Acknowledgment We thank E. A. Rhveda (IQUIOS) and H. Duddeck (Bochum University) for helpful discussions, and CONICET for our fellowships.
Constants of 1:l Complexes from NMR or Spectrophotometric Measurements Valeria Nurchi and Guido Crisponi Universith Cagliari Via Ospedale 72 09124 Cagliari, Italy NMR and spectrophotometry are widely used in the determination of formation constants of complexes; we have recently discussed ( 3 5 3 7 ) the grounds for obtaining reliable estimates of K and spectral paramters for the 1:l complex formation, and along with these lines we have written the BASIC program KEPSll for an IBM XT3 personal computer. When two species A and B react to form an adduct X according to the reaction and all the species absorb, the measured absorbauces of N solutions of different total concentrations, CAand CB,of the two reagents at Q wavelengths hl.. .Ag are given by the relation
,,
(2, [C*i - Cl, + f,[CBi - CXJ + f., c,; which depends on the unknown parameters K (formation constant of complex X, according to eq 1)) and ~ ~ ( a b s o r p t i vities of complex a t various wavelengths), where subscript i refers to the N solutions and subscript j to the Q wavelengths. A formally identical equation is obtained for NMR measurements of chemicalshifts of Q nuclei on the Amolecule in rapid exchange with the adduct X. In such a case Yjj = 6 ~ j f ~ i 6 x j . f ~where , the saturation fractions (A;, fx, are the ratios between the actual concentrations of A and X and the initial concentration of s~eciesA. and 6 ~ ;6xi . are the chemical shifts of nucleus j in pure A A d in G r d k , respectively. The reasoning on s~ectro~hotometric data can therefore he easily extendid to E MR. l'he program processes by a graphdata, arranged in maical analysis the N X Q. experimental . trix form
y..= 6
-
+
in order to verify the correctness of the 1:l assumed model, accordineto literature methods (38.39). Once the model has been corkrmed, a least-squares procedure is used to obtain the association constant, the absomtivities of the complex, and various error parameters, whiih give a measure of the reliability of the final results as well as of thegoodness of the experimental design. The least-squares procedure employed is based on the mathematical propertiesofeq 2, whichshows a linear dependence of absorbance on e but a nonlinear one on K. Therefore we used an iterative procedure based on a linear least-squares optimization of the r's and on the Gauss-Newton nonlinear least-squares for K, regarding 6 as a function of K in the numerical calculatiou of the dYIaK derivatives; this remarkably improves convergence of the program. Our approach, compared with different literature methods ( 4 0 4 2 ) , has several advantages: it only requires an initial estimate of K and not of the absorptivities (to give the 54
Journal of Chemical Education
latter values at all the examined wavelengths would be exceedingly troublesome); it assures convergence even when starting with a bad estimate of K, and it generally takes a shorter computing time. This program, routinely used for scientific work, can also be of great utility in chemical education because of its easy handling and its didactic implications: it indicates how students of inorganic and analytical laboratory courses can extract from the set of experimental data in matrix 3 hoth the equilibrium and the spectral data concerning reaction 1;and above all it gives an approach to the use of least-squares procedures, avoiding the limits implied by linear approximation methods and pointing out the importance of a good experimental design to achieve reliable parameter estimates. In fact, from literature and simulated examples, it can be stressed how also for such a simple model of complex formation certain experimental designs give a high correlation between the parameters, i.e., the errors oriK can be compensated by a proper correction of absorptivities, and the experimental data can be fitted equally well with different sets of K and 6's. Evidence is therefore given that the reproducibility of experimental data is not an indicator of the reliability of estimated parameters: in absence of a good experimental design even precise and accurate measurements may give results of no or poor reliability. A detailed description of the algorithm, the listing and the floppy disk copy of the program (IBM version), and several examples of its use are available from Project SERAPHIM. The same things, as well as a listing of a version with matrix notations for an HP-85 Hewlett-Packard personal computer can be requested from the authors.
