 99.968 00.031 6 �–� �–� �–�3.4 99.960 00.039 8 �–� �–� �–�3.3 99.950 00.050 1 0.6 20.07 79.933.2 99.937 00.063 0 0.7 16.63 83.373.1 99.921 00.079 4 0.8 13.70 86.30�–� �–� 0�–� 0.9 11.19 88.81�–� �–� 0�–� 1.0 09.09 90.913.0 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 92.642.7 99.80 00.199 1 1.2 05.93 94.072.6 99.75 00.250 5 1.3 04.77 95.23�–� �–� 0�–� 1.4 03.83 96.17�–� �–� 0�–� 1.5 03.07 96.932.5 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 97.552.2 99.37 00.627 0 1.7 01.956 98.042.1 99.21 00.787 9 1.8 01.560 98.44�–� �–� 0�–� 1.9 01.243 98.76�–� �–� 0�–� 2.0 00.990 99.012.0 99.01 00.990 �–� 0�–� �–�1.9 98.76 01.243 �–� 0�–� �–�1.8 98.44 01.560 2.1 00.787 9 99.211.7 98.04 01.956 2.2 00.627 0 99.371.6 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 99.681.5 96.93 03.07 �–� 0�–� �–�1.4 96.17 03.83 �–� 0�–� �–�1.3 95.23 04.77 2.6 00.250 5 99.751.2 94.07 05.93 2.7 00.199 1 99.801.1 92.64 07.36 2.8 00.158 2 99.84

2.9 00.125 7 99.871.0 90.91 09.09 3.0 00.099 9 99.900.9 88.81 11.19 �–� 0�–� �–�0.8 86.30 13.70 3.1 00.079 4 99.9210.7 83.37 16.63 3.2 00.063 0 99.9370.6 79.93 20.07 3.3 00.050 1 99.950�–� �–� �–� 3.4 00.039 8 99.9600.5 75.97 24.03 3.5 00.031 6 99.9680.4 71.53 28.47 �–� 0�–� �–�0.3 66.61 33.39 4.0 00.009 999 0 99.990 00.2 61.32 38.68 5.0 00.000 999 9 99.999 000.1 55.73 44.27 6.0 00.000 099 9 99.999 900 50.00 50.00 �–� 0�–� �–�

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● The acidic drug is virtually completelyunionised at pH values up to 2 units belowits pK�a and virtually completely ionised atpH values greater than 2 units above its pK�a.

● Both acidic and basic drugs are exactly 50%ionised at their pK�a values.

3.5.3 Ionisation of amphoteric drugs

Ampholytes (amphoteric electrolytes) canfunction as either weak acids or weak bases inaqueous solution and have pK�a values corres-ponding to the ionisation of each group. Theymay be conveniently divided into two cate-gories – ordinary ampholytes and zwitterionicampholytes – depending on the relative acidityof the two ionisable groups.�12, 13

Ordinary ampholytes

In this category of ampholytes, the pK�a of theacidic group, pK , is higher than that of thebasic group, pK , and consequently the firstgroup that loses its proton as the pH isincreased is the basic group. Table 3.6 includesseveral examples of this type of ampholyte.We will consider, as a simple example, the ion-isation of m-aminophenol (I�), which has

and .The steps of the ionisation on increasing pH

are shown in the following equilibria:

This compound can exist as a cation, as anunionised form, or as an anion depending onthe pH of the solution, but because the differ-ence between and is p2, therewill be no simultaneous ionisation of the twogroups and the distribution of the species willbe as shown in Fig. 3.7. The ionisation patternwill become more complex, however, withdrugs in which the difference in pK�a of the two

groups is much smaller because of overlappingof the two equilibria.

Zwitterionic ampholytes

This group of ampholytes is characterised by therelation . The most commonexamples of zwitterionic ampholytes are theamino acids, peptides and proteins. There areessentially two types of zwitterionic electrolytedepending on the difference between the

and values, ∆pK�a.Large ∆pK�a. The simplest type to consider is

that of compounds having two widely sepa-rated pK�a values, for example glycine. The pK�avalues of the carboxylate and amino groups onglycine are 2.34 and 9.6, respectively and thechanges in ionisation as the pH is increased aredescribed by the following equilibria:

