J. CHEM. SOC. FARADAY TRANS., 1994, 90(21), 3245-3252 New Approach to Sensitivity Analysis of Multiple Equilibria in Solutions llie Fishtik Institute of Chemistry, Academy of Sciences of Moldova ,277028 Kishinev, str. Academiei 3, Republica Moldova lstvan Nagypal and Ivan Gutman? Institute of Physical Chemistry, Attila Jozsef University, P.O. Box 105,H-6701 Szeged, Hungary We consider multiple equilibria in solutions in which the interaction of n chemical species is described by means of m stoichiometrically independent reactions (SIRs). For the study of certain thermodynamic properties of such systems, in particular, for sensitivity analysis, it is important to know the determinant A of the Hessian matrix of the Gibbs energy, as a function of the extent of the SIRs.Any linear combination of SIRs, in which (at least)rn -1 species are not involved, is called a Hessian response reaction (HR): Several properties of the HRs are pointed out, in particular, the equivalence of A to the sum of contributions originating from each HR. The effect of temperature and pressure on chemical equilibria in ideal solutions is analysed. It is shown that the sensitivity coefficient of a chemical species A, may be presented as a sum of contributions coming from all HRs in which A, is involved. Each of these contributions is a product of the stoichiometric coefficient of Ai, the enthalpy or volume change of the respective HR, and a concentration-dependent term which is always positive. It is also shown that the relaxation contribution to the heat capacity is a sum of contributions over all HRs.The position of chemical equilibrium is governed by a series of parameters, such as equilibrium constants, temperature, pressure and initial concentrations. The effect of the change of these parameters on the position of chemical equilibria is the subject of the sensitivity analysis.' Sensitivity analysis requires the calculation of a set of derivatives of the equi- librium concentrations with respect to the above parameters. These derivatives are called sensitivity coefficients. The calculation of the sensitivity coefficients for gas-phase reactions was discussed in general terms and the results are summarized in ref. 1. Complex equilibria in solutions and other special cases were considered elsewhere.2-' By means of sensitivity coefficients, one can predict the response of the system to the change of the respective param- eters.In the case of a single equilibrium process, it is easy to predict the sign of the sensitivity coefficients by means of the Le Chatelier principle. For complex systems, however, the required calculations are complicated and there is no qualit- ative way to predict the sign of the overall effect. The main idea of the present paper is to show how the overall sensitivity may be given as a linear combination of contributions originating from individual reactions, which we call response reactions, and which strictly obey the Le Cha- telier principle. In other words, the Le Chatelier principle is extended to multiple equilibrium systems.The effect of tem- perature and pressure is analysed in this paper, the effect of the other parameters will be discussed elsewhere. We arrive at the concept of response reactions by examin- ing the determinant of the Hessian matrix. Namely, when examining different aspects of the equilibrium conditions in terms of stoichiometric formulation we encounter the second derivatives of the Gibbs energy, with respect to the extents of the reactions occurring in the system ~0nsidered.l~ These derivatives may be arranged into a square matrix of order m, where m is the number of independent reactions, which is called the Hessian matrix of the Gibbs energy. Of special importance is the determinant of this matrix, the Hessian determinant, as well as its minors.In particular, the Hessian 7 Permanent address : Faculty of Science, University of Kraguje-vac, Kragujevac, Yugoslavia. determinant and its diagonal minors are known to appear in the expressions for sensitivity coefficients. l4 The Hessian determinant also plays an important role in different