Calculation of Equilibrium Constants and Their Errors in Redistribution Reactions Studied by Nuclear Magnetic Resonance

SIR: In view of the current interest ( I ) in redistribution reactions and the possible long-term significance of the result- ing data, we would like to comment on the error calculations which underlie much of this work and which were described (2) in this journal several years ago. First it should be noted that except for Examples 1 and 2 of Reference 2, no use has been made in the literature (3) of the section of Reference 2 entitled “Bias of Analytical Results.”

If the analytical instrument is a modern NMR machine, the omission of bias parameters is probably justifiable-that is, one can assume that the strength of the NMR signal is a direct measure of concentration-as long as one important condition is satisfied: There must be no saturation. If in- sufficient time for complete relaxation is allowed between successive runs of a mixture, then the weaker signals will satu- rate before the stronger, and the ratios of NMR signal strengths will no longer be equal to the ratios of concentra- tions, In no case is it justified to treat saturation as introduc- ing a random error; the only satisfactory solution is to try to avoid it entirely.

The errors in equilibrium constants measured in this way by NMR have been calculated from the maximum average error sav in the signal intensities (2) . This upper limit of sav is cal- culated from the deviation of the initial R-values from those calculated from the signal intensities. The R-value of a mix- ture of compounds of the type QZiT,-, is defined as the ratio [Z‘l/[Q]. The use of one average standard error for all signals is justified if all the signals are of equal width, and if the sole error in the signal is due to base line noise. Very often, how- ever, the error in a peak area is not primarily due to noise in the base line, but rather due to a “hump” in the base line under each peak (due to very fine field imperfections) which causes an error approximately proportional to peak size. In such cases the error in the equilibrium constant (which depends on the fractional errors in the different peaks) is not readily ob- tained from sa,, and repeated integration of each sample is probably the most reliable method of finding the random error in the equilibrium constant. The method of Reference 2 also necessitates great care in preparing the NMR tubes if readily hydrolyzable compounds such as the silicon halides or their derivatives are used (especially as the hydrolysis is probably an autocatalytic process due to the hydrogen halide formed). As the initial concentrations of reagents actually in the NMR tubes must be accurately known, in such a situation reagent handling becomes an important potential source of error.

For very large or very small K’s, where signals are of very disparate sizes, the sav procedure naturally emerges with large standard errors, often of the same size as the value of the equilibrium constant K itself. As the quantity of thermody-

(1) K. Rloedritzer, Orgunornet. Chem. Rev., 1 , 179 (1966). (2) L. C. D. Groenweghe, J. R. Van Wazer, and A. W. Dickinson,

ANAL. CHEM., 36, 303 (1964). (3) J. R. Van Wazer, K. Moedritzer, and L. C. D. Groenweghe,

Monsanto Chemical Co., Central Research Dept., St. Louis, Mo., private communications, August and October 1967.

namic interest is In K , if K has a “standard error” of about its own size, it does not mean that K can take negative values, as for these In K is not defined. Thus the mean value of In K may well depart appreciably from In Krnean if the error of K about K,,,, is large. In such cases the quantity AFdev, de- fined as the deviation of the Gibbs free energy change AF of a reaction from the value expected for random exchange, has a wide range of values. Thus for the redistribution reaction ( 4 ) :

MeSiCls + MeSi(NMeJ3 + MeSiC12(NMe2) + MeSiC1(NMe2)2

the equilibrium constants give some value for AFdev between about -3.0 and - m kcal/mole reaction, using the quoted errors on the K’s. Values of A H for the reaction, assumed to be equal to 4Fdev (see, for example, Reference 1, page 190), have been measured calorimetrically as -3.4 and -4.0 kcaljmole reaction. The close agreement between these figures and the value of 4Fdev of -3.9 kcal/mole reaction corresponding to quoted “weighted mean” K’s (4) must be considered coincidental unless the calculated maximum average error was overestimated.

In some cases, because equilibrium is attained so slowly at room temperature, it has been necessary to obtain equilibrium at a high temperature, and subsequently to rapidly quench the systems, before running their spectra at some suitable, lower temperature. If at the equilibration temperature appreciable amounts of both gas and liquid phases are present, then, on quenching, the condensation of the gas phase may alter the measured “equilibrium composition” of the liquid. For example, even if the equilibrium constants are the same in gas and liquid phases for a reaction of the type

2MXY e MX, + MY2

but MX2 is much more volatile than MY2, then the equilibrium gas phase mixture will be richer in MX2 than the liquid phase; on chilling, the liquid phase will therefore be disproportion- ately enriched in MX2 and the observed “equilibrium con- stant” will be too favorable toward MX2 + MY2. In such cases, it is necessary to ensure that the NMR tubes are as full as possible. This however makes sealing off the tubes difficult and increases the chance of decomposition of the contents, a result not welcome if errors are to be calculated by the method of Reference 2. If the NMR tube is three quarters full, then in the 5-mm 0.d. NMR tubes used, there will be about 1.5 cc of liquid and 0.5 cc of gas. For normal temperatures, for one atmosphere pressure, the gas volume will correspond to 0.022 mmole of molecules, while the liquid volume will correspond to ca. 10 mmole of molecules. At 10 atm. pressure then, only about 2 z of the molecules will be in the gas phase, which means that the error in the measured concentration of the liquid due to quenching will probably be no more than a few

(4) J. R. Van Wazer and K. Moedritzer, J . Inorg. Nucl. Chem., 26, 737 (1964).

