Physics
Half Life
In physics, "half-life" refers to the time it takes for half of a substance to undergo decay or transformation. It is commonly used in the context of radioactive decay, where it represents the time required for half of the radioactive atoms in a sample to decay. The concept is fundamental in understanding the behavior of radioactive materials and is widely applied in various scientific fields.
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11 Key excerpts on "Half Life"
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
31.5 Half-Life and Activity • Half-life t 1 / 2 is the time in which there is a 50% chance that a nucleus will decay. The number of nuclei N as a function of time is N = N 0 e −λt , where N 0 is the number present at t = 0 , and λ is the decay constant, related to the half-life by λ = 0.693 t 1 / 2 . • One of the applications of radioactive decay is radioactive dating, in which the age of a material is determined by the amount of radioactive decay that occurs. The rate of decay is called the activity R : R = ΔN Δt . • The SI unit for R is the becquerel (Bq), defined by 1 Bq = 1 decay/s. • R is also expressed in terms of curies (Ci), where 1 Ci = 3.70×10 10 Bq. • The activity R of a source is related to N and t 1 / 2 by R = 0.693N t 1 / 2 . • Since N has an exponential behavior as in the equation N = N 0 e −λt , the activity also has an exponential behavior, given by R = R 0 e −λt , where R 0 is the activity at t = 0 . 31.6 Binding Energy Chapter 31 | Radioactivity and Nuclear Physics 1269 • The binding energy (BE) of a nucleus is the energy needed to separate it into individual protons and neutrons. In terms of atomic masses, BE = {[Zm( 1 H) + Nm n ] − m( A X)}c 2 , where m ⎛ ⎝ 1 H ⎞ ⎠ is the mass of a hydrogen atom, m ⎛ ⎝ A X ⎞ ⎠ is the atomic mass of the nuclide, and m n is the mass of a neutron. Patterns in the binding energy per nucleon, BE / A , reveal details of the nuclear force. The larger the BE / A , the more stable the nucleus. 31.7 Tunneling • Tunneling is a quantum mechanical process of potential energy barrier penetration. The concept was first applied to explain α decay, but tunneling is found to occur in other quantum mechanical systems. Conceptual Questions 31.1 Nuclear Radioactivity 1. Suppose the range for 5.0 MeVα ray is known to be 2.0 mm in a certain material.
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.
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 - ePubNuclear Engineering Fundamentals
A Practical Perspective
- Robert E. Masterson(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
trillions of years and are, for all purposes, completely stable.Example Problem 6.5A sample of radioactive material is found to emit 3.6 × 1012 particles per hour. What is the activity of this sample in becquerels and curies?Solution There are 3600 seconds in one minute, so the number of particles emitted per second is R(t) = 3.6 × 1012 /3.6 × 102 = 1 × 1010 particles per second. This is the activity of the sample in becquerels. Since 1 Bq is the same as one particle emitted per second, the activity of the sample in curies is 1 × 1010 /3.7 × 1010 = 0.27 Ci. [Ans.]6.29 Half-Lives, Average Lives, Decay Constants, and Doubling Times
As we discussed in a previous section, the half-life of a radioactive isotope is the time required for exactly half the atoms of the isotope to decay into something else. That is, if No is the number of the atoms of the isotope at time t = 0, then the half-life T1/2 is defined by the relationshipN( t )=N o2− t/T1 / 2( 6.89 )so that when t = T1/2N(=T)1 / 2N o2=− 1N o2( 6.90 )This relationship is based on writing the number of atoms present at any time t as a power of 2. It also defines the doubling time td as the time required for the concentration of a material (such as an isotope) to double so that whenor after some cancellation of terms,=2 NoN o2R t( 6.91 )
eBook - PDFRadioactivity Measurements
Principles and Practice
- W. B. Mann, A. Rytz, A. Spernol, W. B. Mann, A. Rytz, A. Spernol(Authors)
- 2012(Publication Date)
- Pergamon(Publisher)
Mea n tropica l year , 1 a = 365.2422 days . Ã 1 / 2 = 1ç2/Á, and, finally, N t = N 0 exp( - rin2/7 1/2 ) = N 0 exp( - 0.69315ß/Ã 1 / 2 ) (2-6) The observed half lives of radionuclides range over more than 40 orders of magnitude, from just measurable parts of a femtosecond (about 10~ 1 6 s) up to about 10 20 years. The radionuclides mostly in use have half lives of hours, days, years and up to thousands of years. It is interesting to remember that after 10 half lives the number of radionuclide atoms is reduced by a factor of 2 1 0 = 1024, i.e. about one thousandfold. In health physics and nuclear medicine the biolo-gical Half Life, T b , is of great importance. It is the time required for a biological system to excrete half the amount of any nuclide introduced into it. In health physics or nuclear medicine this is clearly relevant to any injected or ingested