Acid-base Equilibria and CalculationsA Chem1Reference TextStephen K. LowerSimon Fraser UniversityContents1 Proton donor-acceptor equilibria41.1The ion product of water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41.2Acid and base strengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62 The fall of the proton92.1Proton sources and sinks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92.2Leveling effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92.3Dissociation of weak acids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102.4Titration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102.5Strong bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102.6Proton free energy and pH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113 Quantitative treatment of acid-base equilibria123.1Strong acids and bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123.2Concentrated solutions of strong acids. . . . . . . . . . . . . . . . . . . . . . . . . . . . .133.3Weak monoprotic acids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143.4Pure acid in water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143.5Weak bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153.6Carrying out acid-base calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16Selecting the approximation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16Solving quadratic and higher-order polynomials. . . . . . . . . . . . . . . . . . . . . . . .173.7Calculations on mixtures of acids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183.8Mixture of an acid and its conjugate base: buffers. . . . . . . . . . . . . . . . . . . . . .193.9Ionization fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .213.10 Calculations involving mixtures of acids and bases. . . . . . . . . . . . . . . . . . . . . .223.11 Zwitterions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253.12 Diprotic acids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26Solution of an ampholyte salt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284 Acid-base titration284.1Titration curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .294.2Observation of equivalence points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .304.3Detection of the equivalence point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33• CONTENTS5 Acid- and base neutralizing capacity346 Graphical treatment of acid-base problems356.1Log-C vs pH plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35Locating the lines on the graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .356.2Estimating the pH on− log C vs pH diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 37pH of an acid in pure water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37pH of a solution of the conjugate base. . . . . . . . . . . . . . . . . . . . . . . . . . . . .37Titration curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38Polyprotic acids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .387 Acid-base chemistry in physiology407.1Maintenance of acid-base balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .407.2Disturbances of acid-base balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .408 Acid rain419 The carbonate system429.1The geochemical carbon cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .429.2Carbon dioxide in the atmosphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .429.3Dissolution of CO2in water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .439.4Distribution of carbonate species in aqueous solutions. . . . . . . . . . . . . . . . . . . .439.5Calculations on carbonate solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46Chem1General Chemistry Reference Text2Acid-base equilibria and calculations• CONTENTSAcid-base reactions, in which protons are exchanged between donor molecules (acids)and acceptors (bases), form the basis of the most common kinds of equilibrium problems
which you will encounter in almost any application of chemistry.This document provides a reasonably thorough treatment of aquatic-solution acid-baseequilibria. Although it has been used as the principal text for part of a university-level
General Chemistry course, it can also serve as a reference for teachers and advanced
students who seek a more comprehensive treatment of the subject than is likely to be
found in conventional textbooks.As background, we will assume that you already have some understanding of thefollowing topics:• The Arrhenius concept of acids and bases
• the Brønsted-Lowry concept, conjugate acids and bases
• titration
• definition of pH and the pH scale
• strong vs. weak acids and bases
• the names of the common acids and basesChem1General Chemistry Reference Text3Acid-base equilibria and calculations• 1 Proton donor-acceptor equilibria1Proton donor-acceptor equilibriaIn order to describe acid-base equilibria in the most general way, we will often represent an acid by the
formula HA and its conjugate base as A−. The actual electric charges of the species will of course dependon the particular nature of A, but the base will always have one more negative charge than the acid HA.