VOL. 40, NO. 3, MARCH 1966 0 659

tenths of a percent-completely negligible. In a number of cases (5-8) great care has been taken to ensure that the NMR tubes are as full as possible. If, however, the contents of the tube are raised to such a temperature that they are near the critical point, with the densities of the liquid and gas phases ap- proaching each other, then the error due to quenching may be very serious indeed. Such a situation may well obtain in the

( 5 ) N. E. Aubrey and J. R. Van Wazer, J . Am. Chem. Soc., 86,

(6) K. Moedritzer and J. R. Van Wazer, J . Org. Chem., 30, 3920

(7) K. Moedritzer and J. R. Van Wazer, Inorg. Chem., 5, 547

(8) K. Moedritzer and J. R. Van Wazer, J . Phys. Chem., 70, 2025

4380 (1964).

(1965).

(1966).

(1966).

redistribution of GeC14 and GeMe, at 300" C (9), as the equilibration of the methylchlorosilanes, a system not very different as regards boiling points, was considered to be a com- pletely gas phase reaction at 3.50" C (10).

A. R. CONRAD A. G. LEE

University Chemical Laboratory Lensfield Road Cambridge, England

RECEIVED for review October 30, 1967. Accepted December 8, 1967.

(9) G. M. Burch and J. R. Van Wazer, J. Chem. SOC. A , 1966, p 586. (IO) P. D. Zemany and F. P. Price, J . Am. Chem. Soc., 70, 4222

(1948).

Determination of Minor Elements in Rocks by Thin Film X-Ray Fluorescence Techniques

N. B. Price and G . R . Angel1 Grant Institute of Geology, King's Buildings, Edinburgh, Great Britain

IN RECENT YEARS there has been a growing interest in the appli- cation of X-ray fluorescence spectrometry to silicate analysis. Most published work is concerned with the analysis of material in amounts greater than 0.10 gram, although an investigation by Rose, Cuttitta, and Larson ( I ) has shown that the quantita- tive determination of selected major elements in rocks can be undzrtaken using milligram quantities of material. The method of preparation for analysis involves a digestion of the rock in acid which is subsequently absorbed with pulped filter paper.

This paper shows that most minor elements above atomic number 22 can be determined satisfactorily in rock powders using thin film techniques. Rock powders mounted as a thin film have already been used for the detection of certain ele- ments in rocks. The method is at best semiquantitative, and the sensitivity has been surprisingly low-e.g., Rose (2) has shown a detection limit of 0.05% for bromine in certain rocks. The advantages of using thin films for the determination of microgram amounts of an element have been discussed by Rhodin (3) , Gunn (4, and others, who have shown the high sensitivity of the method and, within limits, its freedom from absorption effects. Gunn's (4 ) investigation of the emission and absorption of thin deposits of selected elements produced by evaporation from solution shows a linear relation between instrument response and element concentration, even at mod- erately high levels. To date, no attempt has been made to apply thin films to the quantitative analyses of complex mate- rials, such as rocks, where mass absorption effects differ widely between samples.

A frequently used method of minor element determination

(1) H. J. Rose, F. Cuttitta, and R. R. Larson, U. S. Geol. Suru. Profess. Papers, 5258, 155 (1965).

(2) H. J. Rose, Ibid., 501A, 198 (1964). (3) T. N, Rhodin, ANAL. CHEM., 27,1857 (1955). (4) E. L. Gunn, Ibid., 33, 921 (1961).

of bulk samples of rock is based on the method of Andermann and Kemp (5) who have shown that scattered background and analytical line intensities decrease in a constant ratio as the absorption coefficient of the sample increases. Further, they have shown that a plot of ratio peak intensity/scattered background intensity varies linearly with the concentration of an element even in a multicomponent series of samples, show- ing that this measurement reduces the effects of absorption and instrumental variables in much the same way as the addition of an internal standard. Because of the difficulties in establish- ing the principles for the choice of suitable background wave- lengths, an arbitrary choice was made based purely on an em- pirical approach.

A combination of theories of workers in thin film techniques and the results of Andermann and Kemp (5) appear to be a promising method of analyzing rocks when material is scarce. One complication of this combined approach (2-4) is that the mounting material used to support the micro deposit con- tributes some background to the fluorescence spectra. Al- though this can be greatly reduced by using a mounting mate- rial of low atomic number, such measurements will have to be considered in the compilation of results.

EXPERIMENTAL Procedure. Rock samples and standards (0.05 gram)

were ball-milled in an agate container to pass 200-mesh size. The resulting powder was mounted on the adhesive side of a Sellotape strip, similar in composition to Scotch cellophane adhesive tape, either by light brushing, producing a thin film of powder approximately 3-5 mg, or sprinkling about 10-1.5 mg on the strip which is then pressed into the adhesive at a pressure of 5 tons/inch*. The strip was subsequently mounted on the base of a normal sample holder with its adhesive side up. Natural rock standards were used, where possible, to

( 5 ) G. Andermann and J. W. Kemp, ANAL. CHEW, 30,1306(1958).

660 ANALYTICAL CHEMISTRY