radionuclide in the human body and the resulting dose to various organs in that body. If the decay is not negligible and T efi is the resulting, effective, hal f life of resi -denc e of the nuclide, and T m is that of the injected or ingested nuclide, then IT en =IT b +IT m , (2-7) or, in terms of probabilities, Kft = A b +A. 2.3.6.1. Measurement of decay constant or Half Life As radioactivity is a stochastic phenomenon, Eq. 2-4 is merely a statement that activity is related to the numbers of atoms of a radionuclide by the probability A, from which it follows that the Half Life, as defined, is related to A by In 2. A very common method of measuring Half Life has been simply to follow the activity of a stable radioac-tive source of a nuclide of interest as a function of time, using a stable detector of one or more of the source's radiations in a constant or reproducible geometry (see 1(5.2). As the response of the detector is proportional to activity, the Half Life will be equal to the slope of a semi-logarithmic plot of detector response as a function of time.
- 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.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
4. The half-life T 1/2 and the mean life τ are measures of how quickly radioactive nuclei decay and are related by T 1/2 = ln 2 ____ λ = τ ln 2. After reading this module, you should be able to . . . 42.3.1 Explain what is meant by radioactive decay and identify that it is a random process. 42.3.2 Identify disintegration constant (or decay constant) λ. 42.3.3 Identify that, at any given instant, the rate dN/dt at which radioactive nuclei decay is proportional to the number N of them still present then. 42.3.4 Apply the relationship that gives the number N of radioactive nuclei as a function of time. 42.3.5 Apply the relationship that gives the decay rate R of radioactive nuclei as a function of time. 42.3.6 For any given time, apply the relationship between the decay rate R and the remaining number N of radioactive nuclei. 42.3.7 Identify activity. 42.3.8 Distinguish becquerel (Bq), curie (Ci), and counts per unit time. 42.3.9 Distinguish half-life T 1/2 and mean life τ. 42.3.10 Apply the relationship between half-life T 1/2 , mean life τ, and disintegration constant λ. 42.3.11 Identify that in any nuclear process, including radioactive decay, the charge and the number of nucleons are conserved. LEARNING OBJECTIVES There is absolutely no way to predict whether any given nucleus in a radioactive sample will be among the small number of nuclei that decay during any given second. All have the same chance. Radioactive Decay As Fig. 42.2.1 shows, most nuclides are radioactive. They each spontaneously (randomly) emit a particle and transform into a different nuclide. Thus these decays reveal that the laws for subatomic processes are statistical. For example, in a 1 mg sample of uranium metal, with 2.5 × 10 18 atoms of the very long-lived radionuclide 238 U, only about 12 of the nuclei will decay in a given second by emitting an alpha particle and transforming into a nucleus of 234 Th. However,
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.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
One measure is the half-life T 1/2 of a radionuclide, which is the time at which both N and R have been reduced to one-half their initial values. The other measure is the mean (or average) life τ, which is the time at which both N and R have been reduced to e –1 of their initial values. To relate T 1/2 to the disintegration constant λ, we put R = 1 2 R 0 in Eq. 42-16 and substitute T 1/2 for t. We obtain 1 2 R 0 = R 0 e −λT 1/2 . Taking the natural logarithm of both sides and solving for T 1/2 , we find T 1/2 = ln 2 λ . Similarly, to relate τ to λ, we put R = e –1 R 0 in Eq. 42-16, substitute τ for t, and 1287 42-3 RADIOACTIVE DECAY solve for τ, finding τ = 1 λ . We summarize these results with the following: T 1/2 = ln 2 λ = τ ln 2. (42-18) 1288 CHAPTER 42 NUCLEAR PHYSICS Checkpoint 2 The nuclide 131 I is radioactive, with a half-life of 8.04 days. At noon on January 1, the activity of a certain sample is 600 Bq. Using the concept of half-life, without written cal- culation, determine whether the activity at noon on January 24 will be a little less than 200 Bq, a little more than 200 Bq, a little less than 75 Bq, or a little more than 75 Bq. a function of t. Thus, if we plot ln R (instead of R) versus t, we should get a straight line. Further, the slope of the line should be equal to –λ. Figure 42-9 shows a plot of ln R versus time t for the given measurements. The slope of the straight line that fits through the plotted points is slope = 0 − 6.2 225 min − 0 = −0.0276 min −1 . Thus, –λ = –0.0276 min –1 or λ = 0.0276 min –1 ≈ 1.7 h –1 . (Answer) The time for the decay rate R to decrease by 1/2 is related to the disintegration constant λ via Eq. 42-18 (T 1/2 = (ln 2)/λ). From that equation, we find T 1/2 = ln 2 λ = ln 2 0.0276 min −1 ≈ 25 min.