algo- rithms for calculating the equilibrium compositions or equi- librium constants of systems with simultaneously occurring chemical reactions, as well as in the sensitivity analysis of complex chemical equilibria. '*14 In spite of the importance of the Hessian determinant, it seems that the only thing we know about it, is that the Hessian determinant and its diagonal minors are of positive va1~es.l~It turns out, however, that the Hessian determinant has a number of other interesting properties which, as far as we are aware, have not been observed before.We study the Hessian determinant and introduce the concept of Hessian (response) reactions. We then apply the Hessian reactions to the analysis of certain sensitivity coefficients. Definitions and Notation In this paper we are concerned with the equilibrium state of a system in which chemical reactions take place. We examine the general case in which the interaction of n distinct chemi- cal species A,, A,, . . .,A,, is described by means of m chemi-cal equations: V11A1 + VlzA, + *. . + V1,An = 0 vZ~A,+ v~,A,+ * * + ~2,,A, = 0 ...............................I vmlA1+ vm2 A, + --+ v, A,, = 0 J Throughout this paper it is assumed that the above equations are stoichiometrically independent, i.e. that It is customary to call the relations in eqn. (1) chemical reac- tions; they will be referred to as stoichiometrically indepen- dent chemical reactions (SIRs). Recall that a set of SIRs can be chosen in many different, but (from the point of view of chemical equilibrium analysis) equivalent ways, i:e. the value of the Hessian determinant does not depend on the choice of the SIRS. The extent of the jth SIR, namely of cf=lvjiAi = 0, at equilibrium will be denoted by tj,j = 1, 2,. ..,rn. Let G be the Gibbs energy of the system considered. G can be viewed as a function of the parameters tl, t,, ..., t,.Then the elements of the respective Hessian matrix are d2G/a<,at,. Instead of a2G/at,at, it is cu~tomaryl*'~ to use the quantities G,s, where G,~ RT = -1 a2G/at,at, (3) In this study we examine the determinant (4) which is the Hessian determinant of the system considered. Hessian Reactions It is necessary to define the determinant D (of order m) I V1,il V1,iz .** v1, i,-1 VI, i, I D = D(il, i,, ..., i,,,) = '2, il ... '2, iz . . . ... ... '2, im-1 ... '2, i, I vm,il ... (5) and its minors Djk, where Djk is obtained from D by deleting its jth row and kth column. Now, any linear combination of eqn. (1) leads to a new chemical reaction between the species A,, A,, ...,An,which is acceptable from a stoichiometric point of view. The general form of such a linear combination is m Tn 1 Consider now the special linear combination of the above kind obtained by choosing: Aj = (-ly'+"Djm; j = 1, 2, .. . ,rn Then eqn. (6)becomes m rn 1 n rm 1 and bearing in mind that j=1 is just the expansion of the determinant D with respect to its mth column, we arrive at the reaction n C vimAim= 0; vim= mil, i, ,..,i,,,) (7)i,= 1 The reaction, eqn. (7), has the noteworthy property that vim = 0 whenever i, = i, or i, = i, or ...i, = i,-1, because then two columns in the determinant D(il, i,, ..., i,,,) are equal [see eqn. (5)]. This means that the species A,,, Aiz,..., Aim-l have zero stoichiometric coefficients and are thus not involved in eqn.(7). For reasons that will become evident later, we call eqn. (7) a Hessian response reaction, or shorter J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 a Hessian reaction (HR). We denote eqn. (7) by H = X(i,, i,, ...,im-l), where i,, i,, ...,im-indicate the species which are absent from S. There are (m-1) possible choices of the species A,,, Ai2, ...,Aim-,whose absence defines a particular Hessian reaction H,but not all such reactions need to be distinct and only rn of them can be linearly independent. We can now easily deduce an important property of the HRs. Let X be a state function whose change in thejth SIR is AXj, j = 1,2, ...,rn. Then the change of X in eqn. (6) is m AX = CljAXj j=1 Applying the above formula to a Hessian reaction, eqn.