Equation (9.24), expressing the ratio of the radioactive and inactive atoms at the maximum mixing entropy, has another meaning. When the nominator of Equation
(9.24) is multiplied by the decay constant of the radioactive isotope (A), the denom­inator is divided by Avogadro number (NA) and multiplied by the molar mass (M), the equation is obtained as follows:

0 A’

n a_

n +N M

Na,

where — is the radioactivity of the tracer, m is the mass of the ith component, including the tracer and the inactive carrier, and the -/m ratio is the specific activity.

On the basis of the specific activity, the tracer studies are classified into two groups:

1. The specific activity is constant, which means that the mixing entropy of the system is maximal during the whole period of the studies. In this case, the activity is measured at different places of the system and at different times, the ratio of the activities quantita­tively gives the distribution of the substance. This method is applied, for example, for the determination of the solubility of very insoluble salts (such as in Hevesy’s first tracer experiments, described in Section 8.1) or for the study of the efficiency of electrolysis.

2. The specific activity changes because the radioactive isotope is diluted with the stable isotope of the same element. In these studies, the specific activity has to be deter­mined before and after the dilution. The change in specific activity gives information on the quantity of the diluting substance. This principle is applied to isotope dilution methods, including some important medical applications (such as RIA, described in Section 12.3).

The principle of the isotope dilution methods is discussed here. Let us suppose a radioactive substance with activity -:

— = An (9.26)

where n is the number of radioactive nuclides and A is the decay constant. The ini­tial specific activity before dilution (a0) is:

An 1

a0 = П+N M

Na

where N is the number of the inactive nuclides (carrier), M is the molar mass, and Na is Avogadro number (see also Eq. (9.25)). If N0 inactive carrier nuclides are added to this system (dilution), the total activity (—) remains the same (a closed system); the specific activity (a0), however, decreases:

Since the activity (A) is the same, from Eqs. (9.27) and (9.28), we obtain:

A 5 An 5 aQ(n 1N) = a! (n 1N 1 N0) (9.29)

From here,

N0 5 (n 1 N)0 — l) (9.30)

Since n«N, the number of the radioactive nuclides can be disregarded. The quan­tity of the diluting substance can be calculated when we know N, and the specific activities before (a0) and after (a0) the dilution.

According to the classification by the specific activity, group l (constant specific activity, the mixing entropy is maximal) contains, for example, the determination of solubility, a part of diffusion studies (self-diffusion is not included), radiometric analysis, and autoradiography. Group 2 (specific activity changes, the mixing entropy increases) contains, for example, the determination of specific surface area, the different types of isotopic dilution methods, substoichiometric analysis, RIA, the study of isotopic exchange reactions, self-diffusion, and the determination of exchange current.

The radiotracer methods can also be classified based on the field of applications, such as physicochemical, analytical, biological, medical, and industrial applica­tions. Physicochemical applications include, for example, the determination of sol­ubility, the study of diffusion, the distribution of substance between phases, and the study of reaction mechanisms. It is important to note that radiotracer methods are widely used in the study of interfacial processes because of the high sensitivity of radioindicators. Analytical applications include the radiometric analysis, isotopic dilution methods (including RIA), autoradiography, neutron activation analysis, and all analytical methods based on the interaction of radiation with matter (see the discussion of this in Chapter 10).

Isotope labeling can also be done by stable isotopes, in which case the natural abundance of a given element is altered. In other words, a stable isotope is enriched. The concentration of the stable isotopes can be determined in two ways:

• The samples are activated after the studies and the number of isotopes is determined on

the basis of the produced radioactive nuclides.

• The number of isotopes can be determined by mass spectrometry, infra spectrometry, and

so on.

These methods, however, are much more complicated than simple radioactivity measurement methods.

Of course, the basic concepts of labeling by stable isotopes are just the same as those of radioactive isotopes, only the detection methods are different. Therefore, all of the principles and relation mentioned above for radioactive isotopes apply to stable isotopes.