The stable isotope ratios provide information on the presence and magnitude of important ecological processes. Many ecological processes produce characteristic isotope ratios. The stable isotope ratio value relative to known background values may indicate the presence or absence of such processes. The exact values of the iso­tope ratios make it possible to determine the magnitude of these processes, if any.

As mentioned previously, in the case of carbon isotope ratios, the climatic changes can influence the stable isotope ratios. In addition, the change of other environmental conditions can also affect the isotope ratios. Environmental changes can be studied using some substances (tree rings, hair, and ice cubes) that preserve a record of the isotope ratios for a long time.

The isotope ratios remain the same during the movement of different elements and compounds. As a result, the source of essential elements, resources, or pollutions is easily traced using isotope ratios. The isotope ratios can be very different depend­ing on geographic location. This provides a way to trace the movement or origin of a substance or component in the landscape to continental scales. The origins of envi­ronmental pollutions can be identified in this way. For example, the origin of waste deposits by paint factories can be identified using the lead isotope ratios of the raw material. Lead has four stable isotopes: 204Pb, 206Pb, 207Pb, and 208Pb. 204Pb is a primordial isotope, and the other ones are the final stable members of the radioactive decay series (as discussed in Section 4.2). Since the quantity of 204Pb isotope remains constant and the quantity of 206Pb, 207Pb, and 208Pb changes over time and depends on the uranium and thorium concentrations, the isotope composition of lead strongly depends on its origin, which can then easily be identified. As will be discussed in Section 4.3.1, the isotope ratios of lead can also be used to date rocks.

Further Reading

Choppin, G. R. and Rydberg, J. (1980). Nuclear Chemistry, Theory and Applications. Pergamon Press, Oxford.

Demeny, A. (2004). Stabilizoto’p-geoke’mia (Stable isotope geochemistry). Magyar Kemiai Folyoirat 109-110:192-198.

Friedlander, G., Kennedy, J. W., Macias, E. S. and Miller, J. M. (1981). Nuclear and Radiochemistry. John Wiley and Sons, New York, NY.

Ghosh, P., Adkins, J., Affek, H., Balta, B., Guo, W., Schauble, E. A., et al. (2006). 13C-18O bonds in carbonate minerals: a new kind of paleothermometer. Geochim. Cosmochim. Acta 70:1439-1456.

Haissinsky, M. (1964). Nuclear Chemistry and its Applications. Addison-Wesley Publishing Company, Inc., Reading, MA.

University of Wyoming, USA. <http://www. uwyo. edu/sif/stable-isotopes/index. html.> (accessed 24.03.12.)

Lieser, K. H. (1997). Nuclear and Radiochemistry. Wiley-VCH, Berlin.

McKay, H. A.C. (1971). Principles of Radiochemistry. Butterworths, London.

In the previous sections (Sections 5.4.1—5.4.5), the different interactions (namely, coherent and incoherent scattering), photoelectric effect, and pair formation of the gamma radiation have been discussed. As seen, the cross sections, or the absorption coefficients of all these interactions depends on the energy of gamma radiation and the atomic number of the absorber. The cross sections versus energy or atomic number functions are significantly different for the different processes. The total absorption of the gamma radiation is the sum of the different interactions, expressed by the cross sections:

P pRayleigh + ^Thomson + ^photoelectric + pCompton + Ppair (5:92)

The absorption law (Eq. (5.3)) for the gamma radiation can be expressed as:

/ = /0 e-px (5.93)

Equation (5.93) can be transformed to mass absorption coefficients, as is done in the case of beta radiation (see Section 5.3.4).

Gamma energy (keV)

The scheme of the total absorption of gamma radiation as a function of the gamma energy is shown in Figure 5.25.

As seen in Figure 5.25, the mass absorption coefficients of the individual inter­actions show the range of gamma energy that is characteristic of the given interac­tion. The mass absorption coefficient of the total absorption (д) as a function of gamma energy shows a minimum: the mass absorption coefficient decreases until the gamma energy exceeds 1.02 MeV; it is the start of the pair formation.

In Figure 5.26, the mass absorption coefficient for different gamma energies as a function of the atomic number of the absorbers is shown.