This pair of species constitutes an acid-base system whose two members are related by the reactionHA(aq)−− H++ A−(1)The most fundamental property of a given acid-base system is the extent of the above reaction. Ifthe concentration of undissociated HA is negligible when the reaction is at equilibrium, the acid is said
to be strong. Only a very small number of acids fall into this category; most acids are weak.There are two complications that immediately confront us when we attempt to treat acid-base equi-libria in a quantitative way:1. Since protons cannot exist in solution as independent species, the tendency of an acid or a baseto donate or accept a proton (as in Eq 1) cannot be measured for individual acid or base species
separately; the best we can do is compare two different acid-base systems, and determine the extent
to which the bases are able to compete against each other for the proton.2. Water itself can act both as an acid and a base, and most of the practical applications of acid-basechemistry are those involving aqueous solutions. This means that whenever we are studing an
aqueous solution of an acid HA, we must also contend with the conjugate acid and base of H2O.We can make use of (2) to help us out with (1) by using water as a reference standard for proton-donating
and -accepting power. Thus the strength of an acid HA can be defined by the equilibriumHA + H2O−→ H3O++ A−Ka(2)Similarly, the strength of the base A−is defined byA−+ H2O−→ HA + OH−Kb(3)Note carefully that reaction (3) is not the reverse of (2).1.1The ion product of waterIn pure water, about one H2O molecule out of 109is “dissociated”:H2O −− H++ OH−The actual reaction, of course, is the proton transferH2O + H2O −− H3O++ OH−(4)for which the equilibrium constantKw= [H3O+][OH−](5)is known as the ion product of water. The value ofKwat room temperature is 1.008× 10−14.In pure water, the concentrations of H3O+and OH−must of course be the same:[H3O+] = [OH−] =Kw≈ 10−7a solution in which [H3O+] = [OH−] is said to be neutral.Chem1General Chemistry Reference Text4Acid-base equilibria and calculations• The ion product of waterAs with any equilibrium constant, the value ofKwis affected by the temperature (Kwundergoes a10-fold increase between 0◦C and 60◦C), by the pressure (Kwis about doubled at 1000 atm), andby the presence of ionic species in the solution. Because most practical calculations involvingKwrefer to ionic solutions rather than to pure water, the common practice of using 10−14as if it werea universal constant is unwise; under the conditions commonly encountered in the laboratory, pKwcan vary from about 11 to almost 151. In seawater,Kwis 6.3× 10−12.Notice that under conditions whenKwdiffers significantly from 1.0× 10−14, the pH of a neutralsolution will not be 7.0. For example, at a pressure of 93 kbar and 527◦C,Kw= 10−3.05, the pH ofpure water would be 1.5. Such conditions might conceivably apply to deposits of water in geological
formations and in undersea vents.Problem Example 1At 60◦C, the ion product of water is 9.6E-14. What is the pH of a neutral solution at this tempera-ture?Solution:Under these conditions, [H+][OH−] = 9.6E–14. If the solution is neutral, [H+] =[OH−] = √9.6E–14, corresponding to pH = 6.5.1See Stephen J. Hawkes: “pKwis almost never 14.0”, J. Chem. Education 1995: 72(9) 799-802Chem1General Chemistry Reference Text5Acid-base equilibria and calculations• Acid and base strengths1.2Acid and base strengthsThe equilibrium constants that define the strengths of an acid and of a base areKa= [H3O+][OH−][HA](6)andKb= [HA][OH−][A−](7)How areKaandKbrelated? The answer can be found by adding Equations 2 and 3:HA−− H++ A−(8)A−+ H2O−− HA + OH−(9)H2O−− H++ OH−(10)Since the sum of the first two equations represents the dissociation of water (we are using H+instead ofH3O+for simplicity), the equilibrium constant for the third reaction must be the product of the first twoequilibrium constants:KaKb=Kw(11)Clearly, as the strength of a series of acids increases, the strengths of their conjugate bases will decrease,
hence the inverse relation betweenKaandKb.pK valuesYou will recall that the pH scale serves as a convenient means of compressing a wide rangeof [H+] -values into a small range of numbers. Just as we defined the pH as the negative logarithm of thehydrogen ion concentration, we can definepK =− log Kfor any equilibrium constant. Acid and base strengths are very frequently expressed in terms of pKaandpKb. From Eq 11 it should be apparent thatpKa+ pKb= pKw(= 14.0 at 25◦C)Table 1 on the next page gives the pK values for a number of commonly-encountered acid-base systemswhich are listed in order of decreasing acid strength. Take a moment to locate the H3O+/H2O systemin this table. Notice the value of pKafor the hydronium ion; its value of 0 corresponds toKa= 1. Anyacid whoseKaexceeds that of the hydronium ion is by definition a strong acid. You will also notice thatthe pK’s of the strongest acids and bases are given only approximate values; this is because these speciesare so strongly dissociated that the interactions between the resulting ions make it difficult to accurately
