Physics
Effective Half Life
The effective half-life is a concept used to describe the rate at which a quantity of a radioactive substance decays. It takes into account both the physical decay of the substance and its removal from the system, such as through excretion or metabolism. This concept is important in understanding the behavior and potential hazards of radioactive materials.
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5 Key excerpts on "Effective Half Life"
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
If the number of nuclei present at t 0 = is N 0 , then N N e t 0 = λ − (31.4.2) where e is the exponential function. The half-life T 1 2 / of a particular radioactive isotope is the time required for half of a large collection of these nuclei to decay. The half-life T 1 2 / can be written in terms of the decay constant λ as T ln 2 1 2 / λ = (31.4.3) Carbon dating is a method for establishing the age of archaeological artifacts of bio- logical origin (e.g., bones, wood, or cloth). The technique relies on the isotope C 6 14 (C-14), which has a half-life of 5715 years and decays into N 7 14 by emitting a β − particle. For prob- lems in this text, assume that the activity of C-14 in one gram of carbon in living tissue is 0.251 Bq. If we let A represent activity, then we have A A e t 0 = λ − (31.4.4) where A 0 is the activity at t 0 = . Summary | 877 31.5 The Biological Effects of Radiation Ionizing radiation can be harmful to biological life because it can alter the structure of mol- ecules, which can cause damage or death to living cells. The absorbed dose is the amount of energy per unit mass of absorbing tissue, irrespective of the type of radiation: Absorbed dose Absorbed energy Mass of radiated tissue = (31.5.1) The SI unit of absorbed dose is the gray (Gy), where 1 Gy 1 J kg / = . A popular related unit is the rad, which is equal to 0.01 gray: 1 rad 0.01 Gy = We define the relative biological effectiveness (RBE) as the biological effect of a given dose of 200-keV X-rays divided by the dose of a particular type of radiation that would produce the same effect: RBE The dose of 200-keV X-rays that produces a certain biological effect The dose of a particular type of radiation that produces the same biological effect = (31.5.2) If absorbed dose is measured in grays (Gy), then the biologically equivalent dose is given in units of sieverts (Sv), named after Swedish medical physicist Rolf Sievert (1896–1966).
eBook - PDF- James Cameron(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
If the fraction of the initial amount at time t is designated as F, then F = e~ kt (Eq. 14.5) When F is exactly one-half, (14.5) may be reduced to t = 0.693/k (Eq. 14.6) and the value of t such that F equals one-half is called the half-life for the decay process. If the half-life for a particular isotope is given, (14.5) and (14.6) may be used to calculate the fraction remaining at any given time. For example, 22 Na has a half-life of 2.58 years, so the value of k from (14.6) is 0.269 and the fraction remaining after 4 years would be £-4 *(°· 269 ) or 0.341 of the initial amount. The graph in Fig. 14.10 may also be used to find the fractional amount of radioisotope remaining after any given number of half-lives. The half-lives of some radioisotopes commonly used in physiology are given in Table 14.1, along with information on their decay mode. 14.4. UNITS OF RADIATION In order to have a convenient means of describing quantities of radiation, a variety of different units have been defined. Unhappily, these units are a source of great confusion, and the situation has not been aided by the constant tampering with units undertaken by the international bodies that consecrate unit conventions (such as the SI system). The various units must be carefully divided into those describing radioactivity, those relating to exposure to radioactivity, and those describing the dose of radioactivity received by a body. The earlier unit of radioactivity was the Curie (Ci), defined as that amount of a radioisotope that yielded 3.7 x 10 10 disintegrations per second (dps). This is the number (within 0.5%) of events occurring in 1 g of radium. The 14.4. UNITS OF RADIATION Ρτ= V V 1_J 6 8 Half-Lives 10 12 14 Fig. 14.10 A plot of the radioactivity remaining, /(i)//(0), as a function of the number of half-lives elapsed. SI unit for describing radioactivity is the Becquerel, named for Henri Becquerel, who discovered radiation in 1896.