As mentioned previously in this chapter, high-level nuclear waste (namely, the spent fuel elements) are stored under shielding and cooling in transitional disposals for about 50 years, and then they are deposited in geological repositories for final storage. The spent fuel elements contain the fission products and the transuranium elements. Before final storage, the spent fuel elements have to be treated in differ­ent ways. The aims of these treatments are as follows:

• To utilize the energy of beta and gamma decays,

• To produce additional fuel material (e. g., plutonium),

• To decrease the risk and cost associated with the storage of high-level nuclear waste,

• To decrease the cost of the fuel cycle of nuclear energy production, and

• To gain valuable by-products, e. g., fission products that can be used in other areas.

One possibility of treating the high-level nuclear waste is reprocessing. This is a chemical procedure in which the spent fuel elements are dissolved, and then the fis­sion products, uranium, and transuranium elements are separated. In this way, about 97% of the high-level nuclear waste can be recycled. The steps of reproces­sing are as follows:

• The spent fuel elements are cut into pieces and dissolved in 6—11 mol/dm3 HNO3 solu­tion. If the cladding is zircon or zircalloy, fluoride is also added to the solution. To avoid the chain reaction, neutron absorber (Cd, Gd) is also added.

• The gases released during the dissolution (Kr, Xe, I, T compounds, CO2, etc.) are treated as they would in the normal operation of the nuclear power plant (see Section 7.1.1.1).

• By flowing oxygen gas; if there is any uranium in an oxidation state lower than 6, it is oxidized to uranyl cation (UO2+). As a result of the nitric acidic dissolution, all cations present in the solution are nitrates. The oxidation state of uranium and plutonium is 16 and 14, respectively.

• The uranium and plutonium is extracted by tri-butyl-phosphate dissolved in kerosene. This procedure is called the “PUREX procedure.” The fission products remain dissolved in the aqueous phase.

• The uranium and plutonium are separated by using the reduction of plutonium. For this reason, ferrous(II) sulfamate or U(IV) is added to the kerosene solution. Plutonium is reduced to Pu(III), then extracted by water. The uranium remains in the organic phase (kerosene). If required, this process can be repeated for additional purification.

• The fission products are separated from the aqueous phase using different techniques (precipitation, extraction, ion exchange, etc.). At first, the chemically similar fission pro­ducts are separated, and then the individual isotopes are separated from the groups of the chemically similar elements. An example will be shown in Sections 8.5.2 (Eq. (8.17)) and 8.7.1.4 (Eq. (8.24)).

• The liquid residue of the procedure is solidified in the form of ceramics by the addition of Al(NO3)3 and SiO2, or vitrificated by Al(NO3)3, SiO2, borax, or phosphates.

Besides recycling, isotopes with shorter and longer half-lives may also be sepa­rated during the reprocessing. In this way, both the quantity and the radioactivity of the high-level waste can be significantly reduced, and less-disposal capacity is required.

Another possibility for the treatment of high-level nuclear waste may be the transmutation of the fission products of the spent fuel elements to isotopes with shorter half-lives. During this treatment, the fission products are dissolved in melted salts and bombarded with neutrons with high flux. The neutrons are pro­duced by the spallation reaction of an element with a high atomic number (such as Pb, Bi, or Hg) induced by the bombardment of protons with very high energy (>800 MeV). High-energy protons are generated in linear accelerators. The neu­trons react with the nuclei of the fission products: fission, neutron capture, and then beta decay take place. Finally, radioactive isotopes with shorter half-lives, or even stable isotopes, can be produced. This process is exoergic; about 20% of the released energy is used for the operation of the linear accelerator, and the rest can be utilized for other purposes. Thus, the nuclear energy production becomes more economical. The development of transmutation of spent fuel elements is in the experimental phase at the moment; we may have to wait a long time for the imple­mentation of this process.

Independent of the treatment of spent fuel elements, some amount of high-level nuclear waste is always formed; so final disposal of this waste is always required. Today, the only real option for final disposal is storage in geological repositories; however, presently there is no operating geological repository for high-level radio­active waste. Some countries are researching the construction of such waste reposi­tories, and they are expected to be operational by about 2040. In these repositories, high-level wastes are placed in stainless steel containers surrounded by a bentonite layer and natural geological formations.

