In the Classroom
Ernest Rutherford, Avogadro’s Number, and Chemical Kinetics I. A. Leenson Department of Chemistry, Moscow State University, 119899 Moscow, Russia
Avogadro’s number (NA ) is one of the most important fundamental physical constants. It is the number of identical species (molecules, atoms, ions, etc.) in one mole of a substance. Determination of this number was a real challenge for the physicists of the 19th and 20th centuries. Many ingenious methods have been invented for this purpose. Robert M. Hawthorne reviewed early values in this Journal (1); U. Stille reported the progress achieved in 1945–1956 (2); and Richard D. Deslattes from NBS presented an excellent article on recent developments in the field (3). In the early 1900s French physicist Jean Perrin made the first reasonably precise determination. He counted the number of colloidal particles per unit volume in a suspension and simultaneously measured their masses. Finally he obtained for NA a value between 6.5 and 7.2 × 1023 molecules per mole. W. H. Slabaugh discussed this method and three others (spreading of a monomolecular layer, electrolysis of water, and electroplating) as possible topics for students’ practical work (4). None of the above-cited articles and reviews mention a classical work published in 1911 by the well-known American radiochemist Bertram Boltwood (1870–1927) and the prominent New Zealand–born English physicist Ernest Rutherford (1871–1937) (5 ). Yet it was this work that created a possibility to obtain one of the most precise values of Avogadro’s constant available in the 1910s. It should be noted that the authors themselves did not calculate NA directly from their data. This was done elsewhere—for example, in the textbook by Guggenheim and Prue (6 ). Also, the possibility to obtain a value of Avogadro’s constant using this method was mentioned by Mario Gliozzi in his book on the history of physics (7 ) and by Marie Curie (8). This method is quite straightforward: one should simply count the number of α-particles emitted from a radioactive source (radium salt) and measure the volume of helium obtained. Since one α-particle quickly yields one helium atom and since helium behaves as an ideal gas at NTP, we can easily calculate the number of helium atoms in one mole of the gas—that is, NA . The problem of counting α-particles was complicated by the fact that radium was not the only α-source, so we should briefly consider the radioactive decay chain. Let k k two consecutive first-order reactions A → B → C (k1 and k 2 are rate constants; radiochemists use instead the “decay constant”, λ) represent two radioactive transformations of A and B (C being stable). Concentration of the intermediate product [B] varies with time as 1
2
[B] = [A] [k1/(k2 – k1)](e᎑k t – e᎑k t) ° This equation is a solution of two differential rate equations and can be found in any textbook on chemical kinetics. If k 2 >> k1 the second exponential term is negligible and 1
998
2
[B] = [A] (k1/k2)e᎑k t. This equation can be easily obtained ° without integration if we assume that the rate of accumulation of B equals the rate of its disappearance; that is, k1[A] = k2[B] or d [B]/dt = 0. As [A] = [A] e ᎑k t, [B] = (k 1/k 2)[A] = ° (k 1/k2)[A]oe᎑k t. It is a stationary or a steady (more exactly, quasi-steady because [B] varies with time) state. If 1/k1 is very large compared with the time of the experiment t, e᎑k t = 1 and [B] = const. The steady-state treatment of complex kinetic schemes has been discussed extensively in many articles; see for example refs 9–12. Similarly, an unstable nuclide may reach a steady state through a series of consecutive disintegrations. Such a series of events constitutes a radioactive decay series (several chains occur naturally). If a nuclide having a very small decay constant and therefore a very long half-life (uranium, for example) precedes one or more nuclides with much larger decay constants, after a time—short compared with the halflife of the first nuclide and long compared with that of other nuclides—a state of so called radioactive equilibrium (not to be confused with “ordinary” chemical equilibrium) between these nuclides will be established. The rate of decay of each of the shorter-lived nuclides will equal its rate of production and thus will also equal the rate of decay of the long-lived parent nuclide. In these circumstances the quantity of each short-lived species at equilibrium will be inversely proportional to its decay constant. Part 1 of the article, containing the experimental data, is a quotation from the original articles by Rutherford and his assistants (5, 13). Part 2 includes questions in general chemistry, radiochemistry, and chemical kinetics. Part 3 gives solutions and answers to the questions. 1
1
1
1
Part 1. Original Description of Experiments ( Quoted from Refs 5, 13 [with Permission] )
“Counting of scintillations” “In 1903 Rutherford showed that the α rays emitted by radium consisted of positively charged particles moving with a high velocity. Rutherford and Geiger developed a method of directly counting the a particles emitted by a radioactive substance.… “If a screen coated with the small crystals of phosphorescent zinc sulphide is exposed to α rays a brilliant luminosity is observed. On viewing the surface of the screen with a magnifying glass, the light from the screen is seen…to consist of a number of scintillating points of light scattered over the surface and of short duration.… Each particle which falls on the crystals of zinc sulphide has been found to give a scintillation.… The experiments should be made in a dark room when the eye is well rested.… The scintillation method of counting α particles…has proved a method of very great delicacy and power. By its aid we are able to count directly the number of