Over the pH range 3�–�9, glycine exists in solu-tion predominantly in the form �0��OOCCH�2��NHSuch a structure, having both positive and neg-ative charges on the same molecule, is referredto as a zwitterion and can react both as an acid,

or as a base,

�� e ��HOOCCH�2��NH�+3 + OH��−

��−��OOCCH�2��NH�+3 + H�2��O�� e ��

��−��OOCCH�2��NH�2 + H�3��O��+

��−��OOCCH�2��NH�+3 + H�2��O�� e ��

�+3

�� e ��−��OOCCH�2��NH�2 HOOCCH�2��NH�+3 �� e ��−��OOCCH�2��NH�+3 ��

pK�basicapK�acidic

a

pK�acidica �� ` ��pK�basic

a

pK�basicapK�acidic

a

��NH�2��C�6��H�5��O��−NH�+3C�6��H�4��OH�� e ��NH�2��C�6��H�4��OH�� e ��

pK�basica = 4.4pK�acidic

a = 9.8

�basica

�acidica


Structure I m-Aminophenol

OH

NH2

Figure 3.7 Distribution of ionic species for the ordinaryampholyte m-aminophenol.Redrawn from A. Pagliara, P.-A. Carrupt, G. Caron, P. Gaillard and B. Testa,Chem. Rev., 97, 3385 (1997).

0.01 3 5 7 9 11 13

0.2

0.4

0.6

0.8

1.0

1.2

Mol

e fra

ctio

n

pH

Cation Neutral Anion

09_1439PP_CH03 17/10/05 10:47 pm Page 82


This compound can exist as a cation, as a zwit-terion, or as an anion depending on the pH ofthe solution. The two pK�a values of glycine arep2 pH units apart and hence the distributionof the ionic species will be similar to thatshown in Fig. 3.7.

At a particular pH, known as the isoelectricpH or isoelectric point, pH�i, the effective netcharge on the molecule is zero. pH�i can becalculated from

Small ∆pK�a. In cases where the two pK�avalues are <<2 pH units apart there is overlap ofthe ionisation of the acidic and basic groups,with the result that the zwitterionic electrolytecan exist in four different electrical states – thecation, the unionised form, the zwitterion,and the anion (Scheme 3.3).

In Scheme 3.3, (ABH)�& is the zwitterionand, although possessing both positive andnegative charges, is essentially neutral. Theunionised form, (ABH), is of course also neutraland can be regarded as a tautomer of thezwitterion. Although only two dissociationconstants and (macrodissociationconstants) can be determined experimentally,each of these is composed of individualmicrodissociation constants because of simul-taneous ionisation of the two groups (seesection 3.5.5). These microdissociation con-stants represent equilibria between the cationand zwitterion, the anion and the zwitterion,the cation and the unionised form, and theanion and the unionised form. At pH�i, theunionised form and the zwitterion alwayscoexist, but the ratio of the concentrations ofeach will vary depending on the relativemagnitude of the microdissociation constants.The distribution of the ionic species forlabetalol which has and is shown in Fig. 3.8.

Examples of zwitterionic drugs with bothlarge and small ∆pK�a values are given in Box 3.9; others can be noted in Table 3.6.

3.5.4 Ionisation of polyprotic drugs

In the examples we have considered so far, theacidic drugs have donated a single proton.There are several acids, for example citric,phosphoric and tartaric acids, that are capableof donating more than one proton; these com-pounds are referred to as polyprotic or polybasicacids. Similarly, a polyprotic base is onecapable of accepting two or more protons.Many examples of both types of polyproticdrugs can be found in Table 3.6, including thepolybasic acids amoxicillin and fluorourocil,and the polyacidic bases pilocarpine, doxoru-bicin and aciclovir. Each stage of the dissocia-tion may be represented by an equilibriumexpression and hence each stage has a distinctpK�a or pK�b value. The dissociation of phos-phoric acid, for example, occurs in threestages; thus:

K�3 = 2.1 × 10��−13 �� e ��PO�3−4 + H�3��O��+

HPO�2−4 + H�2��O

K�2 = 6.2 × 10��−8 �� e ��HPO�2−4 + H�3��O��+

H�2��PO�−4 + H�2��O

K�1 = 7.5 × 10��−3 �� e ��H�2��PO�−4 + H�3��O��+

H�3��PO�4 + H�2��O K�basic

a = 9.4pK�acidica = 7.4

K�basicaK�acidic

a

pH�i =pK�acidic

a + pK�basica

2


Scheme 3.3

(ABH2)�K a

acidic K a basic

(ABH)�

(ABH)(AB)�

Figure 3.8 Distribution of ionic species for the zwitterionicampholyte labetalol.Redrawn from A. Pagliara, P.-A. Carrupt, G. Caron, P. Gaillard and B. Testa,Chem. Rev., 97, 3385 (1997).

