Harold Trimm Ramesh C. Patel' and H. Ushio Clarkson College Potsdam, NY 13676
Thermodynamic Analysis of a Coupled Chemical Reaction
Very little information can be found in textbooks regarding the treatment of coupled chemical reactions, despite the fact that a large number of chemical systems encountered by a chemist involve more than one reaction. A general theoretical treatment of such systems employing matrix methods has been used previously ( 1 , 2 ) to describe the "normal modes" of chemical reactions observed hy chemical relaxation methods such as temperature-jump, pressure-jump, ultrasonic absorption, etc. In addition to kinetic parameters of elementary reactions, i t is possible to obtain very accurate thermodynamic parameters such as equilibrium constants, K, thermodynamic enthalpy changes, AHo (via temperature-jump), and reaction stoichiometry from the experimentally measured "normal mode" reaction amplitudes (3,4). In favorable cases it is also possible to determine the total concentration of species participating in the reaction. Partly as a result of the high detection sensitivity of relaxation kinetic instruments (e.g. the measurement of relative absorbance changes as small as it is possible to measure, for example, AHo and other values over very narrow temperature intervals (-1%) (5). This is particularly useful for many biological reactions which can be studied only over a restricted temperature range ( 6 ) . A typical relaxation kinetic experiment using a rectangular steo forcine function. such as a sudden increase in the temperature ot the system, can be completed within a few seconds 1.5,. Dur~nethis oeriod. the time droendenre of the observed (e.g.'absorbancej can be digitized, stored, physical (e.g., up to 4096 individual data points using a Biomation 1010 waveform recorder), and analyzed subsequently by a dedicated minicornouter. It is thus Dossible to time-resolve relatively slow andfast steps in a cokplex reaction using a single ex~eriment.Com~aredto classical titration urocedures, this approach offers the advantage of greater speed, accuracy, sensitivity, as well as the simultaneous determination of several thermodynamic parameters. Particularly for coupled reaction systems in which one or more steps produces a relatively small change in the observed physical property, this method, based on amplitude analysis, has been shown to he very successful, as shown by the study of the formation of a secondary complex between Ca2+ and murexide (7). As a result of high equipment costs, it is difficult to incorporate relaxation kinetic studies in an undergraduate chemistrv curriculum. In this article. we describe exoeriments whfch clearly illustrilte the thermodynamic aspeck and use rmdilv available chemicals and eaui~ment.The theoretical development requires elementary%thematical operations, and it is shown that acom~uterizedtreatment of the data can yield a number of usefuiparameters from a single experiment. Theory
Single Step Reaction Consider a chemical reaction: aA+bB+cC+
...t r X + y Y + z Z +
....
(1)
To whom correspondence regarding the manuscript may be addressed. 762 1 Journal of Chemical Education
We can define a small change in the extent of reaction, as follows:
a[
=
6Cc 6Cx 6Cu ~ CA- 6Ce - - ---a
b
C
X
Y
E,
(2)
or generally 6C; "=x
(3)
where uj is the stoichiometric coefficient of the i t h species, positive for a product and negative for a reactant. The change in concentration given by 6[ would result from a perturbation, such as temperature, applied to reaction (1). If a physical property, P , which is linearly related to concentration, is followed, we can write P = ~ A C A + @ B C S. +. . ~ . X C X + ~ Y C Y + . . . (4) (5) 6P = ~ A ~ +C~AB K B . ..mx6Cx + by6Cv At this point, it isuseful to express 6C.4, ~ C B(6CJ , in terms of St (where SC, = uj@; from eqn. (3)).
+
6P = bn(-a60
+ be(-b6D + .. . b x ( r W + d d y 6 8 = A465
(6) (7)
where Ab = Zuih = x b x + ybv + . . . -@A - bbe - . . . (8) The mi's are simply linear proportionality constants, and if the optical absorption (absorbance A) were followed, $i = e l (c; is the extinction coefficient of the i th s~ecies.and 1 the o&al pathlength). In order to use eqn. (7),8[ had to be exDressed in more familiar terms. which can be achieved hv using the equilibrium constant exp;ession for (1):
6[=r6lnK
(15)
Equation (3) was used to derive eqn. (13) from eqn. (12), and it should be mentioned that there are alternative methods for the derivation of eqn. (14) (2). From eqns. (15) and (7) i t follows that (8) dP=A@61nZC
(16)
I t should he emphasized that eqn. (16) is completely general, and for any given reaction the appropriate 6P can be written down immediatelv. bv The above derivations are . ins~ection. . grnerally valid for small perturbations in the neighhurhood d'euuilihrium. The influence due toartivitv coefficients has been neglected for simplification, and for v&y accurate work changes in the physical property P due to solutiou expansion (as a function of temperature) as well as due to changes in @; (e.g., temperature dependence of the extinction coefficients) have to be taken into account. Some examples of the use of eqn. (16), in conjunction with spectrophotometric detection are given below.
A* = (rc - €A - 4 1 = cc - CA - EB (18) if the pathlength 1 = 1 cm. Note that all the stoichiometric coefficients are one in this case.
If a temperature perturbation is used, the change in the equilibrium constant can be written from the rearranged van't Hoff's equation: AH" AHo A 6lnK=-6T=RT2 RT
(4
From eqns. (16), (la), (19), and (20) the total change in absorbance for reaction (17) is found to be
., ,.
The concentration terms Ci (CA,CBetc.) and the equilibrium constants Ki(K,) in eqns. (13), (19), (21), and related equations refer to the final equilihrium temperature. This particular case. with the hiehlv - ,characteristic chances in AA which are obtainahle, has been previously discussed ( 4 , 7 ) .Although at first sieht it rnavaur~ear . .. that a knowledeeoftheeauilil~rium cons tan