define their concentrations in these solutions.Chem1General Chemistry Reference Text6Acid-base equilibria and calculations• Acid and base strengthsacidpKabasepKbHClO4perchloric acid∼ −7 ClO−4∼ 21HClhydrogen chloride∼ −3 Cl−∼ 17H2SO4sulfuric acid∼ −3 HSO−4∼ 17HNO3nitric acid−1NO−315H3O+hydronium ion0H2O14H2SO3sulfurous acid1.8HSO−312.2HSO−4bisulfate1.9SO2−412.1H3PO4phosphoric acid2.12H2PO−411.88[Fe(H2O)6]3+aquo ferric ion2.10[Fe(H2O)5OH]2+11.90HFhydrofluoric acid3.2F−10.8CH3COOHacetic acid4.7CH3COO−9.3[Al(H2O)6]3+aquo aluminum ion4.9[Al(H2O)5OH]2+9.1H2CO3total dissolved CO2a6.3HCO−37.7H2Shydrogen sulfide7.04HS−6.96H2PO−4dihydrogen phosphate7.2H2PO2−46.8HSO−3bisulfite ion7.21SO2−36.79HOClhypochlorous acid8.0OCl−6.0HCNhydrogen cyanide9.2CN−4.8H3BO4boric acid9.30B(OH)−44.70NH+
4ammonium ion9.25NH34.75Si(OH)4o-silicic acid9.50SiO(OH)−34.50HCO−3bicarbonate10.33CO2−33.67HPO2−4hydrogen phosphate12.32PO3−41.67SiO(OH)−3silicate12.6SiO2(OH)2−21.4H2Owaterb14OH−0HS−bisulfidec∼ 19S2−∼ −5NH3ammonia∼ 23NH−2∼ −9OH−hydroxide ion∼ 24O2−∼ −10aThe acid H2CO3is only a minority species in aqueous carbon dioxide solutions, which contain mainly CO2(aq).The pKaof 6.3 that is commonly given is calculated on the basis of the total CO2in the solution. The true pKaofH2CO3is about 3.5.bIf water is acting as a solute, as it must if the acid strength of H2O is being compared with that of other veryweak acids, then pKa≈ 16 should be used. See J. Chem. Education 1990: 67(5) 386-388.cMany tables still give 14 as pK2for H2S; this is now known to be incorrect.Table 1: pK values of acids and bases in aqueous solutions at 25◦CChem1General Chemistry Reference Text7Acid-base equilibria and calculations• Acid and base strengthsH3O+H2OH2OOHÐHClO4H2SO4HClHNO3HSO4ÐFe(H2O)63+CH3COOHAl(H2O)63+H2CO3HCNNH4+HCO3ÐHPO42ÐNH3OHÐÐ8Ð6Ð4Ð2024681012141618202224Ð40Ð20020406080100120140kJ of free energy released per mole of protons transferred from H3O+DGpHACIDS(proton donors)BASES(proton sinks)ClO4ÐHSO4ÐClÐNO3ÐSO42ÐFe(H2O)5OH2+CH3COOÐAl(H2O)5OH2+CNÐNH3CO32ÐPO43ÐNH2ÐO2ÐHCO3Ðstrong acids(cannot exist in water)strong bases(cannot exist in water)HOClOClÐHCOOHHCOOÐlog [HA][AÐ]4 2 Ð2 Ð4pHvariation of concentration
ratio with pH above and
below pKTable 2: Free energy diagram for acids and bases in aqueous solution.Chem1General Chemistry Reference Text8Acid-base equilibria and calculations• 2 The fall of the proton2The fall of the protonAn acid, being a proton donor, can only act as an acid if there is a suitable base present to accept the
proton. What do we mean by “suitable” in this context? Simply that a base, in order to accept a proton,
must provide a lower-free energy2resting place for the proton than does the acid. Thus you can viewan acid-base reaction as the “fall” of the proton from a higher free energy to a lower free energy.2.1Proton sources and sinksViewed in this way, an acid is a proton source, a base is a proton sink. The tendency for a proton to move
from source to sink depends on how far the proton can fall in energy, and this in turn depends on the
energy difference between the source and the sink. This is entirely analogous to measuring the tendency
of water to flow down from a high elevation to a lower one; this tendency (which is related to the amount
of energy that can be extracted in the form of electrical work if the water flows through a power station at
the bottom of the dam) will be directly proportional to the difference in elevation (difference in potential
energy) between the source (top of the dam) and the sink (bottom of the dam).Now look at Table 2 on the following page and study it carefully. In the center columns of the diagram,you see a list of acids and their conjugate bases. These acid-base pairs are plotted on an energy scale
which is shown at the left side of the diagram. This scale measures the free energy released when one
mole of protons is transferred from a given acid to H2O. Thus if one mole of HCl is added to water, itdissociates completely and heat is released as the protons fall from the source (HCl) to the lower free
energy that they possess in the H3O+ions that are formed when the protons combine with H2O.Any acid shown on the left side of the vertical line running down the center of the diagram candonate protons to any base (on the right side of the line) that appears below it. The greater the vertical
separation, the greater will be the fall in free energy of the proton, and the more complete will be the
