Chemical Education Today
Letters Ernest Rutherford, Avogadro’s Number, and Chemical Kinetics Revisited The article by I. A. Leenson, “Ernest Rutherford, Avogadro’s Number, and Chemical Kinetics” ( J. Chem. Educ. 1998, 75, 998–1003), draws on infrequently cited but relevant pioneering observations from Rutherford’s laboratory. After extensive quotation from the sources cited, Leenson offers 11 study questions for students. Working out these questions provides exercises in the use of fundamental concepts including quantitative understanding of rate terms and an estimate of the Avogadro constant. The author provides “answers and acceptable solutions” to all the questions to complete his presentation. My purpose in this letter is to note that the link between Rutherford’s observations and the Avogadro constant can be expressed succinctly and can be clarified through differential expressions not presented in Leenson’s article. The lengthy quotation of Rutherford’s work, Part 1 of the article, describes determination of his defined quantity x = the initial rate of He gas production from α-decay by 226Ra. The overall treatment recognizes that He gas is also produced by three decay products of 226Ra. His result is x = 0.107 mm3 day᎑1 g᎑1 where the unit “mm3” is the volume of He gas at 0 °C and 1 atm. Suggested here is that x be translated into rate terms with nHe and nRa denoting the macroscopic amounts (e.g., moles) of the species indicated for a given mass m of 226Ra: x = (1/m)(dnHe/dt)VHe° where VHe° is the molar volume of He gas at 0 °C and 1 atm, or (dnHe/dt) = m x/VHe° = ᎑(dnRa/dt) = λnRa = λm/MRa Here, MRa is the molar mass of 226Ra. The macroscopic information from the quoted results from Rutherford’s laboratory relates to the decay constant λ but not to the Avogadro constant! Solving, one finds λ = x MRa/VHe° =
(0.107 mm3 day᎑1 g᎑1)(226 g mol᎑1)/(22.4 × 106 mm3 mol᎑1) = (1.079 × 10 day )(365.25 day year ) = 3.94 × 10 year ᎑6
᎑1
᎑1
᎑4
᎑1
the corresponding half-life t1/2 being 1759 years, a value
1278
consistent with the presently accepted 1622 years in view of the precision of mass measurement involved in determination of x. Therefore, macroscopic information on He formation upon decay of Ra does not allow estimation of the Avogadro constant, NA. To estimate NA requires some further observations, which must include the particle nature of α-decay. Only later, under Question 2 of “Part 2. Questions for Students”, does Leenson’s article offer as “additional information” results of observations reported in 1908 by Rutherford and Geiger. The information provided expresses quantitatively the rate of α-particle emission per unit mass of Ra. Over an interval of 10.0 minutes an average of 49.0 α-decays were detected by scintillations on a fluorescent screen of d = 1.25 mm located at l = 1.50 m from a Ra sample of mass = 0.055 mg. The total rate of Ra decay for this mass can be written in terms of this observation and the geometry of the experiment: ᎑(dNRa/dt) = [4π l 2/π(d/2)2](49.0/10.0 min) = 1.13 × 108 min᎑1 By first-order decay, one can express this same rate in terms of the decay constant λ and NRa, the number of Ra atoms in the given mass: ᎑(dNRa/dt) = λ NRa = λnRa NA = λ(m/MRa )NA This last expression does include NA. All the other quantities are known from Leenson’s Part 1 and the “additional information” given under his Question 2. Solving for NA one has NA = (MRa / λm) [᎑(dNRa/dt)] = (226 g mol᎑1)(24 h day᎑1)(60 min h᎑1)(1.13 × 108 min᎑1)/ (1.079 × 10᎑6 day᎑1)(0.055 × 10᎑3 g) = 6.2 × 1023 mol᎑1 This is a reasonable value for NA in view of the implied precision of Rutherford’s experiments—especially indirect determination of the very small masses of the samples studied. James E. Sturm Department of Chemistry Lehigh University Bethlehem, PA 18015
Journal of Chemical Education • Vol. 77 No. 10 October 2000 • JChemEd.chem.wisc.edu