The mass of the charged particles (protons, nuclei, and electrons) can be deter­mined by injecting them at a high speed into a magnetic field, where depending on their charge and mass, the path of particles deviates from a straight line. Neutrons, however, have no charges, so the mass of a neutron cannot be measured in this way; rather, its mass must be deduced. This can be achieved by the dissociation of the deuterium nucleus (one proton and one neutron) to a proton and neutron under the effect of gamma radiation.

The masses of free protons, neutrons, and electrons are listed in Table 2.1. When comparing the mass of the nucleus of an atom to the total mass of the free protons and neutrons, we can see that the sum of the mass of the free nucleons is always greater than the mass of the corresponding nucleus in the atom.

This difference will be equal to the binding energy of the nucleus (AE). [Note that Einstein’s formula for the equivalence of mass and energy (shown in Eq. (1.1)) can be used to calculate the binding energy.] When the binding energy of the nucleus is divided by the mass number, the binding energy per nucleon is obtained (A E/A):

AE (M — Z X mp — N X mn — Z X me )c2 ^ лл

“A A ( ‘ )

where M is the mass of the atom (not the nucleus!).

The stability of a given nucleus can be characterized by the value of the binding energy per nucleon. The binding energy per nucleon as a function of atomic mass is shown in Figure 2.2.

The characteristic binding energy per nucleon for the most stable nuclei is in the range of 7—9 MeV. The absolute value of the binding energy per nucleon—the mass number function shows a maximum of about mass numbers 50—60. This mass represents the elements of the iron group; thus, these elements are the most stable ones in the periodic table. The smallest nucleus, where the term of the bind­ing energy per nucleon can be defined, is the deuterium, which has the smallest binding energy per nucleon (around 1.8 MeV).

The binding energy is usually expressed in millions of electron volts. One electron volt is the amount of energy gained by an electron (elementary charge, 1.6 X 10219C) when it is accelerated through an electric potential of 1 V. Transferring the electron volt to the SI unit of energy (joule), 1 eV = 1.6 X 10"19 C X 1V = 1.6 X 10"19 J. For every 1 mol of electrons [found by multiplying by the Avogadro’ number (6 X 1023 particles/mol)], about 105 J is obtained. The energy of an atomic mass unit (931 MeV), mentioned in Table 2.1, is в 1013 J. The binding energy per nucleon (7—9 MeV) is about 1011 J that is 108kJ. Therefore, the binding energy of the nuclei is about 108 kJ/mol.

Now let’s compare the binding energy of the nuclei to the energy of chemical bonds. The energy of primary (ionic, covalent) chemical bonds is a few hundred kJ/mol (an amount of electron volts). Thus, the difference is about six orders of magnitude: the binding energy of the nuclei is about a million times higher than the energy of the chemical reactions.

In 1935, Yukawa provided an interpretation of the nature of the forces in the atomic nuclei using quantum mechanics. He constructed a model similar to the one for electrostatic forces, where two charged particles interact through the electro­magnetic field. In Yukawa’s model, the so-called meson field should be substituted for the electromagnetic field. In the case of the meson field, the range of the interaction is very short (about 10_15m), while the electromagnetic field has a much bigger range. The potential between two particles in the nuclei, known as the

Yukawa potential (U), can be expressed as a function of the distance of the two particles (r):

U _ g2 exp(r/R)

In Eq. (2.5), the potential is negative, indicating that the force is attractive. The constant g is a real number; it is equal to the coupling constant between the meson field and the field of the protons and neutrons. R is the range of the nuclear forces, expressed as follows:

where h is Planck’s constant, c is the velocity of light in a vacuum, and mn is the rest mass of the meson. Assuming that the meson field range is about 10-15m, Yukawa suggested that there must be a particle with a rest mass of about 200 times that of an electron. In fact, this particle was observed in the cosmic ray in 1948. It is called п-meson, and its rest mass is 273 times higher than the rest mass of the electron. The meson is a kind of elementary particle (as discussed in Section 2.2).