40.0

0.2

0.4

0.6

0.8

1.0

1.2

6 8 10 12pH

Mol

e fra

ctio

n

Cation Anion

Neutral

Zwitterion

09_1439PP_CH03 17/10/05 10:47 pm Page 83


3.5.5 Microdissociation constants

The experimentally determined dissociationconstants for the various stages of dissociationof polyprotic and zwitterionic drugs arereferred to as macroscopic values. However, itis not always easy to assign macroscopic disso-ciation constants to the ionisation of specificgroups of the molecule, particularly when thepK�a values are close together, as discussed insection 3.5.4. The diprotic drug morphine hasmacroscopic pK�a values of 8.3 and 9.5, arisingfrom ionisation of amino and phenolic

groups. Experience suggests that the first pK�avalue is probably associated with the ionisa-tion of the amino group and the second withthat of the phenolic group; but it is not pos-sible to assign the values of these groupsunequivocally and for a more completepicture of the dissociation it is necessary totake into account all possible ways in whichthe molecule may be ionised and all the pos-sible species present in solution. We mayrepresent the most highly protonated form ofmorphine, �!��HMOH, as !O, where the ‘ ! ’refers to the protonated amino group and the


(Reproduced from reference 12.)

Box 3.9 Chemical structures and pK�a values of some zwitterionic drugs

Azapropazone Cetirizine

Zwitterions with a large �pKa

N

H OH

OH

NH2

O

pK a basic � 9.38

N

NN

N

O

OH

pK a basic

� 6.55

pK a acidic

� 1.8

HO O

N NO

Cl

pK a basic2

� 8.00

pKabasic1

� 2.19pK a acidic

� 2.93

pKaacidic

� 7.44

Labetalol

Zwitterions with a small �pKa

Tenoxicam

SN

S

CH3

H

N

O

N

O

O O

H

pK a basic � 5.34

pK a acidic

� 1.07

09_1439PP_CH03 17/10/05 10:47 pm Page 84


O refers to the uncharged phenolic group.Dissociation of the amino proton only pro-duces an uncharged form MOH, representedby OO, while dissociation of the phenolicproton gives a zwitterion �!��HMO�0, representedby ‘!0’. The completely dissociated form MO�0

is represented as O0. The entire dissociationscheme is given in Scheme 3.4.

The constants K�1, K�2, K�12 and K�21 are termedmicrodissociation constants and are definedby

The micro- and macrodissociation constantsare related by the following expressions:

(3.74)

(3.75)

and

(3.76)

Various methods have been proposed wherebythe microdissociation constants for the mor-phine system may be evaluated.�14 Other drugsfor which microdissociation constants havebeen derived include the tetracyclines,�15 doxo-rubicin,�16 cephalosporin,�17 dopamine�18 andthe group of drugs shown in Box 3.9.�13

3.5.6 pK�a values of proteins

The pK�a values of ionisable groups in proteinsand other macromolecules can be significantlydifferent from those of the correspondinggroups when they are isolated in solution. The

shifts in pK�a values between native and de-natured states or between bound and freeforms of a complex can cause changes inbinding constants or stability of the proteindue to pH effects. For example, the acid denat-uration of proteins may be a consequence ofanomalous shifts of the pK�a values of a smallnumber of amino acids in the native protein.�19

Several possible causes of such shifts havebeen proposed. They may arise from interac-tions among ionisable groups on the proteinmolecule; for example, an acidic group willhave its pK�a lowered by interactions with basicgroups. Other suggested cases of shifts of pK�ainclude hydrogen-bonding interactions withnonionisable groups and the degree of expo-sure to the bulk solvent; for example, an acidicgroup will have its pK�a increased if it is fully orpartially removed from solvent, but the effectmay be reversed by strong hydrogen-bondinginteractions with other groups.

The calculation of pK�a values of proteinmolecules thus requires a detailed consider-ation of the environment of each ionisablegroup, and is consequently highly complex.An additional complication is that a proteinwith N ionisable residues has 2�N� possibleionisation states; the extent of the problem isapparent when it is realised that a moderatelysized protein may contain as many as 50ionisable groups.

3.5.7 Calculation of the pH of drug solutions

We have considered above the effect on theionisation of a drug of buffering the solutionat given pH values. When these weakly acidicor basic drugs are dissolved in water they will,of course, develop a pH value in their ownright. In this section we examine how the pHof drug solutions of known concentration canbe simply calculated from a knowledge of thepK�a of the drug. We will consider the way inwhich one of these expressions may bederived from the expression for the ionisationin solution; the derivation of the other expres-sions follows a similar route and you may wishto derive these for yourselves.

