LIMITATIONS OF THE RADIOACTIVE TRACER METHOD ROBERT R. HENTZ North Carolina State College, Raleigh
Tm
use of radioactive atoms as tracers is based on two characteristics of such atoms which are stated by Wahl and Bonner (1) as follows: "Before a radioactive atom decays, its chemical behavior is essentially the same as that of other atoms isotopic with it; when a radioactive atom does decay, it emits energetic radiation that may be detected." Thus, radioactive atoms serve as tagged or labeled atoms whose behavior in a system serves to indicate the behavior of all isotopic atoms originally in the same chemical form. It is as if certain of the atoms of an element could he painted with a characteristic color for which there is an extremely sensitive detection instrument. The behavior of all the atoms of this element in a system could be determined simply by observation of the colored atoms. Isotopic atoms which are not radioactive have been used for many years as tracers. In 1934 Polanyi and Szabo used oxygen-18 as tracer to elucidate the mechanism of esterification (I?),and Hevesy and Hofer used deuterium to determine the water content of the body by isotopedilution (3). The special virtue of radioactive atoms is the extreme sensitivity with which such atoms may be detected. It is of interest to illustrate the sensitivity of radioisotope measurement by a calculation of the amount of an isotope of promethium (4) necessary for detection and measurement. In terms of the half-life the radioactive decay law becomes -dN/dt = 0.693N/tv2
and the number of atoms required for a given decay rate is N = (-dN/dt) t1/,/0.693
Promethium-149 has a half-life of 47 hours and emits an easily detected beta particle of 1.1 Mev. maximum energy. A counting rate of 30 counts per minute will be assumed as measurable with reasonable accuracy (1-5yo) in a reasonable period of time (w 1 hr.) using a standard end window counting assembly assumed to detect about 20% of all beta particles emitted by the sample. For these conditions N = (30 c./min.)(47 hr.)(60 min./hr.)/(0.'2)(0.69) = 6.1 X 106atoms, N = 6.1 X 1@/6.0 X lonS= gram-atoms, W = 10-IS X 149 = 1.5 X grams
There is no other method which could detect anywhere near as little promethium as 10-l6 grams. There are two general limitations on the use of radioactive atoms as tracers which are implicit in the introductory statement of the characteristics of such VOLUME 35, NO. 12, DECEMBER, 1958
atoms. Although these limitations are usually mentioned in textbooks of radiochemistry, they are rarely emphasized sufficiently or treated in su5cient detail to provide the neophyte with an adequate basis for understanding the fundamental nature of these limitations or the circumstances under which they need to be taken into consideration. Almost all tracer experimentation is based on the assumption of "essentially" identical chemical behavior of the isotopes of an element, the first characteristic. The constancy of isotopic composition of elements in the earth's crust and the difficulty (but not impossibility) of separation of isotopes by chemical means are perhaps sufficient justification for such an assumption in most cases; however, there are differences in chemical behavior of isotopes, and it is of interest and importance to understand how isotopic mass influences chemical behavior and to appreciate the order of magnitude of this isotope effect. Secondly, emission of energetic radiation by a radioactive tracer atom which is the basis for its detection, the second characteristic, produces--except in the case of a nuclear isomer t r a c e n n atom of another element and necessarily results in chemical change. The number of such transmutations is however quite negligible compared to the amount of chemical change produced in the system by the effect of the energetic radiation which accompanies each transmutation. Thus, i t is necessary to consider whether the amount of energetic radiation emitted by the decaying tracer atoms will cause sufficientchemical change in the system to invalidate the interpretation of the tracer experiment. For an interesting quantitative discussion of another possible source of error in tracer studies see the paper by Ropp (5) in THIS JOURNAL. EFFECT O F RADIATION ON A TRACER EXPERIMENT
A method of estimation of the magnitude of the effect of radiation decomposition may be illustrated by a calculation for a simple, hypothetical system. A tracer experiment is to be conducted for a period of t hours using Ti liters of an M molar solution of some compound which contains N carbon atoms per molecule labeled with carbon-14 of specific activity S. This isotope of carbon has a half-life of 5568 years, emits a beta particle of 0.155 Mev. maximum energy with a range of 30 rng./~rn.~ in aluminum, and has a disintegration rate of 4.61 curies/g. (6). The number of curies of carbon-14 present in the system is M X V X N X S X 14.0 X 4.61
=
64.5 M V N S curies
The curie is defined as that amount of a radioisotope which is necessary to provide 3.7 X loL0disintegrations per second (7). The energy of the beta particle emitted may vary from zero to the maximum. The energy difference between t.he maximum energy and that of a beta particle is carried off by a neutrino and is not absorbed in the system. The average energy per beta particle is about one-third the maximum energy; therefore, the beta energy being emitted is 64.5 MVNS X 3.7 X 10"
X 1/3 X
1.55 X 105 = 1.23 X 10" MVNS ev./sec.
Assuming the range of the beta particles to be essentially the same in the system as in aluminum and approximating the density of the system as about 1 g . / ~ r n . the ~ , range in millimeten is 0.03 g./cma/l g./cm.. =
0.03 c m .
=
0.3 mm.