The total nuclear binding energy (ДE) can be given approximately on the basis of nuclear forces, by the summation of the interaction energies of the nucleon pairs (Ur, kl) at the distance r:

ДЕ _-2XX Urk (2.7)

2 k I

where ДЕ is the total nuclear binding energy and k and I are the number of protons and neutrons, respectively. The protons and neutrons are considered to be identical. The factor 1/2 is in Eq. (2.7) because of the two summations for protons and neu­trons, so each nucleon appears twice. The total binding energy of the nucleus is proportional to the product of the number of protons and neutrons:

Z X N _ Z X (A — Z) (2.8)

The function of the total binding energy has a maximum of the atomic number expressed as follows:

From here

(2.10)

In conclusion, those nuclei should be stable, such that the number of protons and neutrons are equal. This is indeed the case for light nuclei (e. g., 4He, 12C, 14N, 16O, 24Mg). However, for heavier nuclei, the number of protons increases, so the electrostatic repulsion of the positively charged protons increases. For this reason, extra neutrons are needed for stability. So Eq. (2.10) is modified as:

A $ 2Z (2.10a)

which means that the nuclei with high atomic numbers are stable at the ratio of neutron/proton = 1.4. The nuclei with medium atomic numbers have a ratio of neu — tron/proton between two values (1 — 1.4), i. e., the ideal neutron/proton ratio of the stable nuclei continuously changes in the periodic table (Figure 2.3).

The energy of the electrostatic repulsion can be calculated as follows:

where e is the elementary charge and Ra is the radius of the nucleus.

The nuclei are classified as isotope, isobar, isoton, or isodiaphere based on the number of nucleons (Table 2.2).

Neutron number

Libby discovered that the radioactive isotope of carbon, 14C isotope, is formed from the nitrogen that is present in air under the effect of neutrons from the cosmic ray. The nuclear reaction is 14N(n, p)14C. (Nuclear reactions will be discussed in Chapter 6.) If the flux of the neutrons is assumed to be constant, the formation and subsequent decay of 14C result in a constant concentration of 14C. Since the living organisms continuously incorporate 14C of the carbon dioxide in the air, the 14C concentration of the living organism (i. e., the ratio of 14C/12C) is the same. When the living organism dies, the continuous uptake of 14C ends, and only the radioac­tive decay of 14C continues. Thus, the concentration and the radioactivity of 14C decrease. From the 14C/12C ratio, the time elapsed from the death of the living organism can be estimated. In radiocarbon dating, 5570 years is traditionally used as the half-life of 14C (the actual half-life is 5736 years). This means that dating is feasible if the living organism lived between 250 and 35,000 years ago. The activ­ity of 14C in the carbon dioxide of the air and the living organisms is 16 dpm/g car­bon. However, more accurate results can be obtained using modern mass spectrometry equipments to determine the 14C/12C isotope ratio.

An interesting application of the 14C/12C ratio of tooth enamel for the estimation of the age of individuals born after 1943 was published. Tooth enamel is formed at well-determined times of childhood and contains 0.4% carbon. After the formation, there is no exchange between the carbon in the enamel and the carbon dioxide in the air. Thus, the 14C/12C of the tooth enamel reflects the ratio in the air at the time of the enamel formation. The 14C/12C ratio in the air was nearly constant until 1955, when aboveground nuclear bomb tests raised it signifiantly. After the Limited Test Ban Treaty in 1963, atmospheric 14C began to drop exponentially,
and it did not return to the level before 1955 until recently. Thus, if the 14C/12C ratio of the tooth enamel is determined, the time of the enamel formation, and from here the age of the individuals, can be estimated.

Besides radiocarbon, tritium is also formed of nitrogen in the air and neutrons in the nuclear reactions 14N(n,3 4He)T and 14N(n, T)12C. Similar to 14C, tritium iso­topes can participate in continuous exchanges between the hydrogen in the air and living organisms. The half-life of tritium is 12.35 years, so it should be suitable for dating in the interval of 10—80 years (e. g., for the dating of wine in bottles). However, thermonuclear explosions in the atmosphere significantly increased the natural tritium concentration, so the dating on the basis of tritium concentration has become quite limited. Tritium activity can be used for the dating of glacier and polar ice in layers, however, because in these cases, the effect of the nuclear explo­sions is neglible.

