Problem Radioactive decay

In a Northern-blot experiment you are trying to detect a specific mRNA by using a radiolabelled probe. This probe contains the radioactive element P(hosphorous), which has a half life of 14.5 days. You know that all radioactive decay follows a first order reaction. From previous experiments you also know that you need at least 10 ng of radiolabelled probe. 3 weeks ago (21 days) you prepared 36 ng of radiolabelled probe. Is there still enough probe left to be used in the experiment, or do you have to make a new probe  

And we can easily solve for P(t). The only problem is that we only have the half life, but not the rate constant k. 

However, we can convert the half life into the rate constant by using the equation  

We can now put this number into our equation (using a table)  

Answer After 21 days there are 13.2 ng of probe left, which should be enough to do the experiment. 

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Our next example concerns the Poisson process, which plays a central role in a variety of problems such as waiting lines, inventory control, electrical noise, the firing of neurons, and radioactive decay. We will discuss the application of the Poisson process to the study of certain kinds of electrical noise in a later section. 

Calorimetry. Radioactive decay produces heat and the rate of heat production can be used to calculate half-life. If the heat production from a known quantity of a pure parent, P, is measured by calorimetry, and the energy released by each decay is also known, the half-life can be calculated in a manner similar to that of the specific activity approach. Calorimetry has been widely used to assess half-lives and works particularly well for pure a-emitters (Attree et al. 1962). As with the specific activity approach, calibration of the measurement technique and purity of the nuclide are the two biggest problems to overcome. Calorimetry provides the best estimates of the half lives of several U-series nuclides including Pa, Ra, Ac, and °Po (Holden 1990). 

The conceptual problems start when considering materials such as plutonium, which is a by-product of the nuclear electricity industry. Plutonium is one of the most chemically toxic materials known to humanity, and it is also radioactive. The half-life of 238Pu is so long at 4.5 x 108 years (see Table 8.2) that we say with some certainty that effectively none of it will disappear from the environment by radioactive decay and if none of it decays, then it cannot have emitted ionizing a and f) particles, etc. and, therefore, cannot really be said to be a radioactive hazard. Unfortunately, the long half-life also means that the 238Pu remains more-or-less for ever to pollute the environment with its lethal chemistry. 

Consider the following problem The rate constant for the radioactive decay of thorium-232 is 5.0 x 10 n/year. Determine the half-life of thorium-232. 

This is a radioactive decay process. Radioactive decay follows first-order kinetics. The solution to the problem simply requires the substitution of the / -value into the appropriate equation ... 

A specific example of applications in the second category is the dating of rocks. Age determination is an inverse problem of radioactive decay, which is a first-order reaction (described later). Because radioactive decay follows a specific law relating concentration and time, and the decay rate is independent of temperature and pressure, the extent of decay is a measure of time passed since the radioactive element is entrapped in a crystal, hence its age. In addition to the age, the initial conditions (such as initial isotopic ratios) may also be inferred, which is another example of inverse problems. 

The concentration of the radioactive nuclide (reactant, such as Sm) decreases exponentially, which is referred to as radioactive decay. The concentration of the daughter nuclides (products, including Nd and He) grows, which is referred to as radiogenic growth. Note the difference between Equations l-47b and l-47c. In the former equation, the concentration of Nd at time t is expressed as a function of the initial Sm concentration. Hence, from the initial state, one can calculate how the Nd concentration would evolve. In the latter equation, the concentration of Nd at time t is expressed as a function of the Sm concentration also at time t. Let s now define time t as the present time. Then [ Nd] is related to the present amount of Sm, the age (time since Sm and Nd were fractionated), and the initial amount of Nd. Therefore, Equation l-47b represents forward calculation, and Equation l-47c represents an inverse problem to obtain either the age, or the initial concentration, or both. Equation l-47d assumes that there are no other ot-decay nuclides. However, U and Th are usually present in a rock or mineral, and their contribution to " He usually dominates and must be added to Equation l-47d. 

