ions and anions interfer with the potassium determination in dilute nitric acid medium. When the chloridometer is used in the HIGH range, the titration is carried out a t a relatively fast rate. For example, a lOO-mequiv/l. sample of sodium tetraphenylboron requires about 30 seconds titration time. With the Buchler Model 4-2500 Chloridometer set on the LOW range, this same 100 mequiv/l. sample requires about five minutes titration time. For most titrations, the LOW range setting is more accurate, but for our system because of the lack of stability of tetraphenylboron in acid medium, it is advantageous to use the fast rate of the HIGH range setting. Cation exchange between silver ions and potassium ions is avoided because of the rapidity of our procedure and thus the troublesome titration of potassium tetraphenylborate precipitate or back titration of excess silver ions is avoided.
Potassium, a cation of extreme importance in biological fluids and cells, can be determined by our procedure using a type of coulometer familiar to virtually all clinical chemists. Future efforts might be directed to improving this procedure for the analysis of potassium in blood sera. The increased concentration of potassium in immature cancer cells as compared with normal cells (21) should encourage others to modify and extend our procedure to the analysis of cellular material. Received for review March 23, 1973. Accepted July 16, 1973.
(21) Kenneth Maclean, Biomagnetic Institute, New York, N.Y., personal communication, 1973.
I CORRESPONDENCE Precaution in Computer Simulation of DTA or DSC Curves Sir: Because a computer has no way of ascertaining the preciseness or completeness of its data or instructions, special care must be used in describing the system under analysis to avoid spurious effects. Spurious effects can easily lead to inaccurate conclusions or predictions. For this reason, critical examination of reports of computer simulation of DTA or DSC peaks is important. There is enough to be gained by successful description to justify prompt criticism so that other workers can pursue effective methods as well as avoid errors. Robinson and Scott ( 1 ) have derived a set of equations based upon the characteristics of a differential scanning calorimeter, prepared a relatively simple computer program to describe and use these equations, and have simulated DSC curves for phase diagram studies. Their computed curve for a naphthalene-azulene system showed a double peak. Since the experimental peak for some compositions also showed a double peak, they examined a more extensive range and found that “many of the computed curves showed more or less well defined double peaks.” This behavior is not immediately obvious or predictable from their set of equations, but the reason lies therein. Simplifying the terms used by Robinson and Scott, f, the fraction melted a t temperature, 7, of the sample is as illustrated in Figure 1. This is identical to Robinson and Scott’s
( 1 ) P. M. Robinson and H. G . Scott, Nature (London). 238. 14-15 (1972).
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It is very apparent from the slope of the solidus in Figure 1 that in the early stages of a melting with great separation of the liquidus and solidus, f will increase very rapidly at the start. That is, a low slope for the solidus yields a high df/dT (or dfldr). The result is a sharp break away from the base line followed by an approach to steady state heating and melting; that is, a maximum followed first by decay, then by the normal rise in differential power. In the case shown by Robinson and Scott, the effect is accentuated by the existence of a horizontal section of solidus (eutectic). The quantity f ( T ) is not satisfactorily defined at the beginning of the peak (Figure 2) because at any increment of T above the eutectic temperature, y and z must suddenly assume very real values. The programming would see a pronounced discontinuity and call for an immediate increase in the observed quantity, the differential power. This is because it sees, suddenly. a non-zero (and not very small) value of f and, consequently, a near infinite value of dfldt. The sudden rise of computed temperature does not disappear immediately in the equation given, but it does tend to diminish because the computed df/d.r is small (though increasing). Eventually the spurious input is dissipated and the smooth program is followed as f 1. Because of the relative slopes of the liquidus and solidus for this illustrated composition, the quantity df/dr in1 until the discontinuity at f = 1. creases steadily as f when the df term again drops out. Consequently, a second peak is seen. These effects are in complete agreement with the computed peak, which is not in good agreement with their experimental curve. The double peak in the latter arises from another cause. Experimentally, the supplied power in a differential scanning calorimeter is not related directly to T , - T ~ as .
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-+
-
Liquid
T
Y
0
%B
Eutectic + A
100
Figure 1. Phase diagram of a two-component system showing the fraction of liquid. f = y / z , at the beginning of melting of a solid solution of composition x assumed by Robinson and Scott, where T is the indicated temperature and the subscript n refers to a particular increment of time. This would be the case in a thermal analysis apparatus if it had the control thermocouple in the sample. I t is appropriate t o note here t h a t t h e t e r m DSC has often been used for DTA systems for which better t h a n usual enthalpy data are claimed. S e e t h e report of ICTA's C o m m i t t e e o n Nomenclature ( 2 ) . The supplied power is actually proportional to
where T s and T R are the temperatures of the sample and reference holders, respectively; note that Tn is the programmed as well as indicated temperature and is a linear function of time. During a heating program, Ts,n, - T , n , will typically reach some (small) steady state value while no change is taking place, but the sudden beginning of a near-isothermal process causes a discontinuous increase in T s , , - 1 , - T , , - l , . This in turn causes a decrease in Tsln + 2 , - T 8 n+ 21, T s l n- 3 , - T , , + 3 ) . . ., and, hence, in the average temperature. This results in an increase in power supplied, both total and differential. (The latter is the indicated and recorded quantity ( 3 ) and is proportional to Ts - TR. The power increases until a new steady state, continual melting of varying composition, is approached. Ordinary control equipment will show an apparent overshoot because some extra material must be melted before the new quasi-steady state is established. This differs from the (2) R. C. Mackenzie, in "Thermal Analysis," R. F. Schwenker, Jr.. and P. D. Garn, Ed.. Voi. i . Academic Press, New York, N.Y.. 1969, p 690. ( 3 ) Perkin-Elmer Corp., Norwalk, Conn.. Thermal Analysis Newsletter. No. 9 ( 1 970).
0
%B
Figure 2. Partial phase diagram of a two-component system showing both a eutectic and a solidus for component A . When material of composition x is heated, a computed y increases discontinuously at the eutectic temperature usual melting or transformation in that there is a continuing process (hence demand) following the first peak. The second peak is quite normal. The power supplied is increasing because of the increasing lag behind the programmed temperature, just as in the simple melting of a single component. DTA instruments show similar curves for the same general reasons. In short, computer programming of the derived mathematical statements can produce peaks somewhat resembling experimental peaks, not because the description is justified or accurate but because the model is overly simple. The particular error which led t o double peaks was the inadequate description of the beginning of melting.
Paul D. Garnl Mineralogisch-Petrographisches Institut der Universitat zu Koln 5 Koln 1, Zulpicher Str. 49, B. D. R. Received for review May 7 , 1973. Accepted August 17, 1973. The author is grateful to the U.S. National Science Foundation and to the B.D.R.'s Alexander von Humboldt-Stiftung for making this study possible.
I P e r m a n e n t address, D e p a r t m e n t
of Akron, Akron, Ohio 44325.
of C h e m i s t r y , The U n i v e r s i t y
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