proton transfer at equilibrium.Notice the H3O+/H2O pair shown at zero kJ on the free energy scale. This zero value of free energycorresponds to the proton transfer processH3O++ H2O−→ H2O + H3O+which is really no reaction at all, hence the zero fall in free energy of the proton. Since the proton is
equally likely to attach itself to either of two identical H2O molecules, the equilibrium constant is unity.Now look at the acid/base pairs shown at the top of the table, above the H3O+-H2O line. All of theseacids can act as proton sources to those sinks (bases) that appear below them. Since H2O is a suitablesink for these acids, all such acids will lose protons to H2O in aqueous solutions. These are therefore allstrong acids that are 100% dissociated in aqueous solution; this total dissociation reflects the very large
equilibrium constants that are associated with any reaction that undergoes a fall in free energy of more
than a few kilojoules per mole.2.2Leveling effectBecause H2O serves as a proton sink to any acid in which the proton free energy level is greater thanzero, the strong acids such as HCl and H2SO4cannot “exist” (as acids) in aqueous solution; they existas their conjugate bases instead, and the only proton donor present will be H3O+. This is the basis ofthe leveling effect, which states that the strongest acid that can exist in aqueous solution is H3O+.2You don’t know what free energy is? Don’t worry about it for the time being; just think of it as you would any otherform of potential energy: something that falls when chemical reactions take place. This topic will be covered later in the
course.Chem1General Chemistry Reference Text9Acid-base equilibria and calculations• Dissociation of weak acidsNow consider a weak acid, such as HCN at about 40 kJ mol−1on the scale. This positive free energymeans that in order for a mole of HCN to dissociate (transfer its proton to H2O), the proton must gain40 kJ of free energy per mole. In the absence of a source of energy, the reaction will simply not go; HCN
is dissociated only to a minute extent in water.2.3Dissociation of weak acidsWhy is a weak acid such as HCN dissociated at all? The molecules in solution are continually being
struck and bounced around by the thermal motions of neighboring molecules. Every once in a while,
a series of fortuitous collisions will provide enough kinetic energy to a HCN molecule to knock off the
proton, effectively boosting it to the level required to attach itself to water. This process is called thermal
excitation, and its probability falls off very rapidly as the distance (in kJ mol−1) that the proton mustrise increases. The protons on a “stronger” weak acid such as HSO−4or CH3COOH will be thermallyexcited to the H3O+level much more frequently than will the protons on HCN or HCO−3, hence thedifference in the dissociation constants of these acids.2.4TitrationAlthough a weak acid such as HCN will not react with water to a significant extent, you are well aware
that such an acid can still be titrated with strong base to yield a solution of NaCN at the equivalence
point. To understand this process, find the H2O/OH−pair at about 80 kJ mol−1on the free energyscale. Because the OH−ion can act as a proton sink to just about every acid shown on the diagram, theaddition of strong base in the form of NaOH solution allows the protons at any acid above this level to
fall to the OH−level according to the reactionH++ OH−−→ H2OTitration, in other words, consists simply in introducing a low free energy sink that can drain off the
protons from the acids initially present, converting them all into their conjugate base forms.2.5Strong basesThere are two other aspects of the H2O-H3O+pair that have great chemical significance. First, itslocation at 80 kJ mol−1tells us that for a H2O molecule to transfer its proton to another H2O molecule(which then becomes a H3O+ion whose relative free energy is zero), a whopping 80 kJ/mol of free energymust be supplied by thermal excitation. This is so improbable that only one out of about 10 million H2Omolecules will have its proton elevated to the H3O+level at a given time; this corresponds to the smallvalue of the ion product of water, about 10−14.The other aspect of the H2O-OH−pair is that its location defines the hydroxide ion as the strongestbase that can exist in water. On our diagram only two stronger bases (lower proton free energy sinks)
are shown: the amide ion NH−2, and the oxide ion O2−. What happens if you add a soluble oxide suchas Na2O to water? Since O2−is a proton sink to H2O, it will react with the solvent, leaving OH−as thestrongest base present:Na2O + H2O−→ 2OH−+ Na+This again is the leveling effect; all bases stronger than OH−appear equally strong in water, simplybecause they are all converted to OH−.Chem1General Chemistry Reference Text10Acid-base equilibria and calculations• Proton free energy and pH2.6Proton free energy and pHThe pH of a solution is more than a means of expressing its hydrogen ion concentration on a convenient