K�1��K�2 = k�1��k�12 = k�2��k�21

1

K�2=

1

k�12

+1

k�21

K�1 = k�1 + k�2

k�12 =[O−][H�3��O��+]

[+−]��k�21 =

[O−][H�3��O��+][OO]

k�1 =[+−][H�3��O��+]

[+O]��k�2 =

[OO][H�3��O��+][+O]


Scheme 3.4

��

OO

O�

k1

k2

k12

k21

�O

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Weakly acidic drugs

We saw above that the dissociation of thesetypes of drugs may be represented by equation(3.63). We can now express the concentra-tions of each of the species in terms of thedegree of dissociation, α, which is a numberwith a value between 0 (no dissociation) and 1(complete dissociation):

HA ! H�2��O e A�0 ! H�3��O�!

(1 0 α)c αc αc

where c is the initial concentration of theweakly acidic drug in mol dm�03.

Because the drugs are weak acids, α will bevery small and hence the term (1 0 α) can beapproximated to 1. We may therefore write

(3.77)

Therefore,

To introduce pH into the discussion we note that

(3.78)

We now have an expression which enables usto calculate the pH of any concentration ofthe weakly acidic drug provided that its pK�avalue is known.

EXAMPLE 3 .6 Calculation of the pH of a weakacid

Calculate the pH of a 50 mg cm�03 solution ofascorbic acid (mol. wt # 176.1; pK�a # 4.17).

AnswerFor a weakly acidic drug,

Weakly basic drugs

We can show by a similar derivation to thatabove that the pH of aqueous solutions ofweakly basic drugs will be given by

(3.79)

EXAMPLE 3 .7 Calculation of the pH of aweakly basic drug

Calculate the pH of a saturated solution ofcodeine monohydrate (mol. wt. # 317.4) giventhat its pK�a # 8.2 and its solubility at roomtemperature is 1 g in 120 cm�3 water.

AnswerCodeine is a weakly basic drug and hence itspH will be given by

where c # 1 g in 120 cm�3 # 8.33 g dm�03 #0.02633 mol dm�03.

Drug salts

Because of the limited solubility of many weakacids and weak bases in water, drugs of thesetypes are commonly used as their salts; forexample, sodium salicylate is the salt of a weakacid (salicylic acid) and a strong base (sodiumhydroxide). The pH of a solution of this typeof salt is given by

(3.80)

Alternatively, a salt may be formed between aweak base and a strong acid; for example,ephedrine hydrochloride is the salt ofephedrine and hydrochloric acid. Solutions ofsuch drugs have a pH given by

(3.81)

Finally, a salt may be produced by the combi-nation of a weak base and a weak acid, as inthe case of codeine phosphate. Solutions ofsuch drugs have a pH given by

(3.82)pH = �12pK�w + �12pK�a − �12pK�b

pH = �12pK�a − �12log c

pH = �12pK�w + �12pK�a + �12log c

∴��pH = 7.0 + 4.1 − 0.790 = 10.31

pH = �12pK�w + �12pK�a + �12log c

pH = �12pK�w + �12pK�a + �12log c

∴��pH = 2.09 + 0.273 = 2.36

= 0.2839 mol��dm��−3

= 50 g��dm��−3 c = 50 mg��cm��−3

pH = �12pK�a − �12log c

pH = �12pK�a − �12log c ∴��−log[H��+] = −�12log K�a − �12log c ∴��[H��+] = (K�a��c)��1�2

αc = [H�3��O��+] = [H��+]

α��2 = �K�a

c �1�2

K�a =α��2��c��2

(1 − α)c�� Q ��α��2��c


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Notice that this equation does not include aconcentration term and hence the pH is inde-pendent of concentration for such drugs.

EXAMPLE 3 .8 Calculation of the pH of drug salts

Calculate the pH of the following solutions

(a) 5% oxycodone hydrochloride (pK�a # 8.9,mol. wt. # 405.9).

(b) 600 mg of benzylpenicillin sodium (pK�a #2.76, mol. wt. # 356.4) in 2 cm�3 of waterfor injection.

(c) 100 mg cm�03 chlorphenamine maleate(mol. wt. # 390.8) in water for injection(pK�b chlorphenamine # 5.0, pK�a maleicacid # 1.9).

Answer(a) Oxycodone hydrochloride is the salt of a

weak base and a strong acid, hence

where c # 50 g dm�03 # 0.1232 mol dm�03.

(b) Benzylpenicillin sodium is the salt of aweak acid and a strong base, hence

where c # 600 mg in 2 cm�3 # 0.842 moldm�03.