(7), we obtain: m AX(H) = 1(-1)j+'"DjmAXj j=1 Property of the Hessian Determinant In the subsequent section we demonstrate that the Hessian determinant A, eqn. (4), is equal to the sum of certain increments, each being associated with a particular Hessian response reaction : where the summation is over all HRs, and T(X),the contri- bution of the reaction H = S(il, i,, . . .,im-1), has the form UJf) (10) In the above formulae and later on, [A,] denotes the (equilibrium) concentration of the species Ai. D(il, i, ,. . .,i,) is defined by eqn. (5). Note that owing to eqn. (l), at least one of the determi- nants D(il, i,, ..., i,,,) must be non-zero. Consequently, at least one HR must have a non-zero contribution and, there- fore, A is necessarily of a positive value.Proof of Eqn.(9) Our starting point is the thermodynamic relation14 for the quantity G,, eqn. (3), where n, is the amount of the species A, in the system con- sidered, and n, the total amount of all species present in the system [including also those which do not participate in eqn. (l)]. Introducing the auxiliary quantity psi n J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 eqn. (1 1) can be rewritten as i Vri Psi Grs = i=l ni (13) We shall now exploit the following property of determi- nants: If alj = aaij + /?a';j+ yayj + * * -holds for all j = 1, 2, ....m,then a;, a,, ... a;, a,, *.-Q2m=a ............ amm a';, a,, *-. a2m + p/ ............a?, a,, ... a1m a;, a22 ... a2m + ...+y ............ a;, am2 *.-amm Let A' be the transposition of A, i.e. Then A' = A. From eqn. (13) and applying eqn. (14) to all of the columns of A', we obtain D*(il, i,, ...,i,,,) = P2, il Pz. i2 ' * P2. i, ............ IPm,il Pm,i2 *** If any two of the indices i,, i, ..... i, are mutually equal, then the respective columns in eqn. (16) coincide, and, there- fore, the corresponding summand in eqn. (15) is equal to zero. We thus have to examine only those summands on the right- hand side of eqn. (15), in which the indices i,, i, ..... i, are all mutually distinct. Let (hl, h,, ....h,) be an ordered m-tuple of integers, such that 1 < h, < h, < -< h, < n.Then one of the summands in eqn. (1 5) will be of the form There will be additional m!-1 summands in which the indices i,, i,, .... i, coincide with h,, h,, ....h,, but not in that order. Then the determinant D*(i,, i,, .... i,) can be brought into the form D*(h,, h,, .... hd by a number of transpositions of columns of D*(i,, i2..... id. Each transposi- tion of two columns of a determinant causes the change of its sign. Consequently, it will be D*(i,, i,, .... i,) = (-ly'D*(h,, h,, ....h,,,) (17) where # is the number transpositions required in the mapping (il, i,, .... i,,,) +(h,, h,, .... hm). As known from algebra, fi is just the parity of the permutation (i,, i, ..... i,,,) relative to (hl, h,, .... h,,,). If we sum the m!terms on the right-hand side of eqn.(15) in which the indices i,, i,, .... i, (when appropriately ordered) coincide with (h,, h, ..... h,,,), and take into account eqn. (17),then we obtain nn D(i,, i,, .... i,,,)D*(i,, i,, .... i,)A= (rn!)-' 11 il=l i2=l i,=l nil ni2 -* nim This can also be expressed as Eqn. (19a)-(19c) are based on eqn. (1 1) and, thus, they rep- resent thermodynamic identities, whose range of applicability coincides with that of eqn. (ll), i.e. eqn. (19) holds for ideal systems. The first summation on the right-hand side of (19c) can be interpreted as being over all Hessian reactions, and thus we may write (19c)as where Y(=@)=-1 D(il, i2, .... im)D*(il,i2, ....i,) (21)nil ni2 -- - nimm im=l In the case of solutions, the terms (ndn,) in eqn.(12) can be considered as negligibly small. Then psi = vSi, and as a special case of eqn. (19b) and eqn. (21) we have and 1 D(i1, i2, .., i,)2Y(=@)=-m im=l (22)ni, ni2-nim In the case of solutions it is usual to exchange ni with [Ai], which can be done in view of the fact that the volume of a solution may be considered as constant. Then eqn. (20) and (22) simply reduce to eqn. (9) and (10). 