The basic law of radioactive decay of a radionuclide. Half-life of radioactive elements - what is it and how is it determined? Half-life formula
Laws of radioactive decay of nuclei
The ability of nuclei to spontaneously decay, emitting particles, is called radioactivity. Radioactive decay is a statistical process. Each radioactive nucleus can decay at any moment and the pattern is observed only on average; in the case of decay, it is enough large quantity cores.
Decay constantλ is the probability of nuclear decay per unit time.
If there are N radioactive nuclei in the sample at time t, then the number of nuclei dN that decayed during time dt is proportional to N.
dN = -λNdt. (13.1)
By integrating (1) we obtain the law of radioactive decay
N(t) = N 0 e -λt . (13.2)
N 0 is the number of radioactive nuclei at time t = 0.
Average life time τ –
. (13.3)
Half life T 1/2 - time during which the initial number of radioactive nuclei will decrease by half
T 1/2 = ln2/λ=0.693/λ = τln2. (13.4)
Activity A - average number of nuclei decaying per unit time
A(t) = λN(t). (13.5)
Activity is measured in curies (Ci) and becquerels (Bq)
1 Ki = 3.7*10 10 decays/s, 1 Bq = 1 decay/s.
The decay of the original nucleus 1 into nucleus 2, followed by its decay into nucleus 3, is described by a system of differential equations
(13.6)
where N 1 (t) and N 2 (t) are the number of nuclei, and λ 1 and λ 2 are the decay constants of nuclei 1 and 2, respectively. Solution of system (6) with initial conditions N 1 (0) = N 10 ; N 2 (0) = 0 will be
, (13.7a)
. (13.7b)
| Figure 13. 1 |
The number of cores reaches 2 maximum value at .
If λ 2< λ 1 (), суммарная активностьN 1 (t)λ 1 + N 2 (t)λ 2 будет монотонно уменьшаться.
If λ 2 >λ 1 ()), the total activity initially increases due to the accumulation of nuclei 2.
If λ 2 >> λ 1 , with sufficient big times the contribution of the second exponential in (7b) becomes negligible compared to the contribution of the first and the activities of the second A 2 = λ 2 N 2 and the first isotopes A 1 = λ 1 N 1 are almost equal. In the future, the activities of both the first and second isotopes will change over time in the same way.
A 1 (t) = N 10 λ 1 = N 1 (t)λ 1 = A 2 (t) = N 2 (t)λ 2 .(13.8)
That is, the so-called age-old balance, in which the number of isotope nuclei in the decay chain is related to the decay constants (half-lives) by a simple relationship.
. (13.9)
Therefore, in natural state all isotopes genetically related in radioactive series are usually found in certain quantitative ratios depending on their half-lives.
In the general case, when there is a chain of decays 1→2→...n, the process is described by a system of differential equations
dN i /dt = -λ i N i +λ i-1 N i-1 .(13.10)
The solution to system (10) for activities with initial conditions N 1 (0) = N 10 ; N i (0) = 0 will be
(13.12)
The prime means that in the product that is in the denominator, the factor with i = m is omitted.
Isotopes
ISOTOPES-varieties of the same chemical element, similar in their physical properties chemical properties, but having different atomic masses. The name “isotopes” was proposed in 1912 by the English radiochemist Frederick Soddy, who formed it from two Greek words: isos - identical and topos - place. Isotopes occupy the same place in the cell of Mendeleev's periodic table of elements.
An atom of any chemical element consists of a positively charged nucleus and a cloud of negatively charged electrons surrounding it ( cm.Also ATOM NUCLEUS). The position of a chemical element in the periodic table of Mendeleev (its atomic number) is determined by the charge of the nucleus of its atoms. Isotopes are therefore called varieties of the same chemical element, the atoms of which have the same nuclear charge (and, therefore, practically the same electronic shells), but differ in core mass values. According to the figurative expression of F. Soddy, the atoms of isotopes are the same “outside”, but different “inside”.
The neutron was discovered in 1932 – a particle that has no charge, with a mass close to the mass of the nucleus of a hydrogen atom - a proton , and a proton-neutron model of the nucleus was created. As a result, science established the final modern definition concepts of isotopes: isotopes are substances whose atomic nuclei consist of the same number protons and differ only in the number of neutrons in the nucleus . Each isotope is usually denoted by a set of symbols, where X is the symbol of the chemical element, Z is the charge of the atomic nucleus (number of protons), A is mass number isotope (total number of nucleons - protons and neutrons in the nucleus, A = Z + N). Since the charge of the nucleus appears to be uniquely associated with the symbol of the chemical element, simply the notation A X is often used for abbreviation.
Of all the isotopes known to us, only hydrogen isotopes have proper names. Thus, the isotopes 2 H and 3 H are called deuterium and tritium and are designated D and T, respectively (the isotope 1 H is sometimes called protium).
Occurs in nature as stable isotopes , and unstable - radioactive, the nuclei of atoms of which are subject to spontaneous transformation into other nuclei with the emission of various particles (or processes of so-called radioactive decay). About 270 stable isotopes are now known, and stable isotopes are found only in elements with atomic number Z Ј 83. The number of unstable isotopes exceeds 2000, the vast majority of them have been obtained artificially as a result of various nuclear reactions. The number of radioactive isotopes of many elements is very large and can exceed two dozen. The number of stable isotopes is significantly smaller. Some chemical elements consist of only one stable isotope (beryllium, fluorine, sodium, aluminum, phosphorus, manganese, gold and a number of other elements). Largest number stable isotopes - 10 were found in tin, in iron, for example, there are 4 of them, in mercury - 7.
Discovery of isotopes, historical background. In 1808, the English scientist naturalist John Dalton first introduced the definition of a chemical element as a substance consisting of atoms of the same type. In 1869, chemist D.I. was discovered by Mendeleev periodic law chemical elements. One of the difficulties in substantiating the concept of an element as a substance occupying a certain place in a cell of the periodic table was the experimentally observed non-integer atomic weights of elements. In 1866, the English physicist and chemist Sir William Crookes hypothesized that every natural chemical element represents a certain mixture of substances that are identical in their properties, but have different atomic masses, but at that time such an assumption did not yet have experimental confirmation and therefore went little noticed.
An important step towards the discovery of isotopes was the discovery of the phenomenon of radioactivity and the hypothesis of radioactive decay formulated by Ernst Rutherford and Frederick Soddy: radioactivity is nothing more than the decay of an atom into a charged particle and an atom of another element, different in its chemical properties from the original one. As a result, the idea of radioactive series or radioactive families arose , at the beginning of which there is the first parent element, which is radioactive, and at the end - the last stable element. Analysis of the chains of transformations showed that during their course, the same radioactive elements, differing only in atomic masses, can appear in one cell of the periodic system. In fact, this meant the introduction of the concept of isotopes.
Independent confirmation of the existence of stable isotopes of chemical elements was then obtained in the experiments of J. J. Thomson and Aston in 1912–1920 with beams of positively charged particles (or so-called channel beams ) emanating from the discharge tube.
In 1919, Aston designed an instrument called a mass spectrograph (or mass spectrometer). . A discharge tube was still used as an ion source, but Aston found a method in which the sequential deflