chemical principle1 Edited by

MVRm BOYD BISHOP Clemson University

Ciemson. SC 29631

Chemical Equilibrium

11. Deriving an Exact Equilibrium Equation

Adon A. Gordus The University of Michigan. Ann Arbor, MI 48109

In this article in the series1 on chemical equilibrium we consider how to derive a completely general equation for any chemical mixture. We use as one example the titration of a polyprotic acid with a weak base and show how the expres- sion can be simplified for use with specific acid-base combi- nations. Another example is the precipitation titratiou of Ag+ or C1-.

An exact equation can always be derived for the equilibri- um existing for any chemical mixture even if the mixture includes precipitates, gases, as well as ionic species and even if the solutions are nonideal. The procedure involves first determining how many concentration (and/or pressure) variables exist in the equilibrium mixture and then writing equations equal in number to the number of concentration variables.

Tltratlon ol a Trlprotlc Acld The first example is a fairly complicated hut very general

and useful case in which V. mL of a C, M triprotic acid H3X is mixed with Vb mL of a Cb M base2 ROH. This could correspond to a stage in the titration of H3X with ROH. I t is not necessary that the acid or base he weak; this can be soecified later.

There are eight concentration variables for this equilihri- um mixture.Thev are: IH'I. IOH-I. IH.Xl, IHjX-I, IHX2-1. . . - [X '-1, [HOH], and [R'I heref fore; &ht equat~ons describe theequilibr~ummixture.The equations w~ll always mcludea charge-halance expression' (i.e., elertroneutrality), mass- balanceesuations'equal in number to the number ofchemi- cal compo&ds added to the mixture, and the equations for the various equilibrium constants. In this case, the equations are:

1. Charge balance: total molarity of + charge = total molar- ity of - charge

[Ht] + [Rt] = [OH-] + [H&] + 2[HX2-] + 3[XS-]

2. Mass balance for X: mmol X added = mmol X in all forms in equilibrium mixture

V.C. = (V. + Vb)l[H3X] + [H&] + [HX2-I + [XP-1)

3. Mass balance forR: mmol R added = mmolR in all forms in equilibrium mixture

VhCb = (V, + Vb)IIROHl + [R'II

The charge-balance and mass-balance equations are based on co&entrarions. They do not includeectir,ity coeffi- cients, no matter how nonideal the solution may be. That is

not the case for the remaining five equations that involve equilibrium constants. The reason is that activity is defined to maintain the simplicity of the ideal-gas thermodynamic eadlibrium-constant eauation and. therefore. actiuitv coef- " , f k z n t s occur only in equilibrium constant eipressions.

The five equations for the equilibrium constants are writ- ten in terms of Kc expressions based on concentrations in M. By doing it this way, the final result is an expression having K, terms. But, as discussed in the first article in this series1, K, = KIKf, where K is the thermodynamic equilibrium con- stant (for which values are given in handbooks and texts) and Kf is the "constant" for the activity coefficients. If we assume that the solution is ideal, in which case Kt is equal to 1.00, we can use the values for the thermodynamic K for Kc. However, if we include nonideal effects, we simply substi- tuteKIKr in the final equation because it is only the Kvalues that are effected by nonideality.

The five equilibrium constants are:

These equations are algebraically combined so that the

' The previous article is: Gordus, A. A. J. Chem. Educ. 1991, 68, 138~ ~ - - .

ROH is used to describe the base, implying that the ionization is ROH - R+ + OH-. It could also have been described as a base R (which is typical of amines) with the reaction being R + H,O - RH' OH-. We also write the hydrogen ion as H+ rather than as H,O+; this results in less clutter in algebraic expressions.

The charooebalance exmession involves summina the tolal mo- larityof posili;echargeandkqdat~ng it to the tolal moiaityof negative charge in tne solul'on. The concentrations of multiply charged ions are multip ed by the absolute values of their charges.

'Mass balance calculations are usually performed in terms of moles(or millimoies) since the mass of a given species is proportion- al to the number of moles. The species usually chosen for mass balance exmessions are those that are uniaue to the oarticular mixture. ~ h k . H+ and OH- are not chosen because the" ilso arise . ~ ~ ~ -~~~~~~ , - -~ ~ -~

from tne water equiliorium. If a mixture conlains an ion arising from more than one solute (for example. Na', when two or more Na' salts are added) tnen the mass oalance express on for that species lakes into account the multiple sources of the ion. In cases where some of lhe added ions result in a precipitate, we try to avoid using those ions in the mass balance equations and base the calculations on ions that are not involved in the precipitation.

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final expression has only one variable; [H+] is usually cho- sen. Usually, the best procedure is to use all but the charge- balance expression to solve for each of the concentration variables in terms of [Hf]. Then the expressions for the concentration terms are inserted into the charge-balance equation. In this case the resultant expression is sixth power in [H+]. However, it is only first power in Va or Vb and it is more useful to write i t so that the volume terms are com- bined. The result is:

This equation may look complicated, but i t can be easily programmed in a microcomputer or even a pocket calcula- tor, where the quantities: V., C., Cb, K1,, K2, K3$! Kbo and K,, are stored separately as constants to be spectfied for a particular application. We then specify a pH (or [H+]) and calculate5 the value of Vb required to produce this pH. With a microcomputer and a spread-sheet program with graphing capabilities, we can also automatically generate and plot a full titration range of Vb values.