Journal of Chemical Education • Vol. 75 No. 8 August 1998 • JChemEd.chem.wisc.edu
In the Classroom α particles falling on the screen in a given time.… The zinc sulphide screen has the great advantage that it is comparatively insensitive to β and γ rays.… In the paper an account is given of a method…for counting the number of α particles emitted from 1 g of Ra.… The general method employed was to allow the α particles to be fired through a small opening into a detecting vessel.… The number of α particles passing through the opening of known area at a known distance from the active source was counted for a definite interval. From this the total number of a particles expelled per second from the source was detected.… It is the first time that it has been found possible to detect a single atom of matter.” “Production of Helium from Radium” “Following the discovery and separation of helium from the mineral cleveite by Ramsay in 1895, it was noted that helium appeared in quantity only in minerals containing uranium and thorium. This association was remarkable as there appeared to be no obvious reason why an inert gaseous element should be present in minerals. A new light was thrown on this subject by the discovery of α radioactivity.… “It was suggested by Rutherford and Soddy that the helium might prove to be a disintegration product of the radioelements. This suggestion was soon verified by the experiments of Ramsay and Soddy in 1904, who found that helium was present in the gases released by heating or dissolving a radium salt. The proof of the identity of the α particle with a charged helium atom at once gives us a method of estimating the rate of production of helium by any active substance where the rate of expulsion of α particles is known.… “Since it is important that the rate of production of helium by radium should be known as accurately as possible, an independent determination of this magnitude has been carried out by the writers. Two separate determinations of the rate of production of helium have been made using for the purpose a considerable portion of the specimen of radium salt kindly loaned to one of the writers by the Vienna Academy of Sciences.… “It was quite essential that the radium salt used in these experiments should be relatively free from radioactive substances other than radium. The salt consisted of a quantity of barium–radium chloride, containing about 7 per cent of the latter element. The dry salt was placed in the cylindrical platinum capsule which was closed by a perforated cover. The capsule was next sealed up in a tube of Jena ‘combustion’ glass. The tube was completely exhausted of air.… “The amount of radium present in the salt contained in the capsule was determined by the measurement of the γ radiation emitted after a period of over two months from the time of sealing up the tube. The γ ray activity of the salt in the tube was compared with the γ ray activity of the 3.69 milligram radium bromide standard of this laboratory. The comparisons of the radium quantities were made at various distances and under different conditions, and gave the amount of radium in the salt used in the experiment as 191 milligrams. When the experiments on the determination of the production of helium has been completed a further measurement of the amount of radium present in the salt used was made. The result of this comparison gave the amount of radium present in the salt used as 193 mg. “The barium–radium salt was sealed up in the tube for eighty-three days. The capillary tip of the radium tube was
broken off and the gases were pumped out. During the pumping process, the lower end of the radium tube and the platinum capsule were heated to a red heat. The water of crystallization remaining in the salt was driven out and condensed in the KOH and P 2O5 bulbs. The radium emanation in the gaseous mixture was condensed in the tube which was immersed in liquid air. The volume and pressure of the gas trapped was measured. The results [of several measurements] were in satisfactory agreement within the limits of 1 per cent. and indicated that the purification of the helium had been carried as far as it could be by treatment with the cooled charcoal. The helium was then introduced into the spectrum tube and its spectrum examined. It was found to be essentially pure helium.