Decay and Half-life Help (page 2)
Decay and Half-Life
Radioactive substances gradually lose “potency” as time passes. Unstable nuclei degenerate one by one. Sometimes an unstable nucleus decays into a stable one in a single event. In other cases, an unstable nucleus changes into another unstable nucleus, which later degenerates into a stable one. Suppose that you have an extremely large number of radioactive nuclei, and you measure the length of time required for each one to degenerate and then average all the results. The average decay time is called the mean life and is symbolized by the lowercase Greek letter tau (τ).
Some radioactive materials give off more than one form of emission. For any given ionizing radiation form (alpha particles, beta particles, gamma rays, or other), there is a separate decay curve , or function of intensity versus time. A radioactive decay curve always has a characteristic shape: It starts out at a certain value and tapers down toward zero. Some decay curves decrease rapidly, and others decrease slowly, but the characteristic shape is always the same and can be defined in terms of a time span known as the half-life , symbolized t 1/2 .
Suppose that the intensity of radiation of a particular sort is measured at time t 0 . After a period of time t 1/2 has passed, the intensity of that form of radiation decreases to half the level it was at t 0 . After the half-life passes again (total elapsed time 2 t 1/2 ), the intensity goes down to one-quarter of its original value. After yet another half-life passes (total elapsed time 3 t 1/2 ), the intensity goes down to one-eighth its original value. In general, after n half-lives pass from the initial time t 0 (total elapsed time nt 1/2 ), the intensity goes down to 1/(2 n ), or 0.5 n , times its original value. If the original intensity is x 0 units and the final intensity is x f units, then
x f = 0.5 n x 0
The general form of a radioactive decay curve is shown in Fig. 18-9. The half-life t 1/2 can vary tremendously depending on the particular radioactive substance involved. Sometimes t 1/2 is a tiny fraction of 1 second; in other cases it is millions of years. For each type of radiation emitted by a material, there is a separate value of t 1/2 and therefore a separate decay curve.
Fig. 18-9 . General form of a radioactive decay curve.
Another way to define radioactive decay is in terms of a number called the decay constant , symbolized by the lowercase Greek letter lambda (λ). The decay constant is equal to the natural logarithm of 2 (approximately 0.69315) divided by the half-life in seconds. This is expressed as follows:
λ = 0.69315/ t 1/2
The symbol for the radioactive decay constant happens to be the same as the symbol for EM wavelength. Don’t confuse them; they are entirely different and independent quantities. Also, when determining the decay constant, be sure that t 1/2 is expressed in seconds. This will ensure that the decay constant is expressed in the proper units ( s −1 ). If you start with t 1/2 expressed in units other than seconds, you’ll get a decay constant that is the wrong number because it is expressed inappropriately.
The decay constant is the reciprocal of the mean life in seconds. Therefore, we can state these equations:
λ = 1/τ and τ = 1/λ
From these equations we can see that the mean life τ is related to the half-life t 1/2 as follows:
τ = t 1/2 /0.69315
= 1.4427 t 1/2
t 1/2 = 0.69315τ
Units And Effects
There are several different units employed to define overall radiation exposure. The unit of radiation in the International System of units is the becquerel (Bq), representing one nuclear transition per second (1 s −1 ). Exposure to radiation is measured according to the amount necessary to produce a coulomb of electric charge, in the form of ions, in a kilogram of pure dry air. The SI unit for this quantity is the coulomb per kilogram (C/kg). An older unit, known as the roentgen (R), is equivalent to 2.58 × 10 −4 C/kg.
When matter such as human tissue is exposed to radiation, the standard unit of dose equivalent is the sievert (Sv), equivalent to 1 joule per kilogram (1 J/kg). Sometimes you’ll hear about the rem (an acronym for roentgen equivalent man); 1 rem = 0.01 Sv.
