T a r s JOURNAL will appreciate learning the origin of Mendeleev's famous designation "eka." Your readers may find it amusing that in a lecture I gave in November, 1969 on the subject of the superheavy elements, I mentioned the concept of element 164 being called ekeeka-lead, but, on the basis of a suggestion from a colleague, suggested as an alternative in a humorous vein the name zwei-blei. In this case, however, I had in mind the interesting coincidence that 164 is twice 82 (the atomic number of lead). Nevertheless, I suppose this appellation could be considered to be equivalent to dvi-lead.
GLENNT.
treated more systematically using linear algebra (see, cg., E. J. HENLEY and E. M. ROSEN,"Mrtterial and Energy Balance Computations," John Wiley & Sons, Inc., New York, 1969, pp. 363Ri i -,-,
3) The equations writtensbove havenothing necessarily to do wit,h what is actually taking place in a.reaction-mechanism sense. As a set of stoichiometric eouabions. thev vrovide an answer to bhe question: given values of certain (m~n~mum) number of molenumber changes (in bhis case three), what is the overall campasition of the system subsequent to a. given initial state? Establishing values of these three mole-number changes (or stoichiometric degrees of freedom) is a. matter either of kinetics (for s. nonequilibrium state) or of thermodynamics (for an equilibrium state), but this goes beyond the question as posed. However, writing an appropriate set of stoichiometric equations, which is what "balancing" implies, does not require a. knowledge either of kinetics SEABORG or of thermodynamics.
a
C H A I ~ A U.S. N , ATOMIC ENERGY COMMISSION WASHINGTON, D. C. 20545
Balancing Equations
T o the Editor: In a recent Chemical Queries column (J. CHEM. Enuc., 47, 281 (1970)), t,here was a question about balancing an equation involving the oxidation of tbutyl alcohol and a specified set of products. The answer given by Robert B. Smith invoked ideas of kinetics, and he suggested that for complicated reactions the general outline of a plausible reaction mechanism is needed. This point of view is valid if a particular product distribution is to be interpreted, but is mi* leading if the question is taken at face value. Since only "balancing" is at stake in the question (although the word "acceptable" may be ambiguous), a different kind of answer seems more appropriate, one that explicitly brings out the essential point that three equations are required to obviate the apparent variable stoichiometry. In outlining an answer to the question, I would then make the following comments. 1) An "equation" involving the six species indicated, made up of three elements, can be balanced not just in "several ways" but in an infinite number of ways. Perhaps the most direct way to see this is to "balance" the equstionwitb (initially) unknown coefficients, and then to write the three algebraic atom-balance q u a t,ions in terms of the six unknown coefficients. The existence of three linearly independent equations in six unknowns shows that a unique (relative) set of ooefficientsdoes not exist, and hence one chemical equation is insufficient to describe the stoichiometry of the system. 2) The existence of three equations in six unknowns further implies that three stoichiometric equations are required. The set of equations used is not unique. A permissible set for this system is
(CH8)&OH CHsCOCHa CHICOOH
+ 60s = 4CO. + 5H2O
+ 401 = 3C0s + 3 H O
+ 202 = 2C0z + 2H10
Many other equations could be written, eaoh of which could replace one of the three above, but each could also be shown to be s linear combination of at least two of the three above. (General requirements for such a set in any circumstance can be set down, and writing a set can be done systemetically (see, e.g., K. G. DENBIGH,'*ThePrinciples of Chemical Equilibrium," (2nd ed.), Cambridge, 1966, pp. 169-72); certain kinds of degeneracy and special situations must be guarded against. If the size of the system is large enough to warrant it, the whole problem csn be
To the Editor: If one accepts the premise that a question is to be "taken at face value," that is, literally, then Professor Missen's comments are indeed to the point. Clearly, in the classroom to extend an answer to a student's question further than the questioner intended can lead to confusion rather than to clarification (though this is not always the case). In preparing answers to queries, however, our efforts have been purposely directed toward extensions to the questions on the premise that direct, to the point answers tend to imply completeness, a finished piece of instruction. On the other hand, as Professor Missen's excellent comments demonstrate, in chemistry there is ample room for amplification, even when the prior published answer went beyond the point at hand. It is doubtful if any answer to a query will ever strike the right note between literal and extended discussion. Perhaps this is an exemplification of an aspect of chemistry that makes it so exciting to learn and to teach.