… “A considerable proportion of the α particles resulting from the disintegration of free emanation might be expected to embed themselves in the walls of the tube and introduce an error which could not easily be avoided. Measurements of the γ radiation from the upper portion of the tube when the lower portion containing the radium salt was screened by a thick lead block, showed that the amount of emanation in the upper part of the tube was too small to be detected. It was therefore apparent that the escape of the emanation from the solid salt did not need to be taken into consideration. A further series of measurements indicated that the emanation had been completely separated from the salt by the heating process. It was therefore reasonable to presume that the helium was quite completely removed at the same time. “In order to avoid all possibility of incompleteness in the removal of the helium from the salt, a second determination of helium was made under distinctly different conditions. The radium salt was sealed up for a further period of 132 days. The salt was then removed from the glass tube and about 30 cubic centimeters of a dilute solution of hydrochloric acid was introduced in such a manner as to preclude the admission of any traces of air. The radium salt was now completely dissolved by gentle warming and the gases were pumped out through small KOH and P2O5 bulbs, and finally through a thin-walled U-tube cooled in liquid air (for removing the radium emanation at this point). The results obtained in both determinations were as follows: — “First determination.—Period of accumulation 83 days. Volume of helium at 0 °C and 760 mm. pressure 6.58 cubic millimeters. “Second determination.—Period of accumulation 132 days. Volume of helium at 0 °C and 760 mm. pressure 10.38 cubic millimeters.… “If x is the volume of helium produced per day by the amount of radium (element) present in the salt, and y is the volume of helium produced per day by the emanation and the two α -ray products (radium A and radium C) in equilibrium with the radium; then when the three latter products are present in equilibrium amounts y = 3x, or, in other words, the amount of helium produced by the three α products will be three times that produced by the radium itself. At the beginning of each period of accumulation, however, all emanation had been removed from the radium salt. The amount of helium produced during a subsequent period of T days would therefore be equal to Tx + y ∫0T(1 – e᎑λ t )dt = Tx + (T – e᎑λ t/λ )y = Tx + (T – 1/λ )y
JChemEd.chem.wisc.edu • Vol. 75 No. 8 August 1998 • Journal of Chemical Education
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In the Classroom
where λ is the constant of change of the radium emanation the unit of time being taken as the day; for a value of T greater than 40 days, e᎑λ t is very nearly 1. If Q is the total quantity of helium produced in the time T and the value 3x is substituted for y in the above expression we obtain Q = [T + 3(T – 1/λ)]x. Substituting in this equation the values of Q and T from the two determinations, it follows that for First determination x = 0.0209 cub. mm.; Second determination x = 0.0203 cub. mm., or an average production of 0.0206 cubic millimeters of helium per day by the radium (element) present in the salt used for these experiments. The quantity of radium contained in the salt was 192 milligrams (average of 191 and 193). The production of helium per gram of radium was therefore (0.02061/.192) = 0.107 cub. mm. per day.…” Part 2. Questions for Students2 The recommended procedure would be to discuss with the students the details of the experiments described and then to consider the questions one by one. The teacher may, of course, pose some additional questions (e.g., calculate the number of radon atoms in the apparatus by the end of the experiment; calculate and plot curves C and D using a suitable computer program). Question 1. How are α-particles converted into helium atoms in the experiment entitled “Production of helium from radium”? Question 2. Calculate (to the third significant digit) the number of α-particles emitted by 1.00 g of Ra in 1.00 s according to the data of Rutherford–Geiger experiment “Counting of scintillations”. Additional information. In 1908 Ernest Rutherford and German physicist Hans Geiger used the “scintillation method” in measuring the rate of α-particles emission by radium. (In nature this element is represented by a single nuclide, 226Ra.) A sample containing radioactive source with the activity equivalent to m1 = 0.055 mg of radium3 was placed in a long evacuated “firing tube” at a distance of l = 1.50 m of “a short glass tube, in the end of which was a circular hole”, d = 1.25 mm. A thin sheet of mica was attached to this hole with wax (one can find a description of this tube in the easily available Mellor’s encyclopedia [14 ]). A fluorescent zinc sulfide screen with a microscope (“spintariscope”) was placed just behind the hole and scintillations from α-particles were counted. In one such experiment 49.0 decays (as an average) were detected in t1 = 10.0 min.
Question 3. Why should the space between the radiation source and the exit hole of “firing tube” be evacuated? Question 4. Write all the equations of radioactive decays in the Boltwood-Rutherford experiment; use a modern notation; indicate atomic number and mass number of all nuclei involved. Additional information. Below you will find the Ra decay series in old notation of nuclides. It is taken from Rutherford’s works (the modern values for half-lives, except those of Ra, are indicated above the arrows; below are the types of decay): >1000 y
3.82 d
3.05 min
26.8 min
α
α
α
β
1.63 × 10 ᎑ 4 s
22.3 y
5d
138 d
α
β
β
α
→ RaD → RaE → RaF → RaG The last nuclide is stable.4 Question 5. Calculate Avogadro’s constant as the rough first approximation using the data given in Part 1. Additional information. In this particular calculation, half-lives of all nuclides in the series between Ra itself and RaD, including Rn, may be assumed as negligible compared to the time of measurement t2 (let t2 be 83 days; see Part 1).