logarithmic scale3. The pH as we commonly use it nowadays indicates the availability of protons in thesolution; that is, the ability of the solution to supply protons to a base such as H2O. This is the same asthe hydrogen ion concentration [H+] only in rather dilute solutions; at ionic concentrations (whether ofH+or other ions) greater than about 0.01 M, electrostatic interactions between the ions cause the relationbetween the pH (as measured by direct independent means) and [H+] to break down. Thus we wouldnot expect the pH of a 0.100 M solution of HCl to be exactly 1.00.On the right side of Figure Table 2 is a pH scale. At the pH value corresponding to a given acid-basepair, the acid and base forms will be present at equal concentrations. For example, if you dissolve some
solid sodium sulfate in pure water and then adjust the pH to 2.0, about half of the SO−4 2will be convertedinto HSO−4. Similarly, a solution of Na2CO3in water will not contain a very large fraction of CO2−3unlessthe pH is kept above 10.Suppose we have a mixture of many different weak acid-base systems, such as exists in most biologicalfluids or natural waters, including the ocean. The available protons will fall to the lowest free energy
levels possible, first filling the lowest-energy sink, then the next, and so on until there are no more
proton-vacant bases below the highest proton-filled (acid) level. Some of the highest protonated species
will donate protons to H2O through thermal excitation, giving rise to a concentration of H3O+that willdepend on the concentrations of the various species. The equilibrium pH of the solution is a measure of
this H3O+concentration, but this in turn reflects the relative free energy of protons required to keep thehighest protonated species in its acid form; it is in this sense that pH is a direct measure of proton free
energy.In order to predict the actual pH of any given solution, we must of course know something about thenominal concentrations (Ca) of the various acid-base species, since this will strongly affect the distributionof protons. Thus if one proton-vacant level is present at twice the concentration of another, it will cause
twice as many acid species from a higher level to become deprotonated. In spite of this limitation, the
proton free energy diagram provides a clear picture of the relationships between the various acid and base
species in a complex solution.3The concept of pH was suggested by the Swedish chemist Sørensen in 1909 as a means of compressing the wide rangeof [H+] values encountered in aqueous solutions into a convenient range. The modern definition of pH replaces [H+] with{H+} in which the curly brackets signify the effective concentration of the hydrogen ion, which chemists refer to as thehydrogen ion activity:pH =− log{H+}Chem1General Chemistry Reference Text11Acid-base equilibria and calculations• 3 Quantitative treatment of acid-base equilibria3Quantitative treatment of acid-base equilibria3.1Strong acids and basesThe usual definition of a “strong” acid or base is one that is completely dissociated in aqueous solution.
Hydrochloric acid is a common example of a strong acid. When HCl gas is dissolved in water, the
resulting solution contains the ions H3O+, OH−, and Cl−, but except in very concentrated solutions, theconcentration of HCl is negligible; for all practical purposes, molecules of “hydrochloric acid”, HCl, do
not exist in dilute aqueous solutions.In order to specify the concentrations of the three species present in an aqueous solution of HCl, weneed three independent relations between them. These relations are obtained by observing that certain
conditions must always be true in any solution of HCl. These are:1. The dissociation equilibrium of water must always be satisfied:[H3O+][OH−] =Kw(12)2. For any acid-base system, one can write a mass balance equation that relates the concentrationsof the various dissociation products of the substance to its “nominal concentration”, which we
designate here asCa. For a solution of HCl, this equation would be[HCl] + [Cl−] =Cabut since HCl is a strong acid, we can neglect the first term and write the trivial mass balance
equation[Cl−] =Ca(13)3. In any ionic solution, the sum of the positive and negative electric charges must be zero; in otherwords, all solutions are electrically neutral. This is known as the electroneutrality principle.[H3O+] = [OH−] + [Cl−](14)The next step is to combine these three equations into a single expression that relates the hydroniumion concentration toCa. This is best done by starting with an equation that relates several quantities,such as Eq 14 , and substituting the terms that we want to eliminate. Thus we can get rid of the [Cl−]term by substituting Eq 13 into Eq 14 :[H3O+] = [OH−] +Ca(15)A [OH−]-term can always be eliminated by use of Eq 12 :[H3O+] =Ca+Kw[H3O+](16)This equation tells us that the hydronium ion concentration will be the same as the nominal concentration
