Half-life, decay rates

Scientists frequently express Equation 3-4 in terms of half-life ( 1/2), which is the time required for one half of the population of unstable atoms to decay. In other words, after one half-life, the rate of particles emitted has decreased to one half the rate of the starting population of unstable atoms. Substitution in Equation 3-4 yields Equation 3-5 ... 

The amount of time required for one-half of the atoms of a radionuclide to transform is called its radioactive half-life. The rate of decay, and thus the half-life, for each radionuclide is unique. The half-life of U is very long, 4.5x10 years the half-lives of U and U are orders of magnitude lower,... 

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 ... 

In 1898, in Cambridge, England, a New Zealander, Ernest Rutherford, demonstrated that there were at least two different types of radiation with different penetrating power. He called these alpha and beta radiation. He subsequentiy worked at McGill University in Montreal, Canada, and found more radioactive elements different types of radium and thorium, and actinium. He proposed that these were links in chains of radioactive materials, called the transformation theory. Rutherford and his colleague, Frederic Soddy, described that the rate of decay of radioactive elements were characteristic of the element, and came to be known as half-life. Decay follows the law of probability. Over a given period of time, each atom has a certain probability of decaying, a process that results from the random movements of the subatomic components of the radioactive atoms. This was the first instance in physics of a truly unpredictable phenomenon. The decay of a radioactive atom was probabilistic. 

Knowledge Required (1) The knowledge that radioactive decay is a first-order process. (2) The integrated form of the first-order rate equation. (3) The relationship between half-life and rate constant... 

Thin layer activation coupons have been used to continuously monitor corrosion rates in continuous digesters, and to verify the effectiveness of anodic protection systems [180], The surface of a thin layer activation coupon is irradiated to a shallow depth and monitoring is performed with a Geiger counter 6x>m outside the digester wall. Subtracting for effects of half-life decay, the corrosion rate can be estimated from the decreased activity of the coupon. 

The half-life of a first-order reaction is inversely proportional to its rate constant, so a short half-life corresponds to a large rate constant. Consider, for example, two radioactive isotopes used in nuclear medicine Na txn = 14.7 h) and Co (tin 5.3 yr). Sodium-24, with the shorter half-life, decays faster. If we started with an equal number of moles of each isotope, most of the sodium-24 would be gone in a week whereas most of the cobalt-60 would remain unchanged. 

In a nuclear reactor, different isotopes are produced by fission at varying rates, and each of these isotopes has their own specific half-life decay periods. Thus, identifying the different atomic species present inside a nuclear reactor at a given time is very complicated. For instance, in a reactor of the starting mass of LP will decay into neptunium in 24 min. For a fixed mass, % of the starting mass of this same amount will have decayed into neptunium in 48 min. The additional complication comes from the continued and simultaneous production of IP via the neutron capture process. There are plenty of neutrons in an operating reactor, so additional IP is continually produced from the large number of IP present. 

Initiators. The degree of polymerization is controlled by the addition rate of initiator(s). Initiators (qv) are chosen primarily on the basis of half-life, the time required for one-half of the initiator to decay at a specified temperature. In general, initiators of longer half-Hves are chosen as the desired reaction temperature increases they must be well dispersed in the reactor prior to the time any substantial reaction takes place. When choosing an initiator, several factors must be considered. For the autoclave reactor, these factors include the time permitted for completion of reaction in each zone, how well the reactor is stirred, the desired reaction temperature, initiator solubiUty in the carrier, and the cost of initiator in terms of active oxygen content. For the tubular reactors, an additional factor to take into account is the position of the peak temperature along the length of the tube (9). 

Decay products of the principal radionuclides used in tracer technology (see Table 1) are not themselves radioactive. Therefore, the primary decomposition events of isotopes in molecules labeled with only one radionuclide / molecule result in unlabeled impurities at a rate proportional to the half-life of the isotope. Eor and H, impurities arising from the decay process are in relatively small amounts. Eor the shorter half-life isotopes the relative amounts of these impurities caused by primary decomposition are larger, but usually not problematic because they are not radioactive and do not interfere with the application of the tracer compounds. Eor multilabeled tritiated compounds the rate of accumulation of labeled impurities owing to tritium decay can be significant. This increases with the number of radioactive atoms per molecule. 

The same chemical separation research was done on thorium ores, leading to the discovery of a completely different set of radioactivities. Although the chemists made fundamental distinctions among the radioactivities based on chemical properties, it was often simpler to distinguish the radiation by the rate at which the radioactivity decayed. For uranium and thorium the level of radioactivity was independent of time. For most of the radioactivities separated from these elements, however, the activity showed an observable decrease with time and it was found that the rate of decrease was characteristic of each radioactive species. Each species had a unique half-life, ie, the time during which the activity was reduced to half of its initial value. 

Approximately 25—30% of a reactor s fuel is removed and replaced during plaimed refueling outages, which normally occur every 12 to 18 months. Spent fuel is highly radioactive because it contains by-products from nuclear fission created during reactor operation. A characteristic of these radioactive materials is that they gradually decay, losing their radioactive properties at a set rate. Each radioactive component has a different rate of decay known as its half-life, which is the time it takes for a material to lose half of its radioactivity. The radioactive components in spent nuclear fuel include cobalt-60 (5-yr half-Hfe), cesium-137 (30-yr half-Hfe), and plutonium-239 (24,400-yr half-Hfe). 