3248 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Note that the term In order to interpret the Hessian determinant in terms of contributions from Hessian reactions, we first have to con- A(i,, i,, ...,i,-,) = ~(i,,i,, . . . ,iJ2 (23) struct the HRs. To do this, two of the chemical species from a i,=l [Aim] total of five are to be excluded from the above reaction occurring in eqn.(10) for r(X)can be understood as the scheme. If these two species are, for example, A, and A,, then Hessian determinant of the Hessian reaction X(il,i,, ..., by applying eqn. (7) we have: i, -l). To see this, recall that, in the case of a single reaction -1 -1 1 -1 -1 0 (m= l), the Hessian determinant is just G,,. On the other -1 -2 0 A3+ -1 -2 1 A4hand, eqn. (1l), when appropriately modified for solutions, yields -1 -3 0 -1 -3 0 -1 -1 0 + -1 -2 0 A, = 0 -1 -3 1 If the single reaction considered is a HR, then vli is given by eqn. (7) and the right-hand sides of eqn. (23) and (24) which is equivalent to: coincide. A, + A, = 2A4 When this procedure is repeated for all possible combinations Example 1 of two species, one arrives at the following set of HRs (in In order to illustrate the above results we consider a solution brackets are indicated the species which are eliminated from in which five chemical species A,, A,, A,, A, and A, react the initial SIRS): according to the following reaction scheme: (A,, A,) A, +A5 = 2A4 (i) A, + A2=A3 (AlY A31 A2 + A4 = A, (ii) A, + 2A2 = A, I (25) (iii) A, + 3A2 = A, (4, A4) 2A2 + A3 = A, The above scheme may, for example, represent a stepwise (A,, A,) A2 + A3 = A4 complex formation or a protonation process : (A2, A,) A, + 2A5 = 3A4 (i) M+ L=ML (A2, A4) 2A1 + A5 = 3A3 (ii) M + 2L = ML2 (A23 A,) A, + A4 = 2A3 (iii) M + 3L = ML, (A3, A4) A, + 3A2 = A, The Hessian determinant for eqn. (25) is evidently of the third order : (A33 A,) A1 + 2A2 = A4 (A4, + = Gll G12 G13 To each of these reaction equations we associate a function G12 G22 G23 A(il, i2), eqn.(23):G13 G23 G33 1 4with A(1, 2) = -+ -+-1 [A31 CA4I CAsI1G --+-1 +-1 1 1 l1 -CAiI [A21 CA3I A(1, 3) = -+ -+-1 CA2I CA4I CAsI1 4G22 = -+ -+-1 4 1CA1I [A21 CA4I A(1, 4) = -+ -+ -1 [A21 [A31 CA5I1 9G33 =-+ -+ -1 1 1[A11 CA2I CASI A(1, 5) = -+ -+-1 CA2I [A31 CA4I1G12= GZ1 = -+ -2 1 9LA11 [A21 A(2, 3) = -+ -+-4 CA1I CA4I CAsI 4 9A(2, 4) = -+ -+-1 CAI] [A31 CA5I 1 4A(2, 5) = -+ -1 + -CA2I CA4I CA3I The stoichiometric matrix for the reaction scheme eqn. (25) 1 9reads : A(3, 4) = -+ -+-1 [A11 CA2I [As] 1 4A(3, 5) = -+ -+-1 ;'-1 -1 1 0 CAI] CA2I CA4I /i -1 -2 0 1 1 1A(4, 5) = -1 + -+ -;I,'-1 -3 0 0 CAiI [A21 CA3I J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Then the Hessian determinant for the reaction scheme eqn. (25) is given by Thus, in this case, the Hessian determinant is equal to the sum of contributions from the ten Hessian reactions. Some Basic Notions of Sensitivity Analysis Consider the system of stoichiometrically independent reac- tions (SIRS) given via eqn. (1). Let tj,Kj, Aj H" and Aj V" stand for the extent, equilibrium constant, standard enthalpy and standard volume change, respectively, of the jth chemical reaction, j = 1, 2, . . . ,rn. Let [Ailo and [Ai] be the initial and equilibrium concentrations of the species Ai, and let pLp be its standard chemical potential, i = 1,2, ... ,n.Then, we have: m cAi1 = CAiIO + vki tk (26) &=1 n n / in The eqn. (27) may be considered as an implicit system of equations in the unknowns tj,[Ailo, Kj, pLp, T and P, i = 1, 2, ..., n, j = 1, 2, ..., rn. The latter five quantities are parameters influencing the position of chemical equilibrium. In what follows we denote them briefly by X. Then, eqn. (27) may be written as : fi(t1,52, ..., tm, j= 1, 2, ..., (28) where we use the abbreviation n in From eqn. (28) we have m aj-. at1 Jk= ah. j = 1, 2, ..., rn k=l atkax ax' Differentiating eqn. (28) with respect to the extents of the reactions (l), we obtain and thus, the basic problem of the sensitivity analysis is to solve the following system of rn linear equations in unknowns atj/ax,j = 1, 2, ...,m: The solution of eqn.