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
An older unit, the curie, is still in common use: 1 curie = 1 Ci = 3.7 × 10 10 Bq. Often a radioactive sample will be placed near a detector that does not record all the disintegrations that occur in the sample. The reading of the detector under these circumstances is proportional to (and smaller than) the true activity of the sample. Such proportional activity measurements are reported not in becquerel units but simply in counts per unit time. Lifetimes There are two common time measures of how long any given type of radionuclides lasts. One measure is the half-life T 1/2 of a radionuclide, which is the time at which both N and R have been reduced to one-half their initial values. The other measure is the mean (or average) life , which is the time at which both N and R have been reduced to e –1 of their initial values. To relate T 1/2 to the disintegration constant , we put R 1 2 R 0 in equation 42.14 and substitute T 1/2 for t. We obtain 1 2 R 0 = R 0 e -T 1∕2 . CHAPTER 42 Nuclear physics 1051 Taking the natural logarithm of both sides and solving for T 1/2 , we find T 1∕2 = ln 2 . Similarly, to relate to , we put R = e –1 R 0 in equation 42.14, substitute for t, and solve for , finding = 1 . We summarise these results with the following: T 1∕2 = ln 2 = ln 2. (42.16) EXAMPLE 42.4 SAMPLE PROBLEM Finding the disintegration constant and half-life from a graph The table that follows shows some measurements of the decay rate of a sample of 128 I, a radionuclide often used medically as a tracer to measure the rate at which iodine is absorbed by the thyroid gland. Time (min) R (counts/s) Time (min) R (counts/s) 4 392.2 132 10.9 36 161.4 164 4.56 68 65.5 196 1.86 100 26.8 218 1.00 Find the disintegration constant and the half-life T 1/2 for this radionuclide. KEY IDEAS The disintegration constant determines the exponential rate at which the decay rate R decreases with time t (as indi- cated by equation 42.14, R = R 0 e –t ).
eBook - PDF- James Shipman, Jerry Wilson, Charles Higgins, Bo Lou, James Shipman(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 278 Chapter 10 ● Nuclear Physics Identify the radioactive nucleus in each pair and state your reasoning. (a) 208 82 Pb and 222 86 Rn (b) 10 10 N and 20 10 Ne (c) 63 29 Cu and 64 29 Cu Thinking It Through Test each nucleus with the three criteria to determine if radioactive. Solution (a) 222 86 Rn (Z above 83) (b) 19 10 Ne (fewer n than p) (c) 64 29 Cu (odd–odd) Confidence Exercise 10.3 Predict which two of the following nuclei are radioactive. 232 90 Th 24 12 M 40 19 K 31 15 P The answers to Confidence Exercises may be found at the back of the book. E X A M P L E 1 0 . 3 Identifying Radioactive Isotopes Half-Life Some samples of radioisotopes take a long time to decay; others decay very rapidly. In a sample of a given isotope, the decay of an individual nucleus is a random event. It is impossible to predict which nucleus will be the next to undergo a nuclear change. However, given a large number of nuclei, it is possible to predict how many will decay in a certain length of time. The rate of decay of a given radioisotope is described by the term half-life, the time it takes for half of the nuclei of a given radioactive sample to decay. In other words, after one half-life has gone by, one-half of the original amount of isotope remains undecayed; after two half-lives, ( 1 2 3 1 2 ) 5 one-fourth ( 1 4 ) of the original amount is undecayed; and so on (● Fig. 10.8).* To determine the half-life of a radioisotope, the activity (the rate of emission of decay particles) is monitored. The activity is commonly measured in counts per minute (cpm).
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