Half-lives can be interpreted in terms of the level of radiation of the corresponding isotopes. Uranium has a very long half-life (4.5 X 109 yr), so it gives off radiation very slowly. At the opposite extreme is fermium-258, which decays with a half-life of 3.8 X 10-4 s. You would expect the rate of decay to be quite high. Within a second virtually all the radiation from fermium-258 is gone. Species such as this produce very high radiation during their brief existence. 

Effective half-life Average neutron energy Neutron emission rate Decay heat... 

FIGURE 13.12 Thu ohange in concentration of the reactant in two first-order reactions plotted on the same graph When the first-order rate constant is large, the half-life of the reactant is short, because the exponential decay of the concentration of the reactant is then fast. 

Cl 13.39 Derive an expression for the half-life of the reactant A that decays by a third-order reaction with rate constant k. 

Suppose that a pollutant is entering the environment at a steady rate R and that, once there, its concentration decays by a first-order reaction. Derive an expression for (a) the concentration of the pollutant at equilibrium in terms of R and (b) the half-life of the pollutant species when R = 0... 

A radioactive sample contains 3.25 X 1018 atoms of a nuclide that decays at a rate of 3.4 X 1013 disintegrations per 15 min. (a) What percentage of the nuclide will have decayed after 150 d (b) How many atoms of the nuclide will remain in the sample (c) What is the half-life of the nuclide ... 

A radioactive isotope X with a half-life of 27.4 d decays into another radioactive isotope Y with a half-life of 18.7 d, which decays into the stable isotope Z. Set up and solve the rate laws for the amounts of the three nuclides as a function of time, and plot your results as a graph. 

If it is certain that the reaction is indeed an irreversible first-order reaction, one can also determine how long it takes before 50 % of the reactant has been converted into products, as for any exponential decay the half-life, ty, is related to the rate constant k as... 

Dark Decay of UDMH in Air, UDMH was observed to undergo a gradual dark decay in the 30,000-liter Teflon chamber at a rate which depended on humidity. Specifically, at 41 C and 4% RH the observed UDMH half-life was " 9 hours (initial UDMH 4.4 ppm) and at 40 C and 15% RH, the half-life was -6 hours (initial UDMH 2.5 ppm). The only observed product of the UDMH dark decay was NH3, which accounted for only -5-10% of the UDMH lost. In particular, no nitrosamine, nitramine, or hydrazone were observed. Formaldehyde dimethyIhydrazone was observed in previous studies which employed higher UDMH concentrations and reaction vessels with relatively high surface/volume ratios (, ) ... 

Dark Decay of UDMH in the Presence of NO, When 1.3 ppm of UDMH in air was reacted in the dark with an approximately equal amount of NO, 0.25 ppm of UDMH was consumed and formation of -0.16 ppm HONO and -0.07 ppm N2O was observed after -3 hours. Throughout the reaction, a broad infrared absorption at -988 cm" corresponding to an unidentified product(s), progressively grew in intensity. The residual infrared spectrum of the unknown product(s) is shown in Figure 2a. It is possible that a very small amount (50.03 ppm) of N-nitrosodimethylamine could also have been formed but the interference by the absorptions of the unknown product(s) made nitrosamine (as well as nitramine) detection difficult. No significant increase in NH3 levels was observed, in contrast to the UDMH dark decay in the absence of NO. Approximately 70% of the UDMH remained at the end of the 3-hour reaction period this corresponds to a half-life of -9 hours which is essentially the same decay rate as that observed in the absence of NO. 

A unique situation is encountered if Fe-M6ssbauer spectroscopy is applied for the study of spin-state transitions in iron complexes. The half-life of the excited state of the Fe nucleus involved in the Mossbauer experiment is tj/2 = 0.977 X 10 s which is related to the decay constant k by tj/2 = ln2/fe. The lifetime t = l//c is therefore = 1.410 x 10 s which value is just at the centre of the range estimated for the spin-state lifetime Tl = I/Zclh- Thus both the situations discussed above are expected to appear under suitable conditions in the Mossbauer spectra. The quantity of importance is here the nuclear Larmor precession frequency co . If the spin-state lifetime Tl = 1/feLH is long relative to the nuclear precession time l/co , i.e. Tl > l/o) , individual and sharp resonance lines for the two spin states are observed. On the other hand, if the spin-state lifetime is short and thus < l/o) , averaged spectra with intermediate values of quadrupole splitting A q and isomer shift 5 are found. For the intermediate case where Tl 1/cl , broadened and asymmetric resonance lines are obtained. These may be the subject of a lineshape analysis that will eventually produce values of rate constants for the dynamic spin-state inter-conversion process. The rate constants extracted from the spectra will be necessarily of the order of 10 -10 s"F... 

Measurement of specific activity. The half-life of a nuclide can be readily calculated if both the number of atoms and their rate of decay can be measured, i.e., if the activity A and the number of atoms of P can be measured, then X is known from A = XP. As instrumentation for both atom counting and decay counting has improved in recent decades, this approach has become the dominant method of assessing half-lives. Potential problems with this technique include the accurate and precise calibration of decay-counter efficiency and ensuring sufficient purity of the nuclide of interest. This technique provides the presently used half-lives for many nuclides, including those for the parents of the three decay chains, U, U (Jaffey et al. 1971), and Th. 

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). 

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