(c) Chlorphenamine maleate is the salt of aweak acid and a weak base, hence

3.5.8 Preparation of buffer solutions

A mixture of a weak acid and its salt (that is, aconjugate base), or a weak base and its con-jugate acid, has the ability to reduce the largechanges in pH which would otherwise resultfrom the addition of small amounts of acid oralkali to the solution. The reason for the

buffering action of a weak acid HA and itsionised salt (for example, NaA) is that the A�0

ions from the salt combine with the added H�! ions, removing them from solution asundissociated weak acid:

Added OH�0 ions are removed by combinationwith the weak acid to form undissociatedwater molecules:

The buffering action of a mixture of a weakbase and its salt arises from the removal of H�!

ions by the base B to form the salt and removalof OH�0 ions by the salt to form undissociatedwater:

The concentration of buffer componentsrequired to maintain a solution at the requiredpH may be calculated using equation (3.70).Since the acid is weak and therefore only veryslightly ionised, the term [HA] in this equationmay be equated with the total acid concen-tration. Similarly, the free A�0 ions in solutionmay be considered to originate entirely fromthe salt and the term [A�0] may be replaced bythe salt concentration.

Equation (3.70) now becomes

(3.83)

By similar reasoning, equation (3.72) may bemodified to facilitate the calculation of the pHof a solution of a weak base and its salt, giving

(3.84)

Equations (3.83) and (3.84) are often referredto as the Henderson�–�Hasselbalch equations.

EXAMPLE 3 .9 Buffer solutions

Calculate the amount of sodium acetate to beadded to 100 cm�3 of a 0.1 mol dm�03 aceticacid solution to prepare a buffer of pH 5.20.

pH = pK�w − pK�b + log [base][salt]

pH = pK�a + log [salt][acid�]

BH��+ + OH��−�� e ��H�2��O + B

B + H�3��O��+�� e ��H�2��O + BH��+

HA + OH��−�� e ��H�2��O + A��−

A��− + H�3��O��+�� e ��H�2��O + HA

∴��pH = 7.00 + 0.95 − 2.5 = 5.45

pH = �12pK�w + �12pK�a − �12pK�b

∴��pH = 7.00 + 1.38 − 0.037 = 8.34

pH = �12pK�w + �12pK�a + �12log c

∴��pH = 4.45 + 0.45 = 4.90

pH = �12pK�a − �12log c


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AnswerThe pK�a of acetic acid is 4.76. Substitution inequation (3.83) gives

The molar ratio of [salt]�[acid] is 2.754. Since100 cm�3 of 0.1 mol dm�03 acetic acid contains0.01 mol, we would require 0.027 54 mol ofsodium acetate (2.258 g), ignoring dilutioneffects.

Equations (3.83) and (3.84) are also useful incalculating the change in pH which resultsfrom the addition of a specific amount of acidor alkali to a given buffer solution, as seen fromthe following calculation in Example 3.10.

EXAMPLE 3 .10 Calculation of the pH changein buffer solutions

Calculate the change in pH following theaddition of 10 cm�3 of 0.1 mol dm�03 NaOH tothe buffer solution described in Example 3.9.

AnswerThe added 10 cm�3 of 0.1 mol dm�03 NaOH (equi-valent to 0.002 mol) combines with 0.001 molof acetic acid to produce 0.001 mol of sodiumacetate. Reapplying equation (3.83) using therevised salt and acid concentrations gives

The pH of the buffer has been increased byonly 0.06 units following the addition of thealkali.

Buffer capacity

The effectiveness of a buffer in reducingchanges in pH is expressed as the buffer capac-ity, β. The buffer capacity is defined by the ratio

(3.85)

where d�c is the number of moles of alkali (or

acid) needed to change the pH of 1 dm�3 ofsolution by an amount d(pH). If the additionof 1 mole of alkali to 1 dm�3 of buffer solutionproduces a pH change of 1 unit, the buffercapacity is unity.

Equation (3.70) may be rewritten in the form

(3.86)

where c�0 is the total initial buffer concentra-tion and c is the amount of alkali added.Rearrangement and subsequent differentia-tion yield

(3.87)

Therefore,

(3.88)

(3.89)

EXAMPLE 3 .11 Calculation of buffer capacity

Calculate the buffer capacity of the aceticacid�–�acetate buffer of Example 3.9 at pH 4.0.

AnswerThe total amount of buffer componentsin 100 cm�3 of solution # 0.01 ! 0.027 54 #0.037 54 moles. Therefore,

pK�a of acetic acid # 4.76. Therefore,

The pH of the solution # 4.0. Therefore,

Substituting in equation (3.89),

The buffer capacity of the acetic acid�–�acetatebuffer is 0.1096 mol dm�03 per pH unit.

