Quantum Mechanical Foundations of Isotope ... - ACS Publications

1 Quantum Mechanical Foundations of Isotope Chemistry

Downloaded by UNIV OF NEBRASKA - LINCOLN on April 4, 2015 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/bk-1975-0011.ch001

JACOB BIGELEISEN Department of Chemistry, University of Rochester, Rochester, Ν. Y. 14627

Introduction I shall open this symposium with a brief overview of the quantum and s t a t i s t i c a l mechanical foundations of isotope chem i s t r y . A number of monographs already exist dealing with the principles and applications of isotope chemistry i n chemical kinetics, geochemistry, isotope separation, and equilibrium processes. This symposium i s directed, in part, toward estab lishing a bridge from the present habitat of isotope chemistry, graduate students and professional scientists, to the under graduate classroom. For this purpose I find it convenient to develop the fundamental principles of isotope chemistry along h i s t o r i c a l lines. I w i l l retain only those ideas that have stood the test of time and omit all of those dead ends and false turns which are part of the development of any science. Even with this restriction, it i s impossible to cover all of the areas of isotope chemistry. The one area which relates most to all disciplines of isotope chemistry i s the equilibrium i n ideal gases. Other papers in this symposium w i l l start from the ideal gas equilibrium to such topics as tunnelling i n chemical kinetics, isotope separation, reaction mechanisms, condensed phase isotope chemistry and isotope biology. Shortly after the discovery of isotopes Fajans (1)recognized that the thermodynamic properties of solids, which depend on the frequencies of atomic and molecular vibrations, must be d i f f e r ent for isotopes. This idea was put into quantitative form independently by Stern (2) and Lindemann (3,4.). * Much of the research reported in this a r t i c l e was supported by the U.S. Atomic Energy Commission. ** A condensed summary of this a r t i c l e was presented at the Division of Chemical Education Symposium on "Isotopes and Chemical Principles",ACS National Meeting 3 April 1974, Los Angeles. The manuscript was prepared during the tenure of a John Simon Guggenheim Memorial Fellowship. 1

In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.

ISOTOPES AND CHEMICAL PRINCIPLES


Stern-Lindemann Formulation of Vapor Pressure Isotope Effect Stern and Lindemann independently gave the f i r s t quantita tive formulation of an isotope effect. They considered the equilibrium between a Debye solid and an ideal gas composed of monatomic substances. Further they introduced the following assumptions: 1) the oscillations i n a solid l a t t i c e are harmon i c , 2) the potential energy i s isotope invariant, and 3) an oscillator has a zero point energy. Some of these assumptions represented a significant insight into physical processes at the time they were put forward. Anharmonic effects are of major importance i n the equation of state of solids. However, neither experiment nor theory had advanced to the point where anharmonic effects needed to be considered i n the analysis of the constant volume heat capacity, the enthalpy, the entropy, and the free energies of solids. The Bohr-Sommerfeld theory of the hydrogen atom shows a small dependence of the potential energy on the mass. The dependence of the Rydberg constant on the mass of the nucleus leads to a small difference i n behavior.of hydrogen and deuterium. The Born-Oppenheimer approximation to the solu tion of the Schrodinger equation for molecules inherently con tains the assumption of an isotope independent potential energy. The small corrections to isotope chemistry from corrections to the Born-Oppenheimer approximation are currently being investi gated by Wolfsberg and are reported in the paper in this sympo sium volume by Wolfsberg and Kleinman. Even before the development of quantum mechanics by Heisenberg and Schrodinger in 1925- 6, in which such concepts as zero point energy follow naturally, d i f f i c u l t i e s with the old quantum theory led scien t i s t s of 1920 to anticipate the existence of a zero point ener gy. It was, therefore, natural for Lindemann to develop the theory of the vapor pressure isotope effect both for the case of the existence of zero point energy and for the non-existence. For the former, he predicted that the isotope effect on vapor pressure would be a small effect, of the order of 0.02 % for P b / P b at 600° Κ (actually Lindemann has an error of a factor of 10 and his result should be reduced to 0.002 %) and that the effect i s a "second order difference". Stern and Lindemann obtained for the monatomic Debye solid under the har monic oscillator, Born-Oppenheimer approximations for a system with zero point energy, Ε = 1/2 h ν » f

206

208

In P7P = (3/40) (Θ/Τ) ο + ...

(1)

2

where P' and Ρ are the vapor pressures of a light and heavy isotope respectively, θ i s the Debye temperature, hv /k, of the heavy isotope and m

Ô - [(H/lO " 1 ]


1.