(29) is given as: j = 1, 2, ..., m (30) where Ajp is the minor of the Hessian determinant obtained by deleting its jth row and pth column. The sensitivity coefficients a[Ai]/aX are now readily calcu- lated by differentiating the mass-balance conditions, eqn. (26): at.aCAi1 =-aCAilo + Cvjil; i = 1, 2, ..., nax ax j=l ax where atj/aX are given by eqn. (30). To obtain more specific results we have to specify the parameter X. The derivation of the expressions for the sensitivity coeffi- cients of the specified parameters X is analogous to that used in the preceding section for the Hessian determinant.There- fore, in the following we present only the main results. Effect of Temperature In the case when X is the temperature T, we have from eqn. (27) and (28) Substituting eqn. (31) into eqn. (30) and using a similar way of reasoning as in the preceding section, we arrive at 1atjaT RT~A where D(il, i, ,...,im-AH")is the following determinant of order rn D(il, i,, ..., im-l, AH") ... and Dimis the minor of the determinant D(il, i,, ..., im-l, AH"), obtained by deleting itsjth row and mth column. Eqn. (33) should be compared with eqn. (7). Through a similar derivation it can be shown that for the sensitivity coefficients of the equilibrium concentrations, the following is valid : (34) where D(il, i,, .. . ,im-1, i) is defined uia eqn. (5). Eqn. (34) has a simple chemical meaning. We have already shown that D(il, i,, ...,im-1, i) is just the stoichiometric coef- ficient of the species Ai in the Hessian response reaction if = &(il, i,, . . ., irn-l),whereas D(il, i,, . . ., im-l, AH") is the enthalpy change of the same response reaction. Thus we can write eqn. (34) in the form 1----Ic(%)vAJf')AH"(Jf') (35)aT RT2, 3250 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 1 Hessian reactions for Example 2 reaction eliminated species HR AHlkJ mol-' H+, OH- C03'-+ H2C03= H+,C032-HC0,-+ H20 = H+, HC03- C032-+ 2H20 = H+, H2C03 C032-+ H20 = OH-, C03'- HC03-+ H+ = OH-, HC03- 2H+ + C032-= OH-, H2C03 H+ + C03'-= C03'-, HC03- H20 = C032-, H2C03 H20 = HC03-, H2C03 H20 = 2HC03-2A2H" -A3 H" = -5.71 H2C03+ OH-A,H" -A2 H" + A, H" = 47.19 H2C03+ 20H-2A,H" + A3 H" = 88.67 HC03-+ OH-A,H" + A2 H" = 41.48 H2C03 -A2 H" + A3H" = -9.37 H2C03 A3 H" = -24.45 HC03-A2 H" = -15.08 H+ + OH-A,H" = 56.56 H+ + OH-A,H" = 56.56 H+OH-A,H" = 56.56 11:: where 4%)= {A x [Ail] x CAiJ x *.* x CAi,-iI)-' (36) Observe that 4%) depends on the equilibrium concentra- tions of the species Al, A,, ..., A,, whereas vi(M) and AH"(&')are concentration-independent. Because A is always positive-valued, it is clear that also 4%)is positive-valued for all Hessian reactions &'. The determinant D(il, i,, ..., i,,,-l, i) = vi(&')is equal to zero whenever i = il or i = i, or .. . i = i,-'. Consequently, vi(&')= 0 whenever the Hessian reaction &' does not involve the species Ai. We thus see that only those HRs in which the chemical species Ai is involved have non-zero contributions to the summations on the right-hand sides of eqn. (34) and (39, i.e. non-zero contributions to the sensitivity coefficient a[AJ/aT. The sign of the contribution of any particular Hessian reaction &' is easy to determine: it is equal to the sign of the product vi(&')AH"(H). Example 2 Consider the proton-carbonate system : (1) H,O = Hf + OH-AIH" = 56.56 kJ mol-' (2) H+ + C03,-= HC03-A, H" = -15.08 kJ mol-' (3) 2H+ + C03,-= H2C03 A3 H" = -24.45 kJ mol-' The stoichiometric matrix for this system reads : H+ OH-H,O C03,-HC03-H2C03 1 1 -1 0 0 0 0 -1 1 110 0 -1 0 1 We first find the Hessian reactions by eliminating all pos- sible groups of rn -1 (= 2) species from the initial set of inde-pendent reactions.Concomitantly, using eqn. (33) the enthalpy changes of these reactions are calculated. For instance, when the species to be eliminated are H+ and OH-, the resulting reaction equation is: 1 1 -1 1 10 -1 0 0 H,O+ -1 O -I C03,--2 0 0 -2 0 -1 110 110 + -1 0 1 HC03-+ -1 0 0 H2C03=0 -2 0 0 -2 0 1 1 1 AIH" AHo(Hf,OH-) = -1 0 A2 H" = -2A2 H" + A3 H" -2 0 ASH" Continuing the procedure we arrive at the 10 HRs given in Table 1, of which only eight are distinct. From this data it follows that the sensitivity coefficients a[H']/dT and a[OH-]/aT are positive.For instance, H+ is involved in reactions (v)-(x) and in each of these reactions the product of its stoichiometric coefficient and the enthalpy change is positive. Analogous considerations reveal that the sensitivity coefficients c?