of a strong acid as long as the solution is not very dilute. As the acid concentration falls below about
10−6M, however, the second term predominates; [H3O+]approaches √Kw, or 10−7M. The hydroniumion concentration can of course never fall below this value; no amount of dilution can make the solution
alkaline!Notice that Eq 16 is a quadratic equation; in regular polynomial form it would be[H3O+]2− Ca[H3O+]− Kw= 0(17)Most practical problems involving strong acids are concerned with more concentrated solutions in which
the second term of Eq 16 can be dropped, yielding the simple relation H3O+= [A−].Chem1General Chemistry Reference Text12Acid-base equilibria and calculations• Concentrated solutions of strong acids0 1 2 3 4 5 6 74321activity coefficientg±HCl concentration, mol/LEffective concentration is greater than
the analytical concentration, owing to
capture of water in ion hydration shells
and electrostatic repulsion between ions.Effective concentration is smaller
than analytical concentration due
to ion pair formation.Figure 1: Mean ionic activity coefficient in HCl as a function of concentration3.2Concentrated solutions of strong acidsIn more concentrated solutions, interactions between ions cause their “effective” concentrations, known
as their activities, to deviate from their “analytical” concentrations.Thus in a solution prepared by adding 0.5 mole of the very strong acid HClO4to sufficient water tomake the volume 1 litre, freezing-point depression measurements indicate that the concentrations of
hydronium and perchlorate ions are only about 0.4 M. This does not mean that the acid is only 80%
dissociated; there is no evidence of HClO4molecules in the solution. What has happened is that about20% of the H3O+and ClO−4ions have formed ion-pair complexes in which the oppositely-chargedspecies are loosely bound by electrostatic forces. Similarly, in a 0.10 M solution of hydrochloric acid,
the activity of H+is 0.081, or only 81% of its concentration.Activities are important because only these work properly in equilibrium calculations. Also, pH is defined
as the negative logarithm of the hydrogen ion activity, not its concentration. The relation between the
concentration of a species and its activity is expressed by the activity coefficientγ:a = γC(18)As a solution becomes more dilute,γ approaches unity. At ionic concentrations below about 0.001 M,concentrations can generally be used in place of activities with negligible error.At very high concentrations, activities can depart wildly from concentrations. This is a practicalconsideration when dealing with strong mineral acids which are available at concentrations of 10 M or
greater. In a 12 M solution of hydrochloric acid, for example, the mean ionic activity coefficient4is 207.This means that under these conditions with [H+] = 12, the activity{H+} = 2500, corresponding to apH of about –3.4, instead of –1.1 as might be predicted if concentrations were being used.These very high activity coefficients also explain another phenomenon: why you can detect the odor ofHCl over a concentrated hydrochloric acid solution even though this acid is supposedly 100% dissociated.4Activities of single ions cannot be determined, so activity coefficients in ionic solutions are always the average, or mean,of those for the ionic species present. This quantity is denoted asγ±.Chem1General Chemistry Reference Text13Acid-base equilibria and calculations• Weak monoprotic acidsWith such high effective concentrations, the “dissociated” ions come into such close contact that the
term begins to lose its meaning; in effect, dissociation is no longer complete. Although the concentration
of HCl(aq)is never very high, its own activity coefficient can be as great as 2000, which means that itsescaping tendency from the solution is extremely high, so that the presence of even a tiny amount is very
noticeable.3.3Weak monoprotic acidsMost acids are weak; there are hundreds of thousands of them, whereas there are no more than a few
dozen strong acids. We can treat weak acid solutions in exactly the same general way as we did for strong
acids. The only difference is that we must now include the equilibrium expression for the acid. We will
start with the simple case of the pure acid in water, and then go from there to the more general one in