Among the reactions involving protons, the (p, Y) and (p, n) reactions are relatively simple because the emitted particle is neutral. The two reactions are competing reactions, however, with the (p, n) reactions (governed by strong interactions) being more frequent. The threshold energy of (p, n) reactions is about 2—4 MeV, and the reactions are endoergic. In the (p, n) reaction, the number of protons increases and the number of neutrons decreases. The product nuclei are rich in protons, so they usually decompose by positive beta decays or electron captures. Carrier-free radio­active isotopes can be produced. For example:

7Be(p, n)7B (6.27)

An example of the (p, Y) nuclear reaction is

6Li(p, Y)7Be (6.28)

The energy of the emitted photon is extremely high—about 17 MeV. Another example is the

12C(p, y)13N (6.29)

reaction. The product nucleus has positive beta decay; it has some medical applications.

The (p, a) has little importance. The Coulomb barrier acts both on the irradiating proton and on the emitting alpha particle. The reaction is always endoergic. The ratio of neutrons increases, so the product nuclei decompose by negative beta decay. Carrier-free isotopes can be produced. For example:

14N(p, a)11C (6.30)

The reaction rate of the isotope molecules may be different. This effect is determined by the reaction mechanism, including thermodynamic properties of the transition state, so the kinetic isotope effects can be applied for the study of the mechanism of the chemical reactions.

The kinetic effects are significant in the case of light elements since the mass of the isotopes of these elements has the greatest differences, resulting in relatively great differences in the rotation, vibration, and electron energies of the isotope molecules and the transition state. The reactions of the molecules containing differ­ent H, C, N, O, and S isotopes are important. Obviously, the reactions of such molecules are interesting mainly in organic chemistry.

The kinetic isotope effects can be classified as primary and secondary effects. In the primary effects, the bond that contains the isotope atoms breaks or forms in the rate-determining step. The primary kinetic isotope effects can be divided further in intermolecular and intramolecular effects. In the intermolecular effect, two molecules react with different rates. In the intramolecular effect, the equivalent sites within the same molecules show different rates because the sites have different isotopes.

A primary intermolecular isotope effect is as follows:

AX 1 BY ——! BX 1 AY (3.19)

AX01 BY ———! BX01 AY (3.20)

In Eqs. (3.19) and (3.20), two identical molecules (AX and AX0) contain differ­ent isotopes of the same element (X and X0). When the reaction constants are different (k1 ф k2), the reaction of the two isotope molecules (AX and AX0) with the molecule BY shows a primary intermolecular isotope effect.

A primary intramolecular isotope effect can be observed in the following process:

AXX’ + 2BY -> BX’ + BX + AYY

(3.21)

£3 k4

where k3 and k4 are the rate of the production of BX0 and BX, respectively. An isotope effect occurs when k3 ф k4.

In the secondary isotope effects, the isotope atom does not directly take part in the reaction. For example,

&5

AXX 1 BY —5—— BX 1 AXY (3.22)

AXX01 BY ——— BX 1 AX’Y (3.23)

where k5 ф k6.

A primary isotope effect can be observed in the thermal decarboxilation of oxa­lic acid if one or both carbon atoms are substituted by the 13C isotope:

12COOH k

I -> 12CO2 + H2O + 12CO (3.24)

(3.26)

13COOH

An intramolecular isotope effect is found when k2/k3, whereas intermolecular isotope effects can be observed in case of k1/(k21 k3), k1/k4, and (k2 1 k3)/k4, respectively.

As a secondary isotope effect, the reaction of carboxyl groups of malonic acid is mentioned when deuterium is substituted for the hydrogen bonded to the (3-carbon atom. The maximum values of the kinetic isotope effects (shown in Table 3.3) are determined using the thermodynamic properties of the isotope molecules.

