4928
J . Phys. Chem. 1989, 93, 4928-4930
Thermodynamics of the Complex Formation Reactions. 1. Mononuctear Complexes I. F. Fishtik Institute f o r Chemistry, Academy of Sciences of the M.S.S.R., 277028, Kishinev. ul. Grosula, 3, U.S.S.R. (Received: September 23, 1987; In Final Form: December 18, 1988)
The thermodynamicsof the complex formation reactions is discussed in terms of generalized interaction equation. An alternative derivation of the Gibbs energy change is presented. By differentiation of Gibbs energy change with respect to temperature the equations for other thermodynamic functions are derived. The dissociation of the complex ML, and the influence of the ligand protonation are also discussed.
Introduction The thermodynamics of the complex formation reactions is considered to be complete, when the following set of data is known:
+ L = ML, P I , AH]' M + 2L = ML2, 02, AH2' M
M
+ nL = ML,,
(l)
where CMis the total metal ion concentration. Equations 3 and 4 are interrelated. It means that the eq 3 can be obtained from eq and vice versa. The derivation Of the eq from eq by differentiating with respect to temperature was done in ref 7 and 8. Here, let us consider the reverse procedure, Le., the derivation of eq 3 from eq 4 by integration. To do this, eq 4 is transformed in the following way:
fin, AH,'
Here pi is the thermodynamic stability constant and AHi' is the standard enthalpy change of the ith reaction in (1). The Gibbs energy and entropy changes are determined by the relation
AGio = -RT In
Pi = AHi' - T S i 0
i= I
(2)
The standard thermodynamic functions are of great importance in the interpretation of the nature of the chemical interaction in the metal ion-ligand system. At the same time, the complex formation reactions usually take place under conditions that are far from standard. In particular, the concentration (activity) of the ligand may be different from the unity. In order to work out a series of important questions, it is frequently necessary to know the thermodynamic functions of the complex formation reactions under conditions different from standard. With respect to Gibbs energy, this problem was extensively d i s c u s ~ e d . ~ -It~was shown that the function
AG = -RT In
CM = 1 + Pl[LI + P2[L12 + ... + Pn[LIn where [L] is the ligand concentration, has the character of a partition function for a macrocanonical ensemble of statistical mechanics and, hence, the following equation is valid:
AG = -RT In
EM= -RT
In (1
+ x-ji[~]i)
(3)
I=]
It is also accepted unanimously6 that the overall enthalpy change of reaction 1 is given by the relation n
AH = EJ.AHio i= I
Here
Here and below the temperature dependence of the ligand concentration (in molarity units) is assumed to be small and is neglected. From the general thermodynamic relations we have
E M + const
When [L] = 0, AG = 0 and const = 0. Hence, eq 3 is obtained. Knowing the AG or AH, all the other thermodynamic functions can be easily derived, It is the purpose of the present paper to do this: however. it is also of interest to give an alternative derivation and interpretation of eq 3. The Goposed approach also permits one to consider the dissociation of a complex ML, and to take into account the ligand protonation.
Generalized Interaction Equation Equation 1 may be substituted by a single, generalized equation
(4) M
+ AL = CJ;:MLi i=O
is the molar fraction of the complex ML, f.=-[MLiI
'
-
Pi[LI'
cM
1
(5)
+ 5Pi[L]' i=l
( I ) Wyman, J. Biophys. Chem. 1981, 14, 135. (2) Poland, D. Cooperafiue Equilibria in Physical Biochemistry; Clarendon: Oxford, U.K., 1978. (3) Schellman, J. A. Biopolymers, 1975, 14, 999. (4) Gill, J.; In Biochemical Thermodynamics;Jones, M. N., Ed.; Elsevier: Amsterdam, 1979; p 224. (5) Braibanti, A.; Dallavalle, F.; Fisicaro, E.; Pascualli, M. lnorg. Chim. Acia 1986, 122, 135. (6) Hartley, F. R.; Burgess, C.; Alcock, R. M. Solution Equilibria; Ellis Horwood: Chichester, U.K., 1980.