[CO,~ -]/dT, d[HCO, -]/dT and d[H2C03]/aT may be both positive and negative. From the 10 Hessian reactions in Table 1 one can easily derive expressions for the sensitivity coefficients. To do this, the product of the stoichiometric coefficient and enthalpy change of an HR has to be divided by the equilibrium concen- trations of those species which were eliminated from that HR, and the results summed over all the Hessian reactions in which the species under consideration is involved.For example: a[H+] 1 9.37 2 x 24.45 [OH-][C032-] [OH-][HCO,-]+ 15.08 56.56 [OH-][H,COJ + [CO, 2-][HC03 -1 56.56 [C032-IL-H,C031 + CHCO,-IL-H,CO,I Some numerical calculations for this system are presented in Fig. 1 and 2. In a solution of strong acid or base and in the absence of carbonate, the temperature dependencies of H+ and OH- are given by dCOH-1 d[H'] -[H+][OH-] AIH"-=--dT dT [H'] + [OH-] RT2 As a function of pH, both d[OH']/dT and d[H']/aT have a maximum at pH = 7. It is seen from Fig. 1 that in the pres- 0.030 10.025 O.OIO 0.008 --. n5 0.020 + 0.006n & 0.015 0, rg Q5 0.010 0.004 5 0.005 0.002 0.000 0.000. ~~~ 2 4 6 8 10 12 PH Fig. 1 Temperature derivatives of [H'] and [OH-] as functions on pH in the C032--H+ system.[C0,2-] + [HC03-]+ [H2C03] = 0.1 mol drn-'. (a)a[H+]/aT; (b)a[OH-]/aT. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 325 1 equilibrium heat capacity, while the second sum is the relax- 3c ation contribution : 2 -2 " 6 7 8 9 10 11 12 PH Fig. 2 Temperature derivatives of [CO,'-], [HCO,-] and [H,CO,] as functions on pH in the C03*--H+ system; notice that their sum is equal to zero. [CO,'-] + [HCO,-] + [H,CO,] = 0.1 mol dm-,. 1, X = HCO,-; 2, X = H'CO,; 3, X = CO,*-. ence of carbonate, the peak of a[H+]/aT is shifted to the acidic region while that of a[OH-]/aT is shifted to the basic region. Note the extremely low value of a[H']/dT. Effect of Pressure In the case when pressure, P,is the parameter influencing the chemical equilibrium, we have from eqn.(27) aln K. AVOafi -afi --1=I ax -ap ap RT It follows that eqn. (31)-(35) may be applied also in the case of pressure: it is only needed to exchange the terms A,H" by -TAj Voin all formulae. Chemical Equilibria Contributions to Heat Capacity Owing to the temperature dependence of the chemical equi- libria, the heat capacity is the sum of two contributions. The first is the conventional compositional contribution (Cp,camp), while the second is the relaxation contribution (Cp,,,J. Recent studies16-" show that Cp,rel may range from negligibly small to several hundreds of J K-' mol-l. The above derived equations can be used to interpret and evaluate this latter contribution.The total enthalpy of a homogeneous system can be expressed in terms of the partial molar enthalpies, H(Ai), of the components and the amounts, n,,of each component at equilibrium: n H = 1niH(Ai) i= 1 We deal with molar concentrations and, assuming that the volume of the solution is temperature independent, we con- sider the enthalpy in a unit volume of solution 4 n H = 1[Ai]H(Ai) (37)v i=l Differentiation of eqn. (37) with respect to temperature at constant pressure leads to a formula for the heat capacity at equilibrium The first sum on the right-hand side of eqn. (38) is identified as the conventional compositional contribution to the total where C,(Ai) stands for the molar heat capacity of the species Ai, i = 1, 2, ..., n.Here we are principally concerned with C, re1 * In order to obtain the relaxation contribution to the heat capacity, the mass-balance conditions, eqn. (26), are to be dif- ferentiated with respect to temperature: a[Ail C vji atj-= aT j=1 Combining eqn. (39) and (40) we obtain where Aj H"is the standard enthalpy change of jth SIR [see eqn. (l)], satisfying the condition n AjH" = 1vijH(Ai) i= 1 Taking into account eqn. (32), the relaxation contribution to heat capacity becomes 1 D(il, i,, ..., i,,,-l, AH")'7Cp,re1 = -1 cRT'A il<i2<...