Rate constants radioactive decay

You know that radioactive decay is first-order, so it is necessary only to find the rate constant for decay, from which the half-life may be calculated by means of Equation 15-11. To make the first-order plot, first convert cpm to log cpm to get... 

Decay Constant. A constant A that relates the instant rate of radioactive decay of a radioactive species to the number of atoms N present at a given time r. 

Ri rate of radioactive decay of nuclei/ the nominally constant ... 

The probability that a radioactive nucleus will decay in a given time is a constant, independent of temperature, pressure, or the decay of other neighboring nuclei. The disintegrations of individual nuclei are statistically independent events and are subject to random fluctuations. In a large number of nuclei, however, the fluctuations average out, and the fraction that decays in unit time is a constant and is numerically equal to the probability that a single nuclei will decay in that time. This rate of radioactive decay is known as the decay constant X, with dimensions of reciprocal time. 

It is customary to describe the specific rate of radioactive decay by the half-life which is the length of time required for half of the nuclei originally present to decay. The relation between the half-life and the decay constant is found from... 

Aii radioactive decay processes foiiow first-order kinetics. What does this mean What happens to the rate of radioactive decay as the number of nuciides is haived Write the first-order rate law and the integrated first-order rate law. Define the terms in each equation. What is the half-life equation for radioactive decay processes How does the half-life depend on how many nuclides are present Are the half-life and rate constant k directly related or inversely related ... 

A radioactive nucleus which emits a particle to become transformed to another nucleus is described as decaying to that nucleus. Such a radioactive event is called radioactive decay. Radionuclides decay at different rates. Some can decay in millionths of a second, others take millions of years. Decay is independent of all the variables which affect chemical reactions such as temperature, pressure, and concentration. This poses particular difficulty with regard to the disposal of nuclear wastes. The rate of radioactive decay is characterized by the loss of a constant percent per unit time, not a constant number of moles per unit time. We therefore characterize the decay rate by specifying the time required for 50 percent of the original material to decay. This period of time is called the half-life, given the symbol, tj/j- The constant percent change means that 50 percent will be lost during the first half-life, 50 percent of what is left after the first half-life will decay over the second half-life, etc. 

The rate of radioactive decay is typically expressed in terms of either the radioactive half-life or the radioactive decay constant. They are related as follows ... 

One of the most important properties of a radioactive nuclide is its lifetime. At present it is not possible to predict theoretically when any particular nucleus in a sample will decay. However, the number of nuclides in a sizeable sample that will decompose in a given time can be measured, and it is found that this rate of decay is characteristic of a given isotope. In fact, the rate of decay of an isotope is constant and unvarying. That is, if a fraction of a radioactive nuclide decays in a certain time interval f, then the same fraction of the remainder will decay in another increment of time f, irrespective of external conditions. Nuclear reactions are not affected by outside influences such as temperature and pressure and it is not possible to significantly alter the constant rate of radioactive decay. For example, radioactive strontium-90, an important... 

A measure of the rate of radioactive decay is the half-life t ii, the time it takes for half of its atoms to disintegrate. The half-Ufe can be related to the decay constant k by noting that after time t = fi/2, N is reduced to Nq. Therefore,... 

Isotopes are also used in the dating of ancient soils, plants, animals, and the tools of early peoples. The isotope of carbon which has a half-life of 5,730 years, can be used to calculate the age of old things. Since the rate of radioactive decay is constant, observing the decay rate allows the calculation of years that have past in relation to the half-life. 

Use the Radioactive Decay simulation (eChapter 21.4) to determine (a) the half-life of Th (b) the rate constant for decay of Th (c) the mass of Th remaining after 6.8 billion years. (Starting mass 20.00 kg)... 

The rate of radioactive decay of some unstable elements is supposed be constant and independent of chemical state and environmental factors such as temperature and... 

Fixing the dates of relics and stone implements or pieces of charcoal from ancient campsites is an application based on radioactive decay rates. Because the rate of radioactive decay of a nuclide is constant, this rate can serve as a clock for dating very old rocks and human implements. Dating wood and similar carbon-containing objects that are several thousand to fifty thousand years old can be done with radioactive carbon, carbon-14, which has a half-life of 5730 y. 