0022-365418912093-4928$01.50/0
(6)
where5 is the molar fraction of the complex ML, and is given by ( 5 ) and ii is the Bjerrum formation function: n
ii
=
x:ifi
i= I
(7)
Equation 6 was derived under the assumption that the complexes MLi are formed in proportion to their molar fractionsL. In other words, 6 is the sum of 1, the latter being in consecutive order multiplied byf,. It is to be noted that the trivial equation M = M, the fraction of which isf,, is also incorporated in the generalized interaction equation in order to satisfy the mass balance (7) Alberty, R. A. lnd. Eng. Chem. Fundam. 1983, 22, 318.
(8) Alberty, R. A. J . Chem. Phys. 1986, 85, 2890.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4929
Thermodynamics of Complex Formation Reactions condition. Evidently, whenfi are integers, eq 6 takes the usual form.
Gibbs Energy Change Let us consider the Gibbs energy change of reaction 6. For this purpose, the following well-known thermodynamic equation will be used: AG = AGO
+ R T In n[i]"i i
dT
AGO = E v i ~ O l
(9)
I
where M~~ is the standard chemical potential of the ith form. Application of eq 9 to 6 gives
= R In
i=o
- (E%)MOI. i=1
Taking into account that p o M L i - ipoL- M O M = -RT In
EM+ RT-
In
(16)
dT
1 , EM + -ZfiAHio Ti- 1
The heat capacity change ACp is calculated in the same way:
where dfi
- =fidT
d In
pi
dT
i=l
n
dT
i= I
It should be noted that very similar equations were obtained in ref 7 and 8 where the thermodynamics of isomer groups is discussed.
Dissociation The dissociation of a complex ML, may be described by the equations:
+ L, + 2L,
ML, = ML,-,
(10)
- MOM
pi
d In
n
- f;Zfi -= ( R P ) - ' KAHjO - fiCfiAHi"]
n
AGO = E f i M o M L i
= R In
(8)
Here AGO is the standard Gibbs energy change; [i] is the concentration of the ith form (for simplicity the activity coefficients are considered to be equal to unity as one can be easily convinced that the results remains the same) and vi is the stoichiometric coefficient which takes positive values for reaction products and negative for reagents. The standard Gibbs energy change AGO can be readily calculated by using the relation
n
dAG
AS = --
ML, = ML,2
xi, AHIO
~ 2 A, H 2 O
(18)
...
pi = AGiO
+ nL,
ML, = M
x,,
AH,'
The generalized interaction equation by analogy is
i=O
eq 10 may be rearranged as follows: Herefpi and Under conditions different from standard the Gibbs energy change of reaction 6 in accordance with (8) and (1 1) is equal to
it
are defined as ( 5 ) and (7): fWi
[ML,J = xPi [ L] i-n =-
cM
(20)
5x,,[L]'-" i=O
n-l
ii =
E ( n - i)fWi i=O
The concentration of the complexes MLi can be determined from ( 5 ) as follows: [MLiI = fic~
(13)
where xo = 1. It may be shown that the thermodynamic functions changes of the reaction 19 under conditions different from standard are equal to n
The substitution of (1 3) into (1 2) after some transformations gives
AG = -RT In (EX,~[L]'-") = -RT In
E$
i=O
fiPP[Llfl
AG = -RT In
(22)
n-I
i= 1 -
AH = Efn-iAHn-io
(14)
(23)
i=O
fiih
n- 1
i=O
TAS = R T In
It may be shown that the following identity is valid:
E$ + Ef,iAHn-io i=O
(24)
AC, = n
i- 1 -- 1 + i-1Epi[L]' = EM i=O fifP
(15)
Thus, eq 3 is obtained.
Enthalpy, Entropy, and Heat Capacity Changes The Gibbs energy being known, all the other thermodynamic functions can be calculated. For the enthalpy change one has
~JAH~O i=l
Le., the well-known result, (4). The entropy change is obtained in the following way:
. .