The study of the isotope effects can be used to elucidate the reaction mecha­nism, as the following example shows. The oxidation of alcohols to carboxylic acid by bromine is made up of two steps:

CH3 — CH2 — OH 1 Br2 ——— CH3 — CHO 1 2HBr (3.27)

fast

CH3 — CHO 1 Br2 1 H2O———— — CH3 — COOH 1 2HBr (3.28)

The rate-determining step is the oxidation of the alcohol, which results in the formation of aldehyde (Eq.(3.27)), a first-order reaction both for alcohol and

bromine. There are two mechanistic possibilities. The first is that bromine reacts with the hydrogen in the hydroxide group and in a rate-determining step:

CH3 — CH2OH 1 Br2 -—! CH3 — CH2 — OBr 1 HBr (3.29)

fast

CH3 — CH2 — OBr———— ! CH3 — CHO 1 HBr (3.30)

This support for this mechanism is that it resembles the fast reaction of alkyl hypochlorites. If this is the right mechanism, secondary isotope effects should be observed if the alcohol CH2 group is labeled by an isotope of the hydrogen. In the case of H — T substitution, this mechanism can decrease the reaction rate by 2.2 times (Table 3.3).

The second possibility is that bromine reacts with the carbon atom of the alcohol CH2, which would result in a much higher (i. e., primary) isotope effect when substituting one of the hydrogen atoms of alcohol CH2 by tritium. In this case, two types of aldehyde would form, an unlabeled and a labeled molecule:

CH3 + CHO + TBr + HBr

CH3 — CHT — OH + Br2 (3.31)

"IT^ CH3 + CTO + HBr

k

CH3 — CH2 — OH 1 Br2 —% CH3CHO 1 2HBR

Because of the two product molecules of the labeled alcohol, the value of the isotope effect has to be calculated as:

2kH

kx1 1 kx2

Thus, examining the relative rates, it can be determined if the reaction starts with the reaction of CH2 and Br2, or if it proceeds via a hypobromite intermediare.

During в+-decay, positrons are emitted. The positron is the antiparticle of the elec­tron, and therefore it is unstable. Its half time is the time of thermalization, which means that the time required for the velocity of the positron decreases to zero. It is about 10-10 s. If the positron encounters an electron in this interval, the two parti­cles (electron and positron) transform to electromagnetic radiation, gamma photons. The process is called “annihilation.” The rest mass of the positron (в — particle) is 0.51 MeV, equal to the rest mass of the electron, so 2 X 0.51 MeV energy is emitted in the annihilation process. Usually, two gamma photons with 0.51 MeV energies are emitted at an angle of 180°. The probability of the formation of two photons is about 90%. (This process is applied in the PET (Section 12.6)). In about 10% of the annihilation process, only one photon with 1.02 MeV is formed.

Interaction with the molecules of matter
Cherenkov radiation

Figure 5.13 Summary of interaction of beta particles with matter.

In some cases, three photons are emitted, and the total energy of them is also 1.02 MeV. The positive beta decay can be detected easily through the detection of the gamma photons with 0.51 MeV.

It is interesting to mention here that before the total thermalization, the positron can interact with an electron, constructing a short-life light element, positronium, whose nucleus is the positron. Positronium can be treated as an atom with an atomic number of zero.

Positronium has two forms: ortho — and para-positronium, depending on the spins of the positron and electron. In ortho-positronium, the spins are parallel; the lifetime in a vacuum is 1.4 X 10_7 s. In para-positronium, the spins are antiparallel; the life­time in a vacuum is 1.25 X 10_10 s. In other media, the chemical reactions (addi­tion, substitution, oxidation, and reduction) decrease the lifetime; thus, the kinetics of chemical reactions can be studied by measuring the lifetime of positronium.

The active zone of the reactors is usually surrounded by a mantle consisting of the moderator (water, graphite, or another substance as previously mentioned). Thus, the number of the escaping neutrons can decrease because this mantle reflects a part of the escaping neutrons. The escaping neutrons influence the effective neutron multiplication factor. The application of the reflector can decrease the size of the reactor, and the fuel can be utilized more economically.

7.1.1.2 Coolants

The greatest part of the energy released in the fission is the kinetic energy of the fission products (Table 7.2). While the fission products are slowing down, the fuel

rods are heated. In operation, the temperature of the fuel rods may rise above 1000° C, and then they have to be cooled permanently. Therefore, the active zone always contains coolant.