Diprotic or MonopmNc Weak Acids Eouation 1 is useful hecause i t can be used for manv other

systdms besides H .X + ROH. For instance, if we c ~ ~ s i d e r a weak d i~ro t i c acid such as H EO.,, we ansian K,, = zero. lfwe - .~ conside; a weak monoprotic acid such as acetic acid, we assign K2= = K3< = zero.

Weak Acid Alone To obtain the exact expression for a C. M solution of a

weak acid, assign Vb = zero. For example, for a C, M solution of a weak monoprotic acid, Vb = Kz, = K3< = zero and eq 1 reduces to:

[H+13 + KI,[H+I' - (C.KI, + Kwc)[Ht1 - KI&, = 0 (2)

Weak Base Alone To obtain the expression for a weak base, ROH, substitute

in eq 2 [OH-] for [H+], and subscript b for a. An alternate equation, where [Hf] is the variable, could be obtained by setting V, = zero in eq. 1. The result is:

Strong Acid Alone The expression for a C. M solution of a strong acid (Vb =

K2. = = zero, KI, = a large number such as 100,000) reduces"

Strong Base Alone The expression for a Cb M solution of a strong base can be

obtained from eq 3 by substituting [OH-] for [Ht] and Cb for C, or from eq 1 by setting V. = zero and&, = a large number such as 100,000. In this case, eq 1 reduces7 to:

R(OM3 + HA Tltratlon An expression analogous to eq 1 can be derived for the case

of a mixture of Vb mL of a base R(OH)3 + V, mL of an acid HA. However, there is no need to derive it because the equation can be obtained by substituting intoeq 1: C,for Cb, Cb for C., V. for Vb, Vb for Va, [OH-] for [Ht], K., for Kb, and assigning KI,, Kze, and K3< to the R(OH)a hase. The same simplifications used for the H3X + ROH mixture will also apply to the R(OH)3 + HA mixture.

The determination of the concentration of CI- by precipi- tation titration with Arr' inastandard lahoratorv exercise. If we have Vct mL of c c l ~ NaCl and titrate it with V A ~ mL of Cns M AgN03, there will be four concentration variables in the equilibrium mixture: [Ag+], [NOs-], [Na+], and [CI-1. The four equations8 required to describe the system are:

1. Charge balance: total molarity of + charge = total molar- ity of - charge

[Apt] + [Na+] = [NOa-] + [CI-]

2. Mass balance for Nos-: mmol NO3- added = mmol NO3- in mixture

VA&A, = (Vag + VCI)[NOB-I

3. Mass balance for Na+: mmol Nat a t start = mmol Naf in mixture

4. Solubility product for the precipitate: K, = K8dKep~ = [AgtlIC1-I

This method of treating Vb, rather than pH as the dependent variable has bean described by: Stairs R. A. J. Chem. Educ. 1978,55, 99and Willis, C. J. J. Chem. Educ. 1981, 58, 659-663.

B i n this case, the whole term in parenqesis following C, in eq 1 equals unity because K,, = KZc = 0 and K,,[H+I2 >> [H+I3.

In this case, Kk[H+] >> KK,, so that the first term in the Vb factor of eo 1 reduces to G.

0 0 10 20 30 40

mL 0.100 M Silver Nitrate

Almost all oreci&tion eauilibriaare more comalex than imalled - ~ ~7~ ~ ~ ~

by me simple solubiky product, even in the C&O~&CI where;o"I such as AgCI; ana soluble molecular species such as AgCl(aq) exist in solution. A s in most texts, we neglect these secondary reactions.

PAg = -log [Agi] for the tltratlon of 20.00 mL of 0.140 M NaCl with 0.100 M %NO*

218 Journal of Chemical Education

We assume that some of the Ag+ and an equivalent amount of CI- are consumed in the formation of the precipi- tate. Therefore, the mass-balance equations were based on the ionic species that do not participate in the precipitation reaction. As in the case of the triprotic acid derivation above, we use an equilibrium constant based on concentration terms. If the solution is ideal, K,,, = K,,, the solubility product given in handbooks and texts, if nonideality is to be included, then K,,, = KsPIK,,r where K,,c = f ~ ~ + f c ~ - , the product of the activity coefficients.

I t is best to treat [Agf] as the principal variable. Combin- ing the equations, the result is found to he a quadratic in [Ag+], but, as with the acid-base example, only first power in the volume of titrant, V A ~ . The result, using [Ag+] as the variable, is:

This equation can he used to generate a titration curve where pAg = -log [Ag+] is plotted versus mL AgN03. The simplest procedure is to specify a value of [Ag+] (or pAg) and to use eq 5 to calculate the value of V A ~ required to produce this value of [Ag+], all other variables having been specified. The titration curve will look like that of the fieure. which is for V, = 20.00mLof C(., = 0.140 M NaCl titraihd k t b c,, = 0.100 M AeNO,: K,. = 1.78 X 10.' '.The euuivalence ~ o i n t is a t V A ~ =-28.6 mi . This type of titraGon curve can be generated experimentally by performing a potentiometric titration using a silver-selective electrode.

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