Question 6. Choose which of the plots given below (Fig. 1) displays the time dependence of the number NRn of radon atoms in the course of Boltwood–Rutherford experiment. A hint: compare half-lives of Ra and Rn and use the concept of “radioactive equilibrium”. Question 7. Choose which of the plots given below (Fig. 2) shows the time dependence of helium volume in the course of the experiment. A hint: consider the number of α-particles emitted by every nuclide in the series and compare these with the answer to the previous question. Question 8. Derive a formula cited in Rutherford’s work; that is, the total quantity of helium atoms produced in the time t2 (the mass of Ra source in this experiment m2 = 0.192 g) and calculate a better approximation of the Avogadro constant, NA . A hint: half-life of radon (t1/2(Rn) = 3.82 days) cannot be neglected here in comparison with the time of experiment t2. You may need to integrate kinetic equations twice to solve this problem, as in fact Boltwood and Rutherford did; nevertheless the problem can be solved without integrations at all!5 Question 9. Calculate the time when a steady state will be reached; that is, the time when the real number of radon atoms NRn in the apparatus will differ from its quasi-steady (“equilibrium”) value, NRn′, by 1% (this is just the accuracy of calculation of the Avogadro constant from the Boltwood– Rutherford experiment). Question 10. Using the value NA obtained in question 8, calculate the half-life of radium-226 from Rutherford’s data (assume 1 year = 365 days) and compare it to the present value 1600 ± 7 years (15). Question 11. Read the following excerpts from the otherwise excellent textbook on General Chemistry (16 ), and try to find an internal incompatibility of data given by the author; use the modern value of Avogadro’s constant (6.02 × 1023): “Measurements (using a Geiger counter or a scintillation counter) have shown that 3.7 × 1010 nuclear disintegrations occur each second in 1 g of radium-226. This rate of disintegration is used as a unit of activity, the curie (Ci), named for (and defined by) Marie Curie…: 1 Ci = 3.7 × 1010 disintegrations per second.” Later, in Table 22.4, we read: “Radium-226, half life t1/2 1.60 × 103 years, activity, Ci/g 1.00”. Additional information. Curie is not a SI unit of radioactivity; the latter one is becquerel (Bq); 1 Ci = 3.7 × 1010 Bq (17), this figure being known as the exact (not rounded) one.
19.9 min
Ra → Rn → RaA → RaB → RaC → RaC′ 1000
β
Journal of Chemical Education • Vol. 75 No. 8 August 1998 • JChemEd.chem.wisc.edu
In the Classroom
A
B
C
D
Figure 1. Accumulation of radon atoms in the Boltwood–Rutherford experiment: only one graph is correct.
E
F
G
H
Figure 2. Accumulation of helium gas in the Boltwood–Rutherford experiment: only one graph is correct.
Part 3. The Answers and Acceptable Solutions Question 1. α -Particle is a helium nucleus or twice-ionized helium atom He2+. The 1st and 2nd ionization potentials (IP) for He are 24.6 and 54.1 eV; that is, so high that this particle easily tears off two electrons from the surrounding molecules (compare with IP = 12.1 eV for O2 and 15.6 eV for N2 molecules). Question 2. The area of a sphere of radius l is 4πl 2, the area of the hole is πd 2/4. If x is the number of α-particles emitted by 1 g of pure radium per second, then x = (49.0/m1 t1) 4πl 2/( πd 2/4) = (784/5.50 × 10᎑5 g ⴢ 600 s)(150 cm/0.125 cm)2 = 3.42 × 1010 g᎑1s᎑1 Question 3. The penetrating power of α-particles in air is rather low (about 5 cm at atmospheric pressure). Question 4. A new element is formed in each succeeding nuclear reaction, while the sum of the mass number of the products equals the mass number of the initial element; the same is true for the atomic numbers or charges of the nuclei (for an electron it is equal to minus 1: ᎑10 e). 226 222 4 88 Ra → 86 Rn + 2 He 222 218 4 86 Rn → 84 Po + 2 He 218 214 4 84 Po → 82 Pb + 2 He 214 214 ᎑ 82 Pb → 83 Bi + e