Because of the strong neutron radiation in the active zone, the coolant becomes radioactive. For this reason, the coolant must be circulated in a closed system, which is called a “primary circuit.” The primary coolant is pumped into a heat exchanger full of tubes. Heat is transferred through the walls of these tubes to the lower-pressure secondary coolant located on the sheet side of the heat exchanger, where it evaporates to pressurized steam (a steam generator). In this way, the cool­ants in the primary and secondary circuits do not touch each other directly, so the secondary coolant remains inactive.

The pressurized steam formed in the steam generator is fed through a steam tur­bine, which drives the electric generator. In the meantime, the secondary coolant (a mixture of water and steam) is cooled down and condensed in a condenser. Then the condensed steam is pumped back into the steam generator (Figure 7.2).

In reactors moderated by light or heavy water, the moderator acts as a coolant too. In graphite-moderated reactors, the coolant is a gas (CO2 or He) or water. In some reactors, molten metals (e. g., sodium), salts, organic solvent, He, or steam is applied as the coolant.

In the water reactors, the coolant is continuously purified by ion exchangers to remove the dissolved radioactive ions (e. g., cesium and iodide ions). The colloidal or greater solid particles are filtered.

As mentioned earlier (Section 2.1.1), the stability of a nucleus (as characterized by the binding energy) is determined by the ratio of protons to neutrons (see Eq. (2.10)). The binding energy of isobar nuclei as a function of proton—neutron ratio forms a parabola or parabolas (see Figure 3.1), where those nuclei close to the minimum are stable while those farther away are undergoing radioactive decay in order to reach the optimal proton/neutron ratio. The radioactive decay is a random process for the individual nucleus so as to describe the kinetics of radioactive decay, a statistical approach has to be applied.

4.1 Kinetics of Radioactive Decay

4.1.1 Statistics of Simple Radioactive Decay

Let us consider that the probability of the decomposition of a radioactive nuclide in a At time interval is p:

p = XAt (4.1)

where A is a factor of proportionality. The probability of the process that the nuclide does not decompose in At is:

1 — p = 1 — AAt (4.2)

The probability that the nuclide does not decompose in another second, third, or more At interval can also be defined by Eq. (4.2). The probability that the nuclide does not decompose in the 2 X At interval is:

(1 — p)2 = (1 — AAt)2 (4.3)

The probability that the nuclide does not decompose in n X At is:

(1 — p)n = (1 — AAt)n (4.4)

Let us divide the total time of the observation (t) into n intervals:

— = At (4.5)

n

Nuclear and Radiochemistry. DOI: http://dx. doi. org/10.1016/B978-0-12-391430-9.00004-4

© 2012 Elsevier Inc. All rights reserved.

Substituting Eq. (4.5) into Eq. (4.4), we obtain:

/ t n

(1 — p)n = 1 — An (4.6)

At n! n:

t n

1 — A — = e-At (4.7)

n

When the initial number of the radioactive nuclides is N0, the number of nuclides that do not undergo radioactive decomposing during t time (N) is:

N = N0 e—A (4.8)

Equation (4.8) describes the kinetics of the simple radioactive decay, i. e., the radioactive decay law, where A is the decay constant. The value of the decay con­stant characterizes the radionuclide; thus, it is independent of physical and chemi­cal conditions (pressure, temperature, chemical environment, etc.).

As seen in Eq. (4.8), the radioactive decay has first-order kinetics, having all characteristics of first-order reactions. It has a well-defined half-life (t1/2), i. e., the time needed to reduce the number of the radioactive nuclides to half:

N0 = N0 e—At1/2 (4.9)

and from here,

A = ^ (4.10)

t1/2

Half-life performance 7 and 10 times over gives the interval when the number of radionuclides decreases below 1% and 0.1% of the initial number, respectively.

The reciprocal of the decay constant (A) is the average lifetime (t) of the radionuclides, which is the time when the number of the radionuclides decreases by a factor of e (i. e., Euler’s constant). This amount of time should be required for the decomposition of all radionuclides if the rate of decay remains constant.

(4.11)