. .
where AHwi' is the enthalpy change of the (n-i)th reaction in (18) and ACpo(,i) is defined usually as
These equations are rather like the corresponding equations of the complex formation reactions. However, they are not identical. Thus, the differences between Gibbs energy and enthalpy changes of the complex formation and dissociation processes are given by AG,m,l - AGdiss= -RT In p,[L]"
(26)
4930
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989
Fishtik
As can be seen, this difference corresponds to the change of the thermodynamic functions of the reaction M
+ nL = ML,
(28)
The same equation is obtained as a result of the subtraction of the eq 19 from 6. Consequently, reaction 28 represents a normalized factor and takes into account the fact that both the complex formation and dissociation processes are thermodynamically possible. In particular, the Gibbs energy change is negative in both cases. Furthermore, the following "boundary conditions" must be satisfied: When [L] m, AGdlSs= 0 and when [L] 0,AGmmPl= 0. In the same conditions one has AHdlss= 0 and AHmmpl = 0, respectively. The functions 22 and 26 were recently introduced in ref 5 as convoluted or saturation functions.
-
-
where
The Ligand Protonation
dii _ - xi-dfi
Usually L is a Bransted base and therefore the following equilibria must be taken into account L
+ j H = HJL, PJ, AHJ"
m
+ iiCfiH,L
n
= CfiML, i=o
/=0
+ AmH
(30)
Here1; and ii are given by (5) and (7). Functionsf, and m are defined by analogy in the following way
fHJL1
PJIHlJ
f,=,-=
w
+ CPj[H]J I
1
"L
(31)
dT
Discussion As appears from the above, the thermodynamic functions of the complex formation reactions have some specific features. One of the most important of them is that the contribution of a separate reaction M + iL = ML, to the total thermodynamic function change depends on the concentration of the ligand, this contribution being different for different thermodynamic functions. Let us consider this aspect in more detail. The contribution of the complex MLi to the total enthalpy change as can be seen from (4) constitutesf,AHio. At the same time, the contribution of the complex ML, to the total Gibbs energy change is notJAG:. In order to find this contribution, eq 14 is written as follows: n
cjr,
where C, is the total ligand concentration. Equation 30 was also derived under the assumption that the species present in the solution react in proportion to their molar fractions. The calculation of the thermodynamic functions in this case is analogous and only the main results will be listed. For Gibbs energy change eq 3 is valid; however, now [L] represents the equilibrium concentration of the ligand and is determined from the mass balance condition: CL
(33)
1 + tPj[L]J
AGj = -RTf; In
Pi - RTJ.
In [L]'
= fiACio - RTA. In [L]'
+ RTf; l n f i
(39)
+ RTfi lnfi
As can be seen, besides the termf;AGio, there is an additional term which depends on the ligand concentration. When [L] =
I , eq 39 takes the form AG, =f;."AGi0
+ RTf;"
lnfio
and AGi = AGio only iff;" = 1. Equation 14 can be also represented as n
AG =
j=1
m
+ i=Cpi[CL(l + cpj[H]')-l]i] 1
(38)
i=O
Hence, the contribution of the ith reaction in (1) to the total Gibbs energy change is
j=I
n
n
i= 1
i= 1
AG = - R T In { l
m
- f,xfiAH,"] Jj-1
n
m
[LI =
dT
AG = -RTCfi. In pi - R T x f i In EL]'+ R T C f ; l n f i
j=
rii =
j=l
df, _ - (RP)-'V;AH,"
(29)
where PJ and AHJ0 represent the cumulative protonation constant and standard enthalpy change of the reaction 29, respectively. It should be noted that all the above relations are also valid for equilibria 29 with the difference that here the ligand plays the role of the central ion and the ligand is hydrogen ion. The influence of the equilibria 29 on the mononuclear complex formation reactions 1 is done on the basis of the generalized interaction equation, which in this case has the form
M
dT
j=1
(34)
From general thermodynamic relations one can readily obtain
-RTC~, In i=O
(40)
Hence, the Gibbs energy fraction of the complex ML, is RTf;. In CM.It is interesting to mention that the sum (40) begins from zero. The first term of this sum, -RTfo In EM = RTfo Info, is related to the reaction M = M and expresses the Gibbs energy change of metal ion concentration decrease as the metal ion is bound to the ligand. The same term appears in the case of entropy change, but it is absent in the case of heat capacity change.