BiGELEiSEN

Quantum Mechanical

Foundations

3

Equation (1) i s the f i r s t term of a Taylor expansion valid for (Θ/Τ) < 2rr. For the case of the non-existence of zero point energy, one predicts an isotope effect for the P b / P b vapor pressure ratio two orders of magnitude larger than the predic tion from Eq. (1) at 600° K. In addition the no zero point energy case predicts the P b to have the larger vapor pressure at 600° K. Stern's estimate of the difference i n vapor pressures of Ne and Ne at 24.6° Κ through Eq. (1) led to the f i r s t separ ation of isotopes on a macro scale by Keesom and van Dijk (2). The same theory, without the approximation (Θ/Τ) < 1, was used by Urey, Brickwedde, and Murphy (5) to design a Raleigh d i s t i l lation concentration procedure to enrich HD i n H five fold above the natural abundance level, which was adequate to demon strate the existence of a heavy isotope of hydrogen of mass 2. It i s interesting to note that as late as 1931 Keesom and van Dijk took their finding that there was a difference i n vapor pressures of the neon isotopes and that the vapor pressure of °Ne i s greater than Ne to be an important experimental proof of the existence of zero point energy. I t i s fortunate that Ur.ey had not yet discovered deuterium at the time of Keesom and van Mjk's work or they might have been misled by the discovery that the vapor pressure of deuterocarbons i s generally larger than hydrocarbons. It i s interesting to speculate what conclu sions would have been drawn from such a finding i n 1931. It i s important to look into the implications of Eq. (1) since the development of the quantum-statistical mechanical theory of isotope chemistry from 1915 u n t i l 1973 centers about the generalization of this equation and the physical interpreta tion of the various terms i n the generalized equations. Accord ing to Eq. (1) the difference i n vapor pressures of isotopes i s a purely quantum mechanical phenomenon. The vapor pressure ^ ratio approaches the classical l i m i t , high temperature, as Τ . The mass dependence of the isotope effect i s 6M/M^ where ôM = M - W . Thus for a unit mass difference i n atomic weights of isotopes of an element, the vapor pressure isotope effect at the same reduced temperature (Θ/Τ) f a l l s off as M~ . Interest ingly the temperature dependence of In P' /P i s T~2 not δλο/Τ where δλ i s the heat of vaporization of the heavy isotope minus that of the light isotope at absolute zero. In fact, i t is the difference between δλ, the difference i n heats of vapor ization at the temperature Τ from δλ. that leads to the Τ law. From Eq. (1) we can write 2 0 6

2 0 8

2 0 8


20

22

2

22

2

β

0

(AG* - AG°' ) RT 0/

,,

m K

1

m

}

(AS° - AS R

P/

)

m

1 (λ - λ') T 2 RT

0/

where (£G° - AG ) (AS° = £S ) , and (λ - λ' ) are respectively the differences i n standard free energies, entropies and heats of vaporization at the temperature T.


1

f 2


4


Within the approximation of Eq. (1) the entropy change i n the process i s equal to the change i n ô(AG)/T and each are equal to one half the isotope effect on T ~ l times the enthalpy change. The competition between entropy and enthalpy i s independent of temperature and the sign of the difference in free energy change never changes. For the system under consideration, equilibrium between a gas and a solid, this conclusion i s v a l i d even i f one includes the higher terms i n Eq. (1). Later we shall see that the consideration of higher order terms i n Eq. (1) leads to a temperature dependence on the enthalpy-entropy balance, but the difference in free energies of formation of any system from i t s atoms i s always of the sign given by Eq. (2). Although Keesom and van Dijk were led to the successful separation of the neon isotopes through predictions made by Professor Otto Stern based on Eq. (1), Keesom and Haantjes (6) interpreted their measurements of the vapor pressure ratio of Ne/ Ne in terms of a δλ/Τ rather than a δλ/Τ temperature dependence. When I replotted their data in 1956 i n accord with expectations from Eq. (1), I found that a linear extrapolation of the Keesom-Haantjes data predicted a significant difference in vapor pressures of the neon isotopes for the hypothetical solids at i n f i n i t e temperature. There were no other direct con firmations of the T~ law predicted by Eq. (1) and this led to a reinvestigation of the vapor pressures of the neon isotopes. Roth and Bigeleisen (7_ 8) not only confirmed Eq. (1) through their investigation of the vapor pressures of the neon isotopes, cf. Fig. 1, but also showed that the systematic investigation of the vapor pressure isotope effect i n simple substances could give important new information on the l a t t i c e energy, the an harmonic vibrations i n crystals, the melting process and the mean square force between molecules in the liquid state. The latter i s related to the intermolecular potential and the radial distribution function i n the liquid. Systematic investigation of the vapor pressures of the argon (9,10) ~ /C 0 - ) for the 18

l6

2

1

2

18

lb

1

3

exchange of one oxygen atom. The table of partition function ratios of Urey and Greiff predicts chemical fractionation factors for the isotopes of carbon and oxygen of the order of 1.0 - 1.1 at 25°C. Oxygen exchange between C0 and H ° ( ) predicted to enrich 0 i n CO over H«0 by a factor of 1.054 at 25°C. When such a fractionation i s cascaded i n a countercurrent column, i t provides the basis of a practical economic isotope separation process. The requirements necessary for such a process are reviewed by Spindel i n this monograph. Chemical exchange processes for the enrichment of ^ L i , ^ C , d 1 % were developed by Lewis and MacDonald (23), Hutchison, Stewart and Urey (24), and Thode and Urey (25), respectively. For the development of nuclear energy for military pur poses i n World War II, the Manhattan Project of the U.S. Army Engineers Corps required large quantities of D«0, highly en riched l^B and the fissionable isotope of uranium, u. The most d i f f i c u l t task was the production of kilograms of 90% U from the natural abundance of 0.7%. Many processes were i s

2

2

g

18

a n

2 3 5

2 3


8


Table I Calculation of Ratios of Distribution Functions for Light Isotopes H. C. Urey and L o t t i J. Greiff (1935)


Ratio 7

Li/ Li

6

7

LiH/ LiH

6

1 8

1 6

c o/c o

Quantum Mechanical Foundations of Isotope ... - ACS Publications

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