Basic Decay Equations

Radioactive decay is what chemists refer to as a first-order reaction that is, the rate of radioactive decay is proportional to the number of each type of radioactive nuclei present in a given sample. So, if we double the number of a given type of radioactive nuclei in a sample, we double the number of particles emitted by the sample per unit time.2 This relation may be expressed as follows  

Note that the foregoing statement is only a proportion. By introducing the decay constant, it is possible to convert this expression into an equation, as follows  

The decay constant A. represents the average probability per nucleus of decay occurring per unit time. Therefore, we are taking the probability of decay per nucleus, and multiplying it by the number of nuclei present so as to get the rate of particle emission. The units of rate are (disintegration of nuclei/time) making the units of the decay constant (1/time), that is, probability/time of decay. 

To convert the preceding word equations to mathematical statements using symbols, let N represent the number of radioactive nuclei present at time t. Then, using differential calculus, the preceding word equations may be written as 

Note that N is constantly reducing in magnitude as a function of time. Rearrangement of Equation (3.2) to separate the variables gives 

---

An approach, similar to that employed in the analysis of tartrate mixtures, has been used for the chiral discrimination of amino acid (M/j/s) mixtures, using an amino acid of defined configuration as reference (S). The proton-bound trimers [S2-M H]+ form [S M H]+ and [S2H]+ fragments upon CID or MIKE decay (equations (9)-(12)). With two independent measurements of the fragmentation ratio [S-M-H] /[S2H] from either [S2-M -H] and [52-M5-H]" , the differences in binding energies can be determined. The relative gas phase basicities (GB) of the molecular pairs [S-M] and [S2] can be derived from equations (13) and (14). 

FPA results obtained at different salt conditions may not be directly comparable because the fluorescence properties of 6-MI, including the lifetime (t), are salt dependent. The salt dependence of the FPA of a helix in a complex construct should thereby be normalized relative to the FPA of a short control duplex of the same sequence of the targeted helix to account for salt effects on the local environment of the 6-MI fluorophore. The normalization ratio, rnoml, can be calculated as the ratio between the apparent rotational correlation time, 9, of the constructs and the control duplex only, rnomi = construct/ control- is related to the rate of anisotropy decay, with larger 9 associated with higher anisotropy. If the basic Perrin equation for a sphere (Eq. (14.3)) is used to simplify calculation, then... 

In cases where energy migration is a dominant feature of luminescence, as in molecular crystals, various forms of decay are expected depending upon circumstances, but relying upon solutions, usually complex, to the basic rate equations where E(t) is the time-dependent population of the initially excited (exciton) state, T(t) the population of the trap state, kg the decay rate constant for band... 

Equation 12.49 is the basic nondimensional equation describing the mole fraction of A in a fixed-bed reactor containing an exponentially decaying catalyst as a function of position and time in terms of two dimensionless parameters, B" and A. The performance of this reactor can be best judged by solving the equation for the reactor exit, that is, for z = 1. The solution for a first-order reaction (m = 1) is given in Table 12.7 (Sadana and Doraiswamy, 1971). It is also possible to assume various other forms of catalyst decay. Solutions are included in the table for two other forms, one of them linear. 

Equation 9.45 is the basic nondimensional equation describing the mole fraction of A in a fixed-bed reactor containing an exponentially decaying catalyst as a function of position and time in terms of two dimensionless parameters j8 and X. The performance of this reactor can be best judged... 

The basic concepts of nuclear structure and isotopes are explained Appendix 2. This section derives the mathematical equation for the rate of radioactive decay of any unstable nucleus, in terms of its half life. 

Equation (9.6) is the basic equation describing the decay of all radioactive particles, and, when plotted out, gives the familiar exponential decay curve. The parameter X is characteristic of the parent nucleus, but is not the most readily visualized measure of the rate of radioactive decay. This is normally expressed as the half life (7/ 2). which is defined as the time taken for half the original amount of the radioactive parent to decay. Substituting N = Na/2 into the Equation (9.6) gives ... 

Equation (8.3) is the basic equation that describes all radioactive decay processes. It gives the number of atoms (N) of a radioactive parent isotope remaining at any time t from a starting number N0 at time t = 0. 

Substituting Equation (8.6) into (8.7) gives the basic equation that is used to calculate a date for a rock or mineral from the decay of a radioactive parent to a stable daughter ... 

An important application of the basic radioactive decay law is that of radionuclide dating. From Equation (3.6), we have... 

Now irreversible processes can be studied on three levels (1) the phenomenological thermodynamic level in which the equations for the macroscopic variables are studied (2) the level of fluctuations, in which we study the nature, growth or decay of small fluctuations either of internal or external origin and (3) the basic level in which we try to identify the microscopic mechanisms of irreversibility. Here I shall be mainly concerned with the first, and to certain extent the second level, through which I believe we can begin to understand how irreversible processes bring about the different aspects of the process of evolution. 

During migration and subsurface storage groundwater comes into contact with crustal rocks that continuously release helium from the decay of uranium and thorium. The basic assumption is made that the water acts as a sink for the helium evolved from the local rocks. The age of groundwater, t, is calculable from the equation given in section 14.2. 

The immediate question, then, is whether this scenario reflects what actually happens. Do the INM theories really work There is, in fact, some evidence on this score (45,52). If we compare the vibrational friction predicted by INM theory, Equation (20), with that revealed by an exact molecular-dynamics evaluation of the force autocorrelation function, Equations (4) and (13), we see some reasonably impressive agreement (Fig. 3) (52). Not only is the few hundred cm-1 spectral range of the friction predicted quite nicely, but the basic form of the response is as well. Each example shows that the friction diminishes as the frequency rises, beginning with a sharp drop from its maximum value at co = 0 and gradually going over to a much slower decay, behavior captured nicely by the INM formulas. 

The spin-lattice relaxation time is found by several techniques, which consist basically of applying an oscillating field, Hx, perpendicular to the static field H0 with a duration shorter than 7, to obtain a maximum signal amplitude. The measurement is made by preparing the system in a state Mx = My = 0, Mz Mq at a time t = 0 and measuring the decay to the equilibrium state M0. According to the Bloch equations the transverse magnetization stays zero and Mz decays exponentially as Mz(t) = M0( 1 — e t/Tl). 

The positron has a short life and will quickly be annihilated in a reaction with an electron, producing y-photons of characteristic energy (0.51 MeV). In addition, the basic nuclear process itself is usually accompanied by the emission of gamma radiation. As in the case of negatron decay a complete energy balance reveals a discrepancy which can be accounted for if the emission of a further particle—the neutrino, v is postulated. Overall, positron emission can be summarized in a general equation... 

This type of problem has recently become of great interest in the field of physics, being directly related to basic problems such as noise-induced phase transitions and the dependence of the relaxation time of the variable of interest on the intensity of external noise terms. As in Chapter X, the reduced equation of motion itself can be assumed to be rehable only when this relaxation time decreases with increasing noise intensity. Further problems of interest are whether or not the threshold of a noise-induced phase transition is characterized by a slowing dowrf " and what is the analytical form of the long-time decay. ... 

---