Saturation Properties at a Given Temperature from Cubic Equations of State Fernando Aguirre-Ode Universidad Tecnica Federico Santa Maria Valparafso, Chile The advantage of cubic equations of state (EOS) lies in the fact that they can he solved analytically for the volume of a real gas when hoth the temperature and the pressure are given. However, for computer calculations an iterative procedure based on the Newton-Raphson method, which shows rapid convergence if the respective ideal gas volume is used as the initial value, is usually recommended. A more interesting problem arises when the chosen temperature is lower than the critical temperature. There are three quantities to calculate: the volume of the gas, the volume of the liquid phase, and the saturation pressure. A computer program is proposed herein t o cover both cases, paying attention mainly to the saturation region through the use of four simple cubic EOS: van der Waals, Berthelot, Redlich-Kwong, and Peng-Robinson. The inputs are the name of the fluid, its critical temperature and pressure, and the temperature chosen for the system. Parameters of any selected EOS are calculated as functions of the critical properties. Figure 5 shows schematically the typical behavior of a cubic EOS a t a low temperature. The thermodynamic stability analysis leads to the condition of equality of areas I and 11, i.e., to the equation
['pdu = P'"(uc - uL)
(1)
in which Ra)is the saturation pressure and P is an explicit function of v. By solving the integral for the given EOS, eq 1 will yield an equation in which Re), v ~and , vc are correlated. The question now is how. to build up hoth the initial and
A Newton-Raphson subroutine is also included for the case T > T,,the pressure of the system being asked for directlv as an input. ~he.proposed program has several interesting features that may be considered useful in a course of either physical chemistry or thermodynamics. Besides illustrating the simple calculation of the volume of a real gas by an iterative procedure, it shows different aspects of the vapor liquid equilibrium of a pure substance. Simple forms of some well. known EOS are used as examples for calculating saturation properties. Initial and corrected guesses of the saturation pressure illustrate the thermodynamic criterion of stability for vapor-liquid equilibrium. Of course, the use of fluid mixtures and the modern views about how to work with more adaptable f o m s of the same EOS would lead t o more complicated calculations. A simple program like this may be enough for illustrating thermodynamic principles and a thermodynamically based way of approaching convergence quite rapidly. Figure 5. P-"diagram showing equality 01 areas I and IIa n d volumes y,m,v~ 86 wiutions of t h e cubic EOS in the saturated region.
Llteraiure Cited Ewig, C. S.;Gerig, J. T.; Harris, D. 0. J. Chem. Educ. 1970.47.97 Menill, J . R. Am. J. Phys. 1372,40,138. Bolemsn, J. S. Am. J. Phys. 1972,40,1511. Sims, J. S.;&ng, G. E. J. Chem. Educ. 1979.56,546. Blukin, U.: Hmell, J. M. J. Chom. Edur. L983,60,207. 6. Kubsch, C. J. Chem. Edue. 1983,60,212. 7. Dunbrack. R. L..Jr. J. C k m . Educ. 1986.63.955. 8. Tellinghuiaen, J.. unpublished. 9. Numerov, B. Publr. obasruofoiireeenfml aslrophys. Rum. 1933.2.188. 10. Cooiey, J. W. Moth Comput. 1H1,15,363. 11. Cashion. J.K. J. Chem.Phy8.1963.39.1872. 12. Zsre, R. N. J Chem Phys. 1964.40,1934. 13. LeR0y.R. J.:Bernatein,R.B. J.Chem.Phya. 1968.49.9.12. 14. Johns0n.B.R. J Chom.Phya.l977,67,4086. A P P I ~ C&idel: ~ ~ ~ ~ ~ : ~ ~hiecho& ~ for n niffelentioi ' ~ ~ ti^^ i 15. IXU~U, L. ~ D d r e e h t , 1984. 16. Kobeissi, H. J P h y ~B:At. . Mol. Phya. 1982,15,693. 17. Leubner, C. Int. J. @onturn Chem. L983.24.127. 18. Tellinghuiaen, 3. Inf. J. Quantum Cham. 1988, in prarr 19. 2sre.R. N.;Ca~hion,J. K."The IBMSAARRPmgrxm 02NUSCHR1072Solutionof the SehrWinger Radial Equstion, by J. W. Cwley". U n i ~ r a i t yof California Berkeley Laborstery Raport UCRL-10881, July. 1963. for Molecular M. Jarmain, W. R.: MeCallum, J. C. "TRAPRB: A Computer ProTransitions? University of Western Ontario Department of Physics. December, 1970. 21. LeRoy,R. J."Further ImprovedComputer Program for SolvingtheRadialSchrbdiwerEquationforBoundsndQumiboundlOrbit~aReaonane)kvels",Univemitvof w a t c r l ~Chrm8rs. Phyws Rrporr zP-2UIR. Frbruav, lshl 1. 2. 3. 4. 5.