which strong cations are present. In this exposition, we will refer to “hydrogen ions” and [H+] for brevity,and will assume that the acid HA dissociates into H+and its conjugate base A−.3.4Pure acid in waterIn addition to the species H+, OH−, and A−which we had in the strong-acid case, we now have theundissociated acid HA; four variables, four equations.1. Equilibria. We now have two equilibrium relations:[H+][OH−] =Kw(19)[H+][A−][HA]=Ka(20)2. Mass balance. The mass balance relation expresses the nominal concentration of the acid in termsof its two conjugate forms:Ca= [HA] + [A−](21)3. Electroneutrality.[H+] = [A−] + OH−(22)We can use Eq 21 to get an expression for [HA], and this can be solved for [A−]; these are substitutedinto Eq 20 to yieldKa= [H+]([H+]− [OH−])Ca− ([H+]− [OH−])(23)This equation is cubic in [H+]when [OH−] is expressed asKw/[H+]; in standard polynomial form itbecomes[H+]3+Ka[H+]2− (Kw+CaKa)[H+])− KwKa= 0(24)For most practical applications, we can make approximations that eliminate the need to solve a cubic
equation.1. Unless the acid is extremely weak or the solution is very dilute, the concentration of OH−can beneglected in comparison to that of [H+]. If we assume that [OH−][H+], then Eq 23 can besimplified toKa=[H+]2Ca− [H+](25)which is a quadratic equation:[H+]2+Ka[H+]− KaCa= 0(26)Chem1General Chemistry Reference Text14Acid-base equilibria and calculations• Weak bases0 2 4 6 8 10 12sstrong acid0 1 2 3 4 5 6Ð1Ð3Ð5Ð7log CpHpKFigure 2: pH as a function of concentration for acids of various strengthsThe shaded area indicates the range of acid strengths and concentrations for which the approximation
Eq 28 is generally valid.2. If the acid is fairly concentrated (usually more than 10−3M), a further simplification can frequentlybe achieved by making the assumption that [H+]Ca. This is justified when most of the acidremains in its protonated form [HA], so that relatively little H+is produced. In this event, Eq 25reduces toKa= [H+]2Ca(27)or[H+]≈ (KaCa)1
2(28)3.5Weak basesThe weak bases most commonly encountered are NH3, amines such as CH3NH2, and anions A−of weakacids. Using the latter as an example, we can write the base constantKb= [HA][OH−][A−]For aCbM solution of NaA in water, the charge balance is[Na+] + [H+] = [A−] + [OH−]Chem1General Chemistry Reference Text15Acid-base equilibria and calculations• Carrying out acid-base calculationsReplacing the [Na+] term byCband combining withKwand the mass balanceCb= [HA] + [A−], arelation is obtained that is analogous to that of Eq 23 for weak acids:Kb= [OH−]([OH−]− [H+])Cb− ([OH−]− [H+])(29)The approximationsKb=[OH−]2Cb− [OH−](30)and[OH−]≈ (KbCb)1
2(31)can be derived in a similar manner.3.6Carrying out acid-base calculationsAcid-base calculations fall into two categories: those that are done as part of a course in which your aim
is to obtain a good grade, and those done for real-world applications in which the goal is to obtain a
useful answer with a minimum of effort. For the latter, there is almost never any need to do anything
other than a graphical estimate as described in Section 6. Unless values ofKwandKa’s that pertainto the actual conditions of temperature and ionic strength are available, carrying out a calculation to a
precicision of more than two significant figures while usingKw= 10−14and theKwvalues commonlyseen in textbooks (which apply only to pure water, which is rarely the subject of our interest) is a waste
of time.Even so, the algebraic approach that is taught in most General Chemistry courses is important in tworespects: the quantitative treatment required to derive the equations provides a clearer view of the equi-
libria involved and serves as a model for the treatment of generalized equilibria, and the approximations
commonly applied to simplify the relations serve as good models of the kinds of judgements that must be
made very commonly in applied mathematics. It is important to bear in mind, however, than the results
are only as valid as the data, and the latter are almost never good enough to yield correct answers for
any but extremely dilute solutions.Selecting the approximation.Eq 25 serves as the starting point for most practical calculations on solutions of weak monoprotic acids.
Don’t memorize this equation; you should be able to derive it (as well as the analogous equation for weak
bases) from the definition of the equilibrium constant. If you know that the system parameters fall into
the shaded region of Fig. 2, then the approximation Eq 28 is usually satisfactory. However, if the acid is
very dilute or very weak, the exact calculation Eq 23 will be required.As an example, consider a 10−6M solution of hypochlorous acid, pKa= 8.0. (HOCl is a decompositionproduct of chlorine in water and is largely responsible for its bactericidal properties.) In this case, the
approximation Eq 23 leads to a quadratic equation whose non-negative root is 9.50E−8, clearly an absurdvalue for a solution of any acid, no matter how dilute or weak. Substitution into Eq 24 and plotting
reveals a root at [H

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