= 0.1096

β =2.303 × 0.3754 × 1.75 x 10��−5 × 10��−4

(10��−4 + 1.75 × 10��−5)��2

[H�3��O��+] = 10��−4

K�a = 1.75 × 10��−5

c�0 = 0.3754 mol��dm��−3

β =2.303 c�0 K�a [H�3��O��+]

([H�3��O��+] + K�a)��2

β =d�c

d(��pH)=

2.303 c�0 exp[2.303 (��pH − pK�a)]

1 + exp[2.303 exp(��pH − pK�a)]��2

c =c�0

1 + exp[ − 2.303(��pH − pK�a)]

pH = pK�a +1

2.303 ln � c

c�0 − c �

β =d�c

d(��pH)

pH = 4.76 + log (0.02754 + 0.001)

(0.01 − 0.001)= 5.26

5.20 = 4.76 + log [salt][acid�]


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Figure 3.9 shows the variation of buffercapacity with pH for the acetic acid�–�acetatebuffer used in the numerical examples above(c�0 # 0.3754 mol dm�03) as calculated fromequation (3.89). It should be noted that β is ata maximum when pH # pK�a (that is, at pH4.76). When selecting a weak acid for thepreparation of a buffer solution, therefore, thechosen acid should have a pK�a as close as pos-sible to the pH required. Substituting pH # pK�ainto equation (3.89) gives the useful result thatthe maximum buffer capacity is β�max # 0.576c�0,where c�0 is the total buffer concentration.

Buffer solutions are widely used in phar-macy to adjust the pH of aqueous solutions tothat required for maximum stability or thatneeded for optimum physiological effect.Solutions for application to delicate tissues,particularly the eye, should also be formulatedat a pH not too far removed from that of theappropriate tissue fluid, as otherwise irritationmay be caused on administration. The pH oftears lies between 7 and 8, with an averagevalue of 7.4. Fortunately, the buffer capacity oftears is high and, provided that the solutionsto be administered have a low buffer capacity,a reasonably wide range of pH may be toler-ated, although there is a difference in the

irritability of the various ionic species that arecommonly used as buffer components. ThepH of blood is maintained at about 7.4 byprimary buffer components in the plasma (car-bonic acid�–�carbonate and the acid�–�sodiumsalts of phosphoric acid) and secondary buffercomponents (oxyhaemoglobin�–�haemoglobinand acid�–�potassium salts of phosphoric acid)in the erythrocytes. Values of 0.025 and0.039 mol dm�03 per pH unit have been quotedfor the buffer capacity of whole blood.Parenteral solutions are not normally buffered,or alternatively are buffered at a very lowcapacity, since the buffers of the blood areusually capable of bringing them within atolerable pH range.

Universal buffers

We have seen from Fig. 3.9 that the buffercapacity is at a maximum at a pH equal to thepK�a of the weak acid used in the formulationof the buffer system and decreases appreciablyas the pH extends more than one unit eitherside of this value. If, instead of a single weakmonobasic acid, a suitable mixture of poly-basic and monobasic acids is used, it is pos-sible to produce a buffer which is effectiveover a wide pH range. Such solutions arereferred to as universal buffers. A typicalexample is a mixture of citric acid (pK�a1 # 3.06,pK�a2 # 4.78, pK�a3 # 5.40), Na�2��HPO�4 (pK�a ofconjugate acid H�2��PO # 7.2), diethylbarbituricacid (pK�a1 # 7.43) and boric acid (pK�a1 # 9.24).Because of the wide range of pK�a valuesinvolved, each associated with a maximumbuffer capacity, this buffer is effective over acorrespondingly wide pH range (pH 2.4�–�12).

3.6 Diffusion of drugs in solution

Diffusion is the process by which a concentra-tion difference is reduced by a spontaneousflow of matter. Consider the simplest case of asolution containing a single solute. The solutewill spontaneously diffuse from a region ofhigh concentration to one of low concentra-tion. Strictly speaking, the driving force for

�−4

Diffusion of drugs in solution 89

Figure 3.9 Buffer capacity of acetic acid�–�acetate buffer(initial concentration # 0.3754 mol dm�03) as a function ofpH.

2.0 3.0 4.0 5.0 6.0 7.0

pH

pKa0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

Buffe

r ca

paci

ty, b

09_1439PP_CH03 17/10/05 10:47 pm Page 89


diffusion is the gradient of chemical potential,but it is more usual to think of the diffusion ofsolutes in terms of the gradient of their con-centration. Imagine the solution to be dividedinto volume elements. Although no indi-vidual solute particle in a particular volumeelement shows a preference for motion in anyparticular direction, a definite fraction of themolecules in this element may be consideredto be moving in, say, the x direction. In anadjacent volume element, the same fractionmay be moving in the reverse direction. If theconcentration in the first volume element isgreater than that in the second, the overalleffect is that more particles are leaving the firstelement for the second and hence there is anet flow of solute in the x direction, thedirection of decreasing concentration. Theexpression which relates the flow of materialto the concentration gradient (d�c�d�x) isreferred to as Fick’s first law:

(3.90)

where J is the flux of a component across aplane of unit area and D is the diffusioncoefficient (or diffusivity). The negative signindicates that the flux is in the direction ofdecreasing concentration. J is in mol m�02 s�01, cis in mol m�03 and x is in metres; therefore, theunits of D are m�2 s�01.