If the ratio of the rate of radioactive decay from the excavation site to the rate of decay from a fresh tree is 0.75, the archaeologists would use the equation for radioactive decay (above) to conclude that the tree that was used to build the house was chopped down about 2,400 years ago, giving the archaeologists an estimate of the age of the house, and perhaps a village of which the house was a part. (Ages are rounded off to allow for experimental error. A key assumption in the calculation is that the concentration of "C in the atmosphere has been constant since the tree died, which may or may not be strictly true.)... 

The phosphorus isotope is used in biochemical studies to determine the pathways of phosphorus atoms in living organisms. Its presence is detected through its emission of j8 particles, (a) What is the decay constant for expressed in the unit s (b) What is the activity of a 1.00 mg sample of (that is, how many atoms disintegrate per second) (c) Approximately what mass of P will remain in the original 1.00 mg sample after 57 days (See Table 25.1.) (d) What will be the rate of radioactive decay after 57 days ... 

Table 13.1 provides a list of several isotopes commonly used as tracers. The half-lives for these isotopes also are listed. What is the rate constant for the radioactive decay of each isotope ... 

Kinetic Equations 3-143 and 3-153 are obeyed by nucleides undergoing radioactive decay, where the rate constant kj is large and kj is small. The reactant A is converted rapidly into the intermediate B, which slowly forms C. Figure 3-13b shows plots of the exponentials g-kit and and of tlieu difference. Since kj is small, tlie exponential g-kit shows a slow decay while e d shows a rapid decline. The... 

From this expression, it is obvious that the rate is proportional to the concentration of A, and k is the proportionality constant, or rate constant, k has the units of (time) usually sec is a function of [A] to the first power, or, in the terminology of kinetics, v is first-order with respect to A. For an elementary reaction, the order for any reactant is given by its exponent in the rate equation. The number of molecules that must simultaneously interact is defined as the molecularity of the reaction. Thus, the simple elementary reaction of A P is a first-order reaction. Figure 14.4 portrays the course of a first-order reaction as a function of time. The rate of decay of a radioactive isotope, like or is a first-order reaction, as is an intramolecular rearrangement, such as A P. Both are unimolecular reactions (the molecularity equals 1). 

Since S/t has units of moles per volume per time and a has units of moles per volume, the rate constant for a first-order reaction has units of reciprocal time e.g., s. The best example of a truly first-order reaction is radioactive decay for example,... 

Radioactive decay provides splendid examples of first-order sequences of this type. The naturally occurring sequence beginning with and ending with ° Pb has 14 consecutive reactions that generate a or /I particles as by-products. The half-lives in Table 2.1—and the corresponding first-order rate constants, see Equation (1.27)—differ by 21 orders of magnitude. 

The normal or constant radioactivity possessed by thorium is an equilibrium value, where the rate of increase of radioactivity due to the production offresh active material is balanced by the rate of decay of radioactivity of that already formed (Rutherford and Soddy 1902). 

The decay constant, X, defines the probability that a particular atom will decay within a given time (X = In 2/t1/2). The half-life (t1/2) describes a time interval after which N = NJ2. The observed counting rate or activity (A) is equal to XN. Another way to describe radioactive decay is in terms of the mean life (t) of a... 

Many scientists thought that Earth must have formed as long as 3.3 billion years ago, but their evidence was confusing and inconsistent. They knew that some of the lead on Earth was primordial, i.e., it dated from the time the planet formed. But they also understood that some lead had formed later from the radioactive decay of uranium and thorium. Different isotopes of uranium decay at different rates into two distinctive forms or isotopes of lead lead-206 and lead-207. In addition, radioactive thorium decays into lead-208. Thus, far from being static, the isotopic composition of lead on Earth was dynamic and constantly changing, and the various proportions of lead isotopes over hundreds of millions of years in different regions of the planet were keys to dating Earth s past. A comparison of the ratio of various lead isotopes in Earth s crust today with the ratio of lead isotopes in meteorites formed at the same time as the solar system would establish Earth s age. Early twentieth century physicists had worked out the equation for the planet s age, but they could not solve it because they did not know the isotopic composition of Earth s primordial lead. Once that number was measured, it could be inserted into the equation and blip, as Patterson put it, out would come the age of the Earth. ... 

Radioactive decay is a stochastic process that occurs at random in a large number of atoms of an isotope (see Textbox 13). The exact time when any particular atom decayed or will decay can be neither established nor predicted. The average rate of decay of any radioactive isotope is, however, constant and predictable. It is usually expressed in terms of a half-life, the amount of time it takes for half of the atoms in a sample of a radioactive isotope to decay to a stable form. 

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