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