The relationship between the radius, a, ofthe diffusing molecule and its diffusion coeffi-cient (assuming spherical particles or mol-ecules) is given by the Stokes�–�Einstein equationas

(3.91)

Table 3.9 shows diffusion coefficients of somep-hydroxybenzoate (paraben) preservativesand typical proteins in aqueous solution.Although the trend for a decrease of D withincrease of molecular size (as predicted byequation 3.91) is clearly seen from the data ofthis table, it is also clear that other factors,such as branching of the p-hydroxybenzoatemolecules (as with the isoalkyl derivatives)and the shape of the protein molecules, alsoaffect the diffusion coefficients. The diffusion

coefficients of more complex molecules suchas proteins will also be affected by the shape ofthe molecule, more asymmetric moleculeshaving a greater resistance to flow.

The diffusional properties of a drug haverelevance in pharmaceutical systems in a con-sideration of such processes as the dissolutionof the drug and transport through artificial(e.g. polymer) or biological membranes. Diffu-sion in tissues such as the skin or in tumours isa process which relies on the same criteria asdiscussed above, even though the diffusiontakes place in complex media.

Summary

● We have looked at the meaning of some ofthe terms commonly used in thermo-dynamics and how these are interrelatedin the three laws of thermodynamics. Inparticular, we have seen that:

● Entropy is a measure of the disorder orchaos of a system and that processes,such as the melting of a crystal, whichresult in an increased disorder are

D =RT

6πηaN�A

J = −D d�cd�x


Table 3.9 Effect of molecular weight on diffusioncoefficient (25°C) in aqueous media

Compound Molecular Dweight (10�010 m�2 s�01)

p-Hydroxybenzoates�a

Methyl hydroxybenzoate 152.2 8.44Ethyl hydroxybenzoate 166.2 7.48n-Propyl hydroxybenzoate 180.2 6.81Isopropyl hydroxybenzoate 180.2 6.94n-Butyl hydroxybenzoate 194.2 6.31Isobutyl hydroxybenzoate 194.2 6.40n-Amyl hydroxybenzoate 208.2 5.70

Proteins�b

Cytochrome c (horse) 12 400 1.28Lysozyme (chicken) 14 400 0.95Trypsin (bovine) 24 000 1.10Albumin (bovine) 66 000 0.46

�a From reference 20.�b From reference 21; see also Chapter 11 for further values of

therapeutic peptides and proteins.

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accompanied by an increase in entropy.Since spontaneous processes alwaysproduce more disordered systems, itfollows that the entropy change for theseprocesses is always positive.

● During a spontaneous change at con-stant temperature and pressure, there is adecrease in the free energy until thesystem reaches an equilibrium state,when the free energy change becomeszero. The free energy is therefore ameasure of the useful work that a systemcan do; when the system has reachedequilibrium it has used up its free energyand consequently no longer has theability to do any further work; thus allspontaneous processes are irreversible.

● There is a simple relationship between freeenergy change and the equilibrium con-stant for a reaction from which an equa-tion has been derived for the change ofequilibrium constant with temperature.

● If there are no interactions between thecomponents of a solution (ideal solu-tion) then the activity equals the concen-tration; in real solutions the ratio of theactivity to the concentration is called theactivity coefficient.

● The chemical potential is the effective freeenergy per mole of each component of asolution. In general, the chemical poten-tial of a component is identical in allphases of a system at equilibrium at afixed temperature and pressure.

● Parenteral solutions should be of approxi-mately the same tonicity as blood serum;the amount of adjusting substance whichmust be added to a formulation to achieveisotonicity can be calculated using the

freezing point depressions of the drug andthe adjusting substance.

● The strengths of weakly acidic or basicdrugs may be expressed by their pK�a andpK�b values; the lower the pK�a the stronger isthe acid; the lower the pK�b the stronger isthe base. Acidic drugs are completelyunionised at pH values up to 2 units belowtheir pK�a and completely ionised at pHvalues greater than 2 units above their pK�a.Conversely, basic drugs are completelyionised at pH values up to 2 units belowtheir pK�a and completely unionised whenthe pH is more than 2 units above their pK�a.Both types of drug are exactly 50% ionisedat their pK�a values. Some drugs can donateor accept more than one proton and so mayhave several pK�a values; other drugs canbehave as both acids and bases, i.e. they areamphoteric drugs. The pH of aqueous solu-tions of each of these types of drug andtheir salts can be calculated from their pK�aand the concentration of the drug.