Since this is the range of a maximum energy beta particle and since even half of the beta particles right a t a solution interface will be emitted back into the solution, it is apparent that for any reasonable volume-in excess of about 1 ml. and not too thin in shape-all the beta energy may he assumed to he absorbed within the volume of solution. The rate of energy absorption by the dolution is then equal to the rate of energy emission, 1.23 X 10" MVNS ev./see.
It is now necessary to know the efficiency with which this energy effects decomposition of the compound under study. The yield of a radiation chemical reaction is denoted by the symbol G which is defined as the number of molecules of reactant decomposed or of a particular product formed for 100 ev. of energy absorbed in the system (8). The number of molecules of the compound which will be decomposed by the absorbed radiation in t hours is then 1.23 X lo1' MVNS X 3600 t X G/100 = 4.44 X MVNSGt molecules. This corresponds to a fraction of the total number of molecules of the compound originally present in the system equal to or in parts per million. 7.36
NSGt p.p.m.
Calculation for a particular set of conditions is instructive. Assume a compound of 6 carbon atoms per molecule in which the specific activity is unity (all carbon atoms in the system are carbon-14) and that the experiment is carried out over a period of 10 hours. G values may vary over a wide range depending on the nature of the chemical system. For example, the G value for total gas production from liquid benzene at room temperature is 0.06 while the G value for the gaseous chain reaction between hydrogen and chlorine may be as large as 100,OM) (8); however, in the absence of chain reactions and unusual stability such as that of aromatic compounds, a G value of the order of unity may reasonably be assumed for most systems under ordinary conditions. Under the postulated conditions the fraction decomposition due t o tracer radiation becomes
Since tracer experimentation is rarely more precise than I%',, it is apparent that the effect of radiation decomposition is negligible even when carbon-14 of unit specific activity is used (which is rarely the case). Only for compounds of very large N or G and for long experiments would the effect become appreciable, especially for the much lower carbon-14 specific activities ordinarily used. Sulfur-35 decays by soft beta emission of maximum energy 0.167 Mev. a t a rate equal to 4.28 X lo4 curies /g. (6). A similar calculation may be made for a system in which this radioisotope is used as tracer and gives for the fraction compound decomposed 1.84 X lo5 NSGt p.p.rn.
=
18.4NSGl%
Assuming only one sulfur atom per molecule, S = 1, G = 1, and t = 10, the fraction decomposition becomes 184%. Obviously, sulfur must be used a t very low specific activities. I n work of the author (9) the maximum specific activity of the sulfur employed was about lo-" which would give a fraction decomposition due to radiation of 1.84 X 10-5 p.p.m. A similar calculation may be made for other soft beta emitters and for alpha emitters. The calculation for hard beta and gamma emitters requires a correction for the fraction of energetic radiation emitted which is absorbed in the system. Discussions of the mechanisms whereby radiation causes chemical change may he found in the review articles of Burton (8) and Hart (10). ISOTOPE E F E C T ON REACTION RATES
The effect of isotopic mass on the rate of a chemical reaction may be illustrated in a fairly elementary manner by application of the collision theory of reaction rates to a very simple type of reaction for which the isotope effectwould he a maximum. Consider a chemical reaction in which the rate determining step is dissociation of a diatomic molecule. A2 = 2A
In accordance with a simple collisional model, the rate of reaction will be given by the number of collisions suffered by these molecules per cubic centimeter per second multiplied by the fraction of such collisions in which the relative kinetic energy along the line of impact exceeds the dissociation energy of the molecules. If the collision diameter of a molecule is represented by a, then one molecule may he imagined to have a radius of a and all other molecules may be considered as points. In one second a cylinder of volume s a 2 P will be swept out by the hypothetical double-sized molecule, and all hypothetical point molecules within this volume will have suffered a collision with it. Therefore the number of collisions per molecule per second is simply s o Z V C where V is the mean relative molecular velocity and C is the concentration of molecules per cc. The total number of collisions between all molecules per cc. per second then becomes s o 2 P C2. The fraction of such collisions in which the relative kinetic energy along the line of impact exceeds the dissociatioil energy, ED, may be shown to be simply the familiar Arrhenius expression e d D I R T . The rate of reaction is then JOURNAL OF CHEMICAL EDUCATION
The rate constant of a chemical reaction may be defined as the coefficient of the concentration terms or the rate a t unit concentration. k =
(L/~),,,P
Pe-En/RT
(2)
Mass Dependence of the Rate Constant. The effect of isotopic mass on the rate constant appears in both r a n d ED. The mean relative velocity is given by where M is the molecular weight. For a diatomic molecule M = 2m (the atomic weight) and the collision frequency at unit concentration becomes
F = oz m4=a/dG (4) where all constants have been combined in the new constant a. A diatomic molecule in its lowest vibrational energy states may be represented approximately as a simple harmonic oscillator. Quantum mechanical calculation shows that such an oscillator must possess a definite residual vibrational energy even a t the absolute zero of temperature. This minimum vibrational energy permitted, called the zero-point energy, is given for one mole as (I/,)ncr2
or substituting equation (6) k/k' = ( 1
+ n/2m)
enEd2m"T
(12)
where Eoand m are both for the lighter isotope. Equations (11)and (12)are only valid for n/m