All these units make it confusing to talk about radiation quantity. To make things worse, some of the older, technically obsolete units such as the roentgen and rem have stuck around, especially in laypeople’s conversations about radiation, whereas the standard units have been slow to gain acceptance. Have you read that “more than 100 roentgens of exposure to ionizing radiation within a few hours will make a person sick” or that “people are typically exposed to a few rems during a lifetime”? Statements like these were common in civil-defense documents in the 1960s after the Cuban missile crisis, when fears of worldwide nuclear war led to the installation of air-raid sirens and fallout shelters all over the United States.
When people are exposed to excessive amounts of radiation in a short time, physical symptoms such as nausea, skin burns, fatigue, and dehydration commonly occur. In extreme cases, internal ulceration and bleeding lead to death. When people get too much radiation gradually over a period of years, cancer rates increase, and genetic mutations also occur, giving rise to increased incidence of birth defects.
Radioactivity has numerous constructive applications in science, industry, and medicine. The most well known is the nuclear fission reactor, which was popular during the middle to late 1900s for generating electricity on a large scale. This type of power plant has fallen into disfavor because of the dangerous waste products it produces.
Radioactive isotopes are used in medicine to aid doctors in diagnosing illness, locating tumors inside the body, measuring rates of metabolism, and examining the structure of internal organs. Controlled doses of radiation are sometimes used in an attempt to destroy cancerous growths. In industry, radiation can be used to measure the dimensions of thin sheets of metal or plastic, to destroy bacteria and viruses that might contaminate food and other matter consumed or handled by people, and to x-ray airline baggage. Other applications include the irradiation of food, freight, and mail to protect the public against the danger of biological attack.
Geologists and biologists use radioactive dating to estimate the ages of fossil samples and archeological artifacts. The element most commonly used for this purpose is carbon. When a sample is created or a specimen is alive, there is believed to exist a certain proportion of 14 C atoms among the total carbon atoms. These gradually decay into 12 C atoms. By measuring the radiation intensity and determining the proportion of 14 C in archeological samples, anthropologists can get an idea of how long ago the world’s great civilizations came into being, thrived, and declined. Climatologists used the technique to discover that the Earth has gone through cycles of generalized global warming and cooling.
Carbon dating has revealed that the dinosaurs suddenly and almost completely disappeared in a short span of time around 65 million years ago. By a process of elimination, it was determined that a large meteorite or comet splashed down in the Gulf of Mexico at that time. The Earth’s climate cooled off for years because of debris injected into the atmosphere following the impact that blocked much of the solar IR that normally reaches the surface. Further research has shown that there have been several major bolide impacts in the distant past, each of which has radically altered the evolutionary course of life on Earth. Based on this knowledge, most scientists agree that it is only a matter of time before another such event takes place. When—not if—it does, the consequences for humanity will be of biblical proportions.
Suppose that the half-life of a certain radioactive substance is 100 days. You measure the radiation intensity and find it to be x 0 units. What will the intensity x 365 be after 365 days?
To determine this, use the equation presented earlier:
x 365 = 0.5 n x 0
where n is the number of half-lives elapsed. In this case, n = 365/100 = 3.65. Therefore:
x 365 = 0.5 3.65 x 0
To determine the value of 0.5 365 , use a calculator with an x y function. This yields the following result to three significant digits:
x 365 = 0.0797 x 0
What is the decay constant of the substance described in Problem 18-5?
Use the preceding formula for decay constant λ in terms of half-life t 1/2 . In this case, t 1/2 = 100 days. This must be converted to seconds to get the proper result for the decay constant. There are 24 × 60 × 60 = 8.64 × 10 4 s in one day. Thus t 1/2 = 8.64 × 10 6 s, and
λ = 0.69315/(8.64 × 10 6 )
= 8.02 × 10 −8 s −1
What is the mean life of the substance described in Problem 18-5? Express the answer in seconds and in days.
The mean life τ is the reciprocal of the decay constant. To obtain τ in seconds, divide the numbers in the preceding equation with the numerator and denominator interchanged:
τ= (8.64 × 10 6 )/0.69315
= 1.25 × 10 7
This is expressed in seconds. To express it in days, divide by 8.64 × 10 4 . This gives an answer of approximately 145 days.
Practice problems of these concepts can be found at: Forms of Radiation Quiz
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