● A solution of a weak acid and its salt (conju-gate base) or a weak base and its conjugateacid acts as a buffer solution. The quantitiesof buffer components required to preparebuffers solutions of known pH can be cal-culated from the Henderson�–�Hasselbalchequation. The buffering capacity of a buffersolution is maximum at the pK�a of the weakacid component of the buffer. Universalbuffers are mixtures of polybasic andmonobasic acids that are effective over awide range of pH.

● The drug molecules in solution will spon-taneously diffuse from a region of highchemical potential to one of low chemicalpotential; the rate of diffusion may becalculated from Fick’s law.

References 91

References

1. The Pharmaceutical Codex, 12th edn, The Pharma-ceutical Press, London, 1994

2. P. W. Atkins. Physical Chemistry, 7th edn, OxfordUniversity Press, 2001

3. D. J. Shaw and H. E. Avery. Physical Chemistry,Macmillan Education Limited, London, 1994

4. J. H. Rytting, S. S. Davis and T. Higuchi. Suggestedthermodynamic standard state for comparing drugmolecules in structure�–�activity studies. J. Pharm.Sci., 61, 816�–�8 (1972)

5. J. H. B. Christian. In Water Activity: Influences onFood Quality (ed. L. B. Rockland and G. F. Stewart),Academic Press, New York, 1981, p. 825

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6. U. Ambrose, K. Middleton and D. Seal. In vitrostudies of water activity and bacterial growthinhibition of sucrose�–�polyethylene glycol 400�–�hydrogen peroxide and xylose�–�polyethyleneglycol 400�–� hydrogen peroxide pastes used to treatinfected wounds. Antimicrob. Agents Chemother.,35, 1799�–�803 (1991)

7. W. H. Streng, H. E. Huber and J. T. Carstensen.Relationship between osmolality and osmolarity. J.Pharm. Sci., 67, 384�–�6 (1978)

8. D. M. Willis, J. Chabot, I. C. Radde and G. W.Chance. Unsuspected hyperosmolality of oral solu-tions contributing to necrotizing enterocolitis invery-low-birth-weight infants. Pediatrics, 60, 535�–�8(1977)

9. K. C. White and K. L. Harkavy. Hypertonic formularesulting from added oral medications. Am. J. Dis.Child., 136, 931�–�3 (1982)

10. J. Lerman, M. M. Willis, G. A. Gregory and E. I.Eger. Osmolarity determines the solubility ofanesthetics in aqueous solutions at 37°C.Anesthesiology, 59, 554�–�8 (1983)

11. The Merck Index, 12th edn, Merck, Rahway, NJ,1996

12. A. Pagliara, P.-A. Carrupt, G. Caron, et al.Lipophilicity profiles of ampholytes. Chem. Rev.,97, 3385�–�400 (1997)

13. G. Bouchard, A. Pagliara, P.-A. Carrupt, et al.Theoretical and experimental exploration of thelipophilicity of zwitterionic drugs in 1,2-dichloro-ethane�water system. Pharm. Res., 19, 1150�–�9(2002)

14. P. J. Niebergall, R. L. Schnaare and E. T. Sugita.Spectral determination of microdissociation con-stants. J. Pharm. Sci., 61, 232 (1972)

15. L. J. Leeson, J. E. Krueger and R. A. Nash. Structuralassignment of the second and third acidity con-stants of tetracycline antibiotics. Tetrahedron Lett.,18, 1155�–�60 (1963)

16. R. J. Sturgeon and S. G. Schulman. Electronicabsorption spectra and protolytic equilibriums ofdoxorubicin: direct spectrophotometric deter-mination of microconstants. J. Pharm. Sci., 66,958�–�61 (1977)

17. W. H. Streng, H. E. Huber, J. L. DeYoung and M. A.Zoglio. Ionization constants of cephalosporinzwitterionic compounds. J. Pharm. Sci., 65, 1034(1976); 66, 1357 (1977)

18. T. Ishimitsu, S. Hirose and H. Sakurai. Microscopicacid dissociation constants of 3,4-dihydrox-yphenethylamine (dopamine). Chem. Pharm. Bull.,26, 74�–�8 (1978)

19. B. Honig and A. Nicholls. Classical electrostatics inbiology and chemistry. Science, 268, 1144�–�9 (1995)

20. T. Seki, M. Okamoto, O. Hosoya and K. Juni.Measurement of diffusion coefficients of parabensby the chromatographic broadening method. J.Pharm. Sci. Technol., Jpn. 60, 114�–�7 (2000)

21. N. Baden and M. Terazima. A novel method formeasurement of diffusion coefficients of proteinsand DNA in solution. Chem. Phys. Lett., 393,539�–�45 (2004)


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