210 210 ᎑ 83 Bi → 84 Po + e 210 206 4 84 Po → 82 Pb + 2 He
Question 5. The calculation here is fairly trivial. It is necessary to know only the definition of Avogadro’s constant and the molar volume of an ideal gas (22.4 L; helium at NTP behaves essentially as an ideal gas). The Avogadro constant NA is the number of particles (helium atoms) in one mole: NA = NHe /ν He , where NHe is the number of helium atoms and νHe is the number of moles of helium gas formed within the time t 2 . If we assume that all radon atoms formed from radium decayed in the course of the experiment (this follows from the assumption that radon’s half-life t1/2(Rn) = 3.82 d can be neglected in comparison with t 2 = 83 d; that introduces an error of about 5%), and if we take into account that each decayed Ra atom emits four α -particles (one α-particle is emitted at once and the other three are emitted from radon with some delay—see the scheme above), then we obtain that in 83 days the number of He atoms emitted is NHe = 4x m 2 t 2
and
NA = 4 xm 2 t 2 /νHe = 4 × 3.42 × 10 g s × 0.192 g × (83 × 24× 3600) s/(6.58× 10᎑6 L/22.4 L mol ᎑1)= 10 ᎑1 ᎑1
6.41 × 1023 mol ᎑1
Question 6. The right answer: plot C. At t = 0 radon atoms are accumulated linearly as a result of radium decay. Then in the course of two consecutive reactions k1
214 214 ᎑ 83 Bi → 84 Po + e 214 210 4 84 Po → 82 Pb + 2 He 210 210 ᎑ 82 Pb → 83 Bi + e
(1)
k2
Ra → Rn → 210Pb the number of Rn atoms reaches a steady state6 where the rate of Rn formation is equal to its rate of decay. Note that all other radionuclides between Rn and 210Pb have too short half-lives to be taken into account in the course of the experiment. Also note that the graph C is a real calculated plot for the experiment described in Part 1, whereas all other graphs
JChemEd.chem.wisc.edu • Vol. 75 No. 8 August 1998 • Journal of Chemical Education
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In the Classroom
are arbitrary curves. Question 7. The right answer: plot F. In the beginning of the experiment helium gas is formed (at the rate xm2) only from α -particles emitted by radium. But at “equilibrium” (steady state) Ra itself and each of its α -active products (Rn, 218 Po, and 214Po) expels the same number of α-particles per second. Consequently the total number of α -particles and He atoms formed per second is four times that of radium itself, or 4xm2. (Half-lives 3 and 27 min for the two α-emitters, 218 Po and 214Po, are negligibly short compared with 3.8 days for Rn; on the other hand, the place of the fifth α-emitter, 210 Po, in the series is after the long-lived 210Pb so we can safely neglect α -particles emitted by this nuclide.) All other radionuclides between Rn and 210 Pb have half-lives too short to be taken into account in the course of the experiment. Note that the graph F is again a real calculated plot for the experiment described in Part 1, whereas all other graphs are arbitrary curves. Question 8. Method 1: it is the method implied by Guggenheim and Prue (6 ). However, they gave only the final formula in their book, as did Rutherford and his coworkers (5, 13), so it would be useful to derive this formula here. Let us represent the number of He atoms produced in the course of the experiment as a sum of He atoms emitted from Ra itself (N1) and fromt Rn (N2): NHe = N1 + N2 . We have N1 = x m2 t2 and N2 = ∫02 x1dt , where x1 is the rate of He atom production from Rn and its daughter nuclides. We should integrate because x1 is variable (see Fig. 1C). As it was shown above, we can safely assume that each decayed Rn atom emits three α-particles and produces three He atoms. Therefore x1 = 3(dNRn/dt) = 3k 2 NRn , where d NRn /d t is the rate of Rn decay; k2 = ln 2/t1/2(Rn) is the corresponding rate constant; and NRn is the number of Rn atoms. Note that x 1 varies with time in the same way as NRn (see Fig. 1C). Now we must find the time dependence of NRn to carry out integration and to determine N2 . This can be done easily if we write kinetic equations for the rate of Rn formation from Ra and the rate of its decay: dNRn /dt = k1NRa – k2NRn = xm2 – k2NRn , where x (remember that it is the rate of Ra decay) is essentially constant in the course of the experiment. Integration with initial conditions NRn = 0 at t = 0 gives k 2 NRn = xm2 (1 – e᎑k2t ) t 3xm2 ∫02(1