<im-l [Ail] x x *** x CAi,,,-,I (41) In order to deduce eqn. (41) we used the identity m l(-l~+"DjmAjH"= D(il, i,, ..., AH") j= 1 In analogy to (35), formula (41) can be written as Hence, also Cp,relis equal to a sum of contributions coming from Hessian response reactions. In contrast with eqn.(35), each summand in eqn. (42) is non-zero (provided that AH"($f) differs from zero). Example 3 We evaluate the relaxation contribution to heat capacity for the system NH,-H+, for which the following data are avail- able: H,O = H+ + OH-; A,H" = 56.56 kJ mol-' NH, + Hf= NH,+; A, H" = -52.22 kJ mol-' As the concentration of H,O is constant we have to consider only four chemical species: H', OH-, NH, and NH4+. The stoichiometric matrix for the above reaction scheme reads : H+ OH-NH, NH,' H20 1 1 0/1 -I-l 0 -1 O1 0 Eliminating consecutively from the above reaction equa- tions H+, OH-, NH, and NH4+, we arrive at the following PH Fig.3 Relaxation contribution to heat capacity in the NH,-H+ system. [NH,] + [NH,'] = 0.1 mol dm-,. Hessian reactions : (i) (H+) NH, + H,O = NH,+ + OH-A,H" + A2H0= 4.34 kJ mol-' (ii) (OH-) NH, + H+ = NH4+ A2 H" = -52.22 kJ mol-' (iii) (NH,) H,O = H+ + OH-A,H" = 56.56 kJ mol-' (iv) (NH4+) H20 = Hf + OH-A,H" = 56.56 kJ mol-' We thus see that the equilibrium in this system is influenced by four HRs [although reactions (iii) and (iv) are identical]. In accordance with eqn. (41), the relaxation contribution to the heat capacity is given by (4.34)2 (-52.22)2 (56.56)2 (56.56)2 +[H'] [OH-] +-[NH,] +-][NH, 3 The dependence of Cp,rel on pH is shown in Fig. 3.Discussion and Concluding Remarks The main finding reported in this paper is that various quan- tities, occurring in the sensitivity analysis of multiple chemi- cal equilibria in solutions, can be expressed as linear combinations of contributions that are associated with certain chemical reactions. Results of this kind apply both to the Hessian determinant [see eqn. (9)] and to the sensitivity coeficients [cf: eqn. (34)]. In view of this we speak about 'Hessian response reactions', HRs. We mention in passing that in the sensitivity analysis of gas-phase equilibria, response reactions other than Hessian are encountered.' The Hessian determinant is shown to be the sum of certain well defined contributions from all HRs. The same applies to the sensitivity coefficients of a chemical species Ai, except that here only those HRs in which Ai participates have non- zero contributions.We applied our general theory to the effects of temperature and pressure on the position of multiple chemical equilibria. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 These effects may be quite complicated. It was shown, however, that the overall effect is equal to the sum of contri- butions from HRs, having very simple forms: In the case of temperature each such contribution is equal to the product of the stoichiometric coefficient of Ai in the respective HR, the enthalpy change of that HR and of a concentration-dependent term; the latter is always positive. Fully analogous results hold also in the case of pressure.This makes it pos- sible to deduce easily the sign of each such contribution. On the basis of the above considerations the following pro- cedure for analysing the effect of temperature (pressure) on the complex chemical equilibria may be proposed. First, we have to derive all the Hessian reactions. Then, by means of eqn. (33) we evaluate the enthalpy change (volume change) of each HR. In order to determine the sensitivity coefficient of the chemical species Ai we select those HRs in which Ai is involved. For each such HR the stoichiometric coefficient of Ai is to be multiplied by the enthalpy change (volume change) and by c(&), eqn. (36). This yields the contribution of the given HR. The sensitivity coefficient is then the sum of such contributions over the selected Hessian reactions.References 1 W. R. Smith and R. W. Missen, Chemical Reaction Equilibrium Analysis: Theory and Algorithms, Wiley, New York, 1982. 2 M. Beck and I. Nagypal, Chemistry of Complex Equilibria, Aka-demiai Kiado, Budapest, 1990. 3 A. A. Bugaevski and B. A. Dunai, Zh. Anal. 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