letters Serni-Classical Linear Model of Atom To the Editor: In their interesting article on linear electron paths [52, 398 (1974)], Dankel and Levy said that one of the reasons for continued presentation of the semi-classical approach is the qualitative agreement of the distances obtained semi-classically with those calculated from the corresponding wave functions. Although they gave values for X,,,, they did not calculate a, the average distance of the electron from the nucleus. The procedure for calculating a is essentially the same for the linear paths as for the wave functions, and the linear paths possibly could serve as a useful introduction to that procedure. When the integral in Dankel and Levy's equation for t is written in atomic units (m = e = 1 and E = -1/2n2), the upper limit is 2nZ and the value of the integral is n3a. Since the inverse of the velocity a t x is proportional to the length of time the electron is within dx of x , the integrand itself gives the relative probability of the electron being at x (analogous to the square of the wave function), hut this probability must be normalized by dividing the integrand by n3a. To obtain a, the integrand is next multiplied by x and then integrated. The integral is now 3n2/2, a formula which gives exactly the same values of a (in Bohr radii) as those obtained from the hydrogen m wave functions. This result is rather remarkable, because the probability density for the linear paths becomes infinite at x = 2nZ and immediately becomes zero for larger values of x, in contrast to the long tails in the radial probability density (4arz112) for the wave functions. This extension of the wave-function probability beyond the semi-classical tuming point is similar to the situation in a harmonic oscillator, and this aspect of the linear-paths problem could be presented along familiar lines. (See, e.g., L. Pauling and E. B. Wilson, "Introduction To Quantum Mechanics," McGraw-Hill Book Co., New York, 1935, pp. 73-77.) James L. Bills Brigham Young University Pmvo, Utah 84602
Defining the Degree of Advancement of a Reaction To the Editor: In a recent article [J. Chem. Educ., 51, 572 (1974)], M. W. Zemansky shows in a convincing way that thermodynamics has subtle features. From chemical thermodynamics an example is taken to illustrate the misuse of symbols. This example is elaborated further in a subsequent article by J. N. Spencer (p. 577). The symbol 2: is used in both articles to denote the degree of advancement of a reaction (the extent of reaction). Spencer's refers ohviously to a reaction, Zemansky's E is a ratio which refers less clearly to a reaction. To illustrate the difference, let an equation for a chemical reaction he expressed by 0 = X v i M i , ui being the stoichiometric coefficient of the substance Mi (negative for (Continued on page 276) 274 / Journal Of Chemical Education
reactants, positive for products). Suppose that n&', nxli and n ~ ; "denote the initial, instantaneous and final number of moles of M,, respectively. Then Spencer's definition of [ is given by
But by defining [ as a ratio u~~ is eliminated and the connection with the reaction now lies hidden. In this case I is defined by
We get, of course, two different (6G/6[), T, P, and comDosition beina constant. S~encer's definition leads to ( 6 ~ / 6 [ ) = Y ~ M : ~ M , . Here the relation to the reaction is clear. Zemansky's definition gives (6G160 = Y(nhq," - n\i,')gv, . . which m i y easily he mistaken f o r ' l ~ . Spencer's definition is identical with IUPAC's recommendations [Pure and A p p l . Chem., 21, No 1 (1970)l. Its advantage: it is beyond doubt that the operator 6/6[ refers tounit advancement of a definite reaction. Bj$m Bergthorsson The Technical University of Denmark DTH DK-2800 Lyngby, Denmark
Relationship of Melting Point to Triple Point
To the Editor: R. C. Parker and D. S. Kristol [J.Chem. Educ., 51, 658 (1974)l have written a valuable explanation of the relationship of the melting point of a substance measured under various conditions to the s-l-u triple point. This concept is not usually discussed clearly in recent textbooks, either elementary or advanced. It may he of interest to those who wish to compare numerical values of the properties to learn that selected values of the melting points of compounds and of the various triple points are compiled, where experimental data are available, in "Selected Values of Properties of Hydrocarbons and Related Compounds," American Petroleum Institute Research Project 44 and "Selected Values of Properties of Chemical Compounds," Thermodynamics Research Center Data Project. Both sets of loose-leaf data sheets are published and distributed by the Thermodynamics Research Center, Texas A & M University. These data are listed in the m-tables which include temperatures, AH, dAH/dT, and AS for various phase transitions. The convention used in these tables for a "fusion" temperature is that, when a pressure of 760 mm is shown the temperature is the melting point in equilibrium with air a t one atmosphere, and when the saturated pressure, or no pressure, is shown, the temperature is the s-l-u triple point. A scan of these values reveals that t,, - tm is generally in the range of 0.01 to 0.1"C fur hydrocarbons in which the triple point pressure is less than one atmosphere. This (Continued on page 278) 276 / Journal of Chemical Education