By contrast, the gas transfer estimates utilizing Rn measurements assumes steady state between Rn production from radioactive decay of nonvolatile Rd and gas transfer with the atmosphere. This assumption is possible because Rn has a half-life of only 3.8 days, so accumulation and lateral ocean fluxes of Rn is assumed to be minimal. Again, a potential problem is the active, versus inactive layer of the ocean in this case, the mixed layer depth that may change during an experiment. 

Experimental investigations of spectroscopic and other physical-chemical properties of actinides are severely hampered by their radioactive decay and radiation which lead to chemical modifications of the systems under study. The diversity of properties of lanthanide and actinide compounds is unique due to the multitude of their valency forms (which can vary over a wide range) and because of the particular importance of relativistic effects. They are, therefore, of great interest, both for fundamental research and for the development of new technologies and materials. The most important practical problems involve storage and processing of radioactive waste and nuclear fuel, as well as pollution of the environment by radioactive waste, where most of the decayed elements are actinides. 

Measurements of the diffusion of other elements at this time after fission is very difficult. Signal-to-noise ratios and radioactive decay problems make the study of "Mo and "Tc very difficult. Other isotopes present in the sample do not exhibit sufficient volatility to be studied... 

The reason why the boundaries in physical problems are often natural becomes obvious by looking at the simple example of radioactive decay in IV.6. The probability for an emission to take place is proportional to the number n of radioactive nuclei, and therefore automatically vanishes at n = 0. The same consideration applies when n is the number of molecules of a certain species in a chemical reaction, or the number of individuals in a population. Whenever by its nature n cannot be negative any reasonable master equation should have r(0) = 0. However, this does not exclude the possibility that something special happens at low n by which the analytic character of r(n) is broken, as in the example of diffusion-controlled reactions. A boundary that is not natural will be called artificial in section 7. 

A problem not mentioned in Chapter 15 is one that is very special for radioactive decay when the elapsed time given in the problem is insignificant in comparison with the half-life. Under such circumstances, Equation 26-2 is totally inappropriate, and the proper equation to use is Equation 26-1. In this case, consider —dN to be the number of atoms that disintegrate in a finite period of time df, which is negligible compared to q consider also that A remains constant during this same period of time. The following problem shows this application of Equation 26-1. 

This is the same model process that we described above for radioactive decay of if we substitute decay constants by rate constants, and amount of substance by concentration, and assume that [A]0 = 1 mol dm-3, we can adapt equation (7.40) derived in Problem 7.5(c) to describe how [B] varies with time ... 

Nuclear chemistry consists of a four-pronged endeavor made up of (a) studies of the chemical and physical properties of the heaviest elements where detection of radioactive decay is an essential part of the work, (b) studies of nuclear properties such as structure, reactions, and radioactive decay by people trained as chemists, (c) studies of macroscopic phenomena (such as geochronology or astrophysics) where nuclear processes are intimately involved, and (d) the application of measurement techniques based upon nuclear phenomena (such as nuclear medicine, activation analysis or radiotracers) to study scientific problems in a variety of fields. The principal activity or mainstream of nuclear chemistry involves those activities listed under part (b). 

Sample Problem 1.2 Because of the conservation of the number of nucleons in the nucleus and conservation of charge during radioactive decay (Table 1.1),... 

A promising approach to this problem has been the use of lead isotope ratios to characterize sources. Chapter 9 by Gale and Stos-Gale is an example of this type of study. The isotopic ratios of lead are variable because some of the isotopes are the daughters from the radioactive decay of uranium and thorium (4), Even though the amount of lead in bronze artifacts is small, Gale has been able to distinguish between sources of the ore on the basis of the ratios of the various lead isotopes. The sources of silver, lead, and copper in the Bronze Age Mediterranean are discussed. 

That brings up the second thing, which is that there are acceptable levels of radiation. We are constantly bombarded with naturally occurring radiation from outer space and natural elements in the Earth. YouVe been bombarded with particles from radioactive decay since you were born. Of course, even naturally occurring radiation can be harmful. In my part of the country, it s wise to check the levels of radon underneath your home because its radioactive particles can cause health problems. 

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