(2) ᎑k2t
We can now take the key integral N2 = – e )dt = 3xm2[t 2 + (1/k 2) e ᎑k2t 2 – 1/k 2 ] ≈ 3xm2(t2 – 1/k2), because the exponential term in the brackets is negligibly small compared with other terms. Indeed, t 2 = 83 d, 1/k 2 = t1/ 2(Rn)/ln 2 = 5.51 d, and (1/k2)e᎑k2t 2 = 5.53 d ⴢ exp(᎑0.18 ⴢ 83) = 1.8 ⴢ 10᎑6 d. Finally we have NHe = xm2 t 2 + 3xm2(t 2 – 1/k 2) = xm2(4t2 – 3/k 2) (3) We can see that this is essentially the same as the formula quoted from Rutherford’s work in Part 1, since it can be presented in the form Q = x(4T – 3/ λ). (We should remember that Rutherford denoted x as the rate of helium production by the whole amount of radium present in the salt, not by 1 g as we specified.) Now we see that helium evolves rather slowly at the beginning of the experiment and its rate of emission progressively increases as the term 3/k 2 becomes less and less in comparison with the term 4t2 (see Fig. 2F). Eventually NHe = 4xm2 t2 when 3/k2 > k1 and t1/2 (Ra) >> t2 ; therefore the number of Ra atoms and consequently the rate of Rn production are constant and the latter is equal to x m2 . 7. A very interesting story about the origin of this standard radium source (courtesy of the Vienna Academy of Science) and fascinating but not always polite correspondence on this matter between E. Rutherford and another prominent British scientist, W. Ramsay, can be found in ref 19.
15.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
16.
17.
18. 19.
20. 21.
Hawthorne, R. M. J. Chem. Educ. 1970, 47, 751–755. Stille, U. Nuovo Cimento 1957, 6 (Suppl.), Ser. 10, 185–213. Deslattes, R. D. Annu. Rev. Phys. Chem. 1980, 31, 435–461. Slabaugh, W. H. J. Chem. Educ. 1969, 46, 40. Boltwood, B. B.; Rutherford, E. Philos. Mag. 1911, 22, 586–604. Guggenheim, E. A.; Prue, M. A. Physicochemical Calculations; Interscience: New York, 1955; Chapter 24. Gliozzi, M. Storia della Fisica; Union Tipografico-Editrice Torinese: Torino, 1965; Chapter 13.8. Curie-Sklodowska, M. Radioactive Substances; Philosophical Library: New York, 1961; Chapter 2.11. Rasiel, Y.; Freeman, W. A. J. Chem. Educ. 1970, 47, 159–160. Chong, W. P. J. Chem. Educ. 1971, 48, 194–196. Volk, L.; Richardson, W.; Lau, K. H.; Hall, M.; Lin, S. H. J. Chem. Educ. 1977, 54, 95–97. Raines, R. T.; Hansen, D. E. J. Chem. Educ. 1988, 65, 757–759. Rutherford, E.; Chadwick, J.; Ellis, C. D. Radiation from Radioactive Substances; reissue of the edition of 1930; Cambridge University Press: Cambridge, 1951; pp 50–55, 60–64, 163–166. Mellor, J. W. A Comprehensive Treatise on Inorganic and Theoretical Chemistry, 2nd ed.; Longmans: London, 1946; Vol. 4, p 81. CRC Handbook of Chemistry and Physics; 69th ed.; CRC: Boca Raton, FL, 1988. Atkins, P. W. General Chemistry; Scientific American Books: New York, 1989; pp 815, 819. The same disagreement can also be found in Atkins, P. W.; Beran, J. A. General Chemistry; Scientific American Books: New York, 1992; pp 840, 842; Chang, R. Chemistry; 4th ed.; McGraw-Hill: New York, 1991; p 985; and others. IUPAC. Quantities, Units and Symbols in Physical Chemistry (usually referred to as “The Green Book”), 2nd ed.; Blackwell Science: Cambridge, MA, 1993. Prescott, F. Intermediate Chemistry. Inorganic and Physical ; 4th ed.; University Press: London, 1949; p 90. Eve, A. S. Rutherford. Being the Life and Letters of the Rt. Hon. Lord Rutherford; Cambridge University Press: Cambridge, 1939; pp 172, 325. Kingzett’s Chemical Encyclopaedia; 9th ed.; Bailliere: London, 1966; p 277. International Encyclopedia of Chemical Science; Van Nostrand: Princeton, NJ, 1964; p 314.
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