successively corrected guesses of PIP',from which provisional values of UL and u~ are obtained, in order to converge rapidly to the final solutionof the problem with the aid of a computer. The key for the answer can beviewed from theobservation of Figure 5. An initial guess which gives no problem when solving the cubic EOS analytically, i.e., with three real solutionsfor the volume, should be in the region P,j. < P < P,.,. Both Pmi, and P, correspond to the conditions
An intermediate pressure for the given temperature can always be found in the indicated pressure region by applying a condition intermediate to the inequalities in eq 2. Initial recommended guesses consistent with both null first and second derivatives are then included in the computer program. Once UL and v~ are found for either the initial or a corrected pressure, these volumes are then used to calculate the corrected pressure of the next iteration. The ratio 4 = (Area IIIArea I) can be used as a factor to reduce either a too high initial guess (4 < 1) or to increase a too low one. (4 > 1). As the solution is approached, that ratio gets closer and closer to unity and convergence is obtained in a few steps. The ratio 4 can be expressed by the following equation:
so that the guess of Pa)for the ( r
+ 1)th iteration is given by
.
21 h r b r e a e . G Ed Thr Oxlord Endlzrh h Ln,ver%rLy:Lonoon, 1953, p 546.
r m a n of Qumonom. 2nd 4 ;Oxlurd
24. The Z ZBasie interactive compiler ia a d d by Zedmr Inc., 4SM) E. Speedruay, Suite 93, Tucson, Ariwne.35712. 25. Wright,J.R.:Robinson, J. J. Chsm.Edue. 1979,56,€43. fo~gonie 26. Silverstein, Silverstein. R. M.; Bassiel, Bassiel. B. C.; Morril, T. C. Spectromalric Spectrometric Idonfificofion oof01gonie Compounds. 4th ed.:Wiiey: NsluYork, 1981: p 225. 27. Breitmsier, E.: Valter, W. 'JC IJC NMR Spectrmeopy, 2nd ed.; Verlag Chcmi.: Weinheim, 1978. 2% Grxnt. D. M.: Paul. E 0.J. Am. Chem. SN. 1964.86.298C2990.
Chom. 1986,M, 552555. 33. 34. 35. 36. 37. 38. 39. 40. 41.
Sends,Y.;Iahiyama, J.; Imaizumi, S.Buil. Chem. Sac. Japan 1979,52,1994-1997. Reierhoek; H.: Ssunden. J . K. Con. J. Chpm. 1980.58,125&1265. C a m , G.;Crisponi, G.; Nurehi, V. Terrohodmn 1981.37.2115. Csrfa, G.; Criaponi, G. J. Chom. Sm.,PerkinII 1982,53. Carts, G.;Crisponi, G.; Lai, A. J. Mogn. Ros. 1982,48,341. W d l a c ~ , RM.;Katz,S.M. . J.Phya. Chom. I984,68,3690. Colaman. J. S.; Vargs,L. P.; Mastin, S. H. Inorg. Chom. 1910,9.1015. Gonrolu,K.;Jahnson,G.D.:Bowen,R.E.J.Am. Chem.Soc. 1964,86,1025. Rooseinakv. 0. R.: Kellawi. H. J. Chem. Soe. A 1969.1207.
Volume 66
Number 1 January 1969
55
