392 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 15 



where q = grams of labeled substance added 



M x = molecular weight of added substance with enriched rarer isotopes 

 M<i = molecular weight of diluent with normal isotopic composition 

 C\, Ci = atom per cent excess of rare isotope in carrier before adding and 

 after recovery, respectively 

 If the molecular weight is large and the tracer isotope occupies only a few 

 atomic positions in the molecule, then M1/M2 ~ 1. 



15.3. Tracer Problems Involving First-order Reactions. Many of the 

 tracer problems encountered at the present time, particularly in biological 

 investigations, are concerned not only with identification of constituents and 

 metabolic routes in systems of widely varying degrees of complexity but 

 also with kinetic properties of systems. Although measurements of the 

 amounts of a labeling agent in various parts of a system serve sufficiently 

 well to identify constituents and chemical or physical processes, some kind of 

 mathematical formulation based on the experimental data is required to 

 describe adequately the rates of processes. The precise method of formulat- 

 ing the appropriate mathematical expression may not, of course, always be 

 immediately obvious since, presumably, it may include any of the processes 

 already known to chemical kinetics. 



When the processes under consideration can be shown to be first-order 

 reactions their mathematical description becomes particularly simple and, 

 needless to say, extremely useful. Reactions of this type proceed at rates 

 proportional to the amount of substance present. They may be described, 

 therefore, by linear first-order differential equations of the type 



dx 



It = klX ~ hf 



where x is the amount of substance present at time t, ki, and k 2 are constants, 

 and /is some arbitrary function (arbitrary in a mathematical sense). The 

 first term on the right is the amount added per unit time, and the second 

 term represents the amount disappearing per unit time. A considerable 

 simplification is usually introduced in practice in that the system with its 

 manifold processes is in dynamic equilibrium during the time it is investi- 

 gated. The differential dx/dt = 0, and the differential equation then 

 becomes an algebraic equation. Within the system substances may be 

 produced and utilized, transferred from one region or tissue to others, or 

 undergo changes in chemical and physical form; but all such processes take 

 place at constant rates. The addition of a small quantity of material, 

 homologous with a normal metabolite already present and labeled with a 

 convenient tracer, will not appreciably disturb the system and presumably 

 will undergo the same processes without discrimination. The tagged 

 molecules themselves, however, will not be in equilibrium, and since they are 