eBook - PDFBasic Physics Of Radiotracers
Volume II
- Earl W. Barnes(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
Of course the activity, which we will denote I(t), is simply the rate of decrease in the number of radioactive atoms, -dN (t)/dt, which is equal to AN(t). If we denote the initial activity by 1(0), then it follows directly from Equation 4 that ( 7 ) (5) ( 6 ) so that, as one would intuitively expect, the activity follows the same exponential decay behavior as the number of radioactive atoms. 2. The Half-Life T 1/2 and the Mean Life r The half-life T, /2 (or, more properly, half-period ) is the time interval over which the chance of survival of a particular radioactive atom is exactly one half. Then, if A is the decay constant, Equation 3 yields 109 For a large initial number N(0) of radioactive atoms, with initial activity N(0)A, the average value of the activity one half-life later, N (T 1/2)A, is N(0)A/2, or one half the initial activity. The actual life of any particular radioactive atom may range between zero and infin ity. The average lifetime of a large number of similar atoms is, however, a definite and useful quantity. For N(0) radioactive atoms present at time zero, we have seen that the number remaining undecayed at the subsequent time t is N(t) = N(0)e_At. All of these remaining have a lifetime longer than t. Those which decay within the short time interval dt following t can be considered to have a lifetime t, and these will be of number N(t)Adt = N(0)Ae-Ar. The total lifetime of all the atoms is obtained by integrat ing the product of the lifetime t with the number having this lifetime over all values of t from 0 to 00, and the average lifetime, which is called the mean life r , is simply this total lifetime divided by the initial number N(0). The partial activity of a sample of N(0) nuclei, if measured by a method susceptible to one particular mode of decay characterized by A,, is and the total activity as a function of time is ( 13 ) ( 8 ) Thus the mean life t is simply the reciprocal of the decay constant A.
- Frederick Bettelheim, William Brown, Mary Campbell, Shawn Farrell(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
The time it takes for one-half of any sample of radioactive ma-terial to decay is called the half-life , t 1/2 . It does not matter how big or small a sample is. For example, in the case of our 40 g of strontium-90, 20 g will be left at the end of 28.1 years (the rest has been converted to yttrium-90). It will then take another 28.1 years for half of the remainder to decay, so that after 56.2 years, we will have 10 g of 9.4 Nuclear Half-Life | 273 Copyright 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. strontium-90. If we wait for a third span of 28.1 years, then 5 g will be left. If we had begun with 100 g, then 50 g would be left after the first 28.1-year period. Figure 9.4 shows the radioactive decay curve of iodine-131. Inspection of this graph shows that at the end of 8 days, half of the original has dis-appeared. Thus, the half-life of iodine-131 is 8 days. It would take a total of 16 days, or two half-lives, for three-fourths of the original amount of iodine-131 to decay. EXAMPLE 9.5 Nuclear Half-Life If 10.0 mg of 131 53 I is administered to a patient, how much is left in the body after 32 days? STRATEGY AND SOLUTION We know from Figure 9.4 that t 1/2 of iodine-131 is eight days. The time span of 32 days corresponds to four half-lives. If we start with 10.0 mg, 5.00 mg remains after one half-life, 2.50 mg after two half-lives, 1.25 mg after three half-lives and 0.625 mg after four half-lives. 10.0 mg H11003 H11005 0.625 mg H11003 32 days (4 half-lives) 1 2 H11003 1 2 H11003 1 2 1 2 ■ QUICK CHECK 9.5 Barium-122 has a half-life of 2 minutes.
eBook - PDFChemistry for Today
General, Organic, and Biochemistry
- Spencer Seager, Michael Slabaugh, Maren Hansen, , Spencer Seager, Spencer Seager, Michael Slabaugh, Maren Hansen(Authors)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
The half-life of an isotope is used to indicate stability, and it is equal to the time required for one-half (50%) of the at- oms of a sample of the isotope to decay. Table 10.2 contains examples showing the wide range of half-lives that have been determined. half-life The time required for one- half the unstable nuclei in a sample to undergo radioactive decay. TABLE 10.2 Examples of Half-Lives Isotope Half-Life Source 238 92 U 4.5 3 10 9 years Naturally occurring 40 19 K 1.3 3 10 9 years Naturally occurring 226 88 Ra 1600 years Naturally occurring 14 6 C 5600 years Naturally occurring 239 94 Pu 24,000 years Synthetically produced 90 38 Sr 28 years Synthetically produced 131 53 I 8 days Synthetically produced 24 11 Na 15 hours Synthetically produced 15 8 O 2 minutes Synthetically produced 5 3 Li 10 221 seconds Synthetically produced Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 310 Chapter 10 Example 10.4 Calculating Half-Lives Rubidium-84 is used to monitor cardiac output. Another isotope, rubidium-79, decays by positron emission and forms krypton-79, which is a gas. A weighed 100.00 mg sample of solid rubidium-79 was allowed to decay for 42 minutes, then weighed again. Its mass was 25.00 mg. What is the half-life of rubidium-79? Solution We assume all the gaseous krypton-79 that was formed escaped into the surrounding air and so was not weighed. In the first half-life, one-half of the original 100.00 mg sample of rubidium would have been lost, so the sample would have a mass of 50.00 mg.
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