Grading the Copper Sulflde Experiment
To the Editor: I n the February issue of THIS JOURNAL (46, 119 (1969)), David P. Dingledy and Lawrence A. Patrie discuss the statistical evaluation of the stoichiometry of sulfides prepared in the general chemistry laboratory. The standard deviation and standard error (S. E.) for a given group of results are calculated using the standard formulas based on the assumption that the student sample is randomly representative of a normal population distribution. In a recent paper on data apalysis by Mosteller and Tukey (MOSTELLER, F.r AND TUKEY,J. W., "Data Volume 47, Number 1 1, November 1970
/
785
Analysis" in "Handbook of Social Psychology" (Editors: LINDZEY AND ARONSON), Addison Wesley, 1968, pp. 80200), it is pointed out that the assumption of a normal population distribution is often unwarranted. They suggest, therefore, that variability should be directly assessed wherever possible. Using the computer, I calculated the 95% confidence limits around the mean of the reported copper sulfide analysis by the direct assessment method. The 266 values were randomly divided (by shuffling) into 17 groups and the mean of each group g,, was calculated. The mean of these values is
5
estimates F , the population mean. The estimated is calculated by the formula variance of ?j sta = (g; - ))//17(17
- 1)
and the 950jo confidence limits around y are given by -
III It&..G
*
The result is 78.40 2.26 X 1.99, or (78.40 + 4.50)% copper. This result is quite surprising, considering the average value of 78.3 0.7 reported by Diugledy and Patrie. Repeating my calculation on 17 reshuffled 3.71. Using the jackknife groups yielded 78.40 method (see Mosteller and Tukey, above), confidence limits were placed around the standard deviation, of the 17 group means in the first calculation
*
8;
= 1.99
8;.
= 2.03 (jackknifed value
The standard deviation of s;:* =0.38. dence limits around si* are 2.03
of
* 2.12 X 0.38
2.03
+ 0.81
The 95% confi-
or the interval from 1.22 to 2.84. It is seen that the value of s; = 1.99, calculated by the "direct assessment" method is well within the 950j0 confidence range, while the reported standard error value of 0.34 is outside this range. These results would suggest that a much more liberal distribution of grades for student analyses is in keeping with the more realistic direct assessment of variability. I wish to thank Dr. Dingledy for supplying the raw data for this analysis.
786 / Journol o f Chemical Education
To the Editor: I t is commonplace to represent atomic and molecular orbitals by horizontal limes in a vertical scale of energies. (See, for example, in J. CHEM.EDUC., 46,679,747,806, 812 (1969)). It is also commonplace to represent the electrons filling these orbitals by arrows pointing up and down. What puzzles me is that it is not equally commonplace to show pairing energies of these electrons on the same vertical scale. This can be done easily by letting the point of the arrow indicate the energy of the electron. The first one in an orbital is drawn pointing down, the second one (of opposite spin) pointing up, and the length of the arrow then shows the extra energy needed to put the second electron in the same orbital. With this convention it becomes obvious how crystalor ligand-field splitting of the d orbitals influences the pairing of the electrons and hence the paramagnetism of the ion in the field. Take as an example a dj ion such as Mn2+ or Pea+ in an octahedral field. If the splitting A. between the e , and hr sets of d orbitals is less than the pairing energy, then to give the electrons the lowest energies they must all be pointing down; the ground state will have five unpaired electrons and "high spin"
8,)
* ltxsla.~X 0.38
2.03
Representing Pairing Energy of Electrons
If, on the other hand, the splitting is greater than the pairing energy, then to put the electrons in the lowest positions four of them must be paired off, giving a "low spin" ground state
If we draw all the arrows the same length we are dealing only with an average pairing energy, but this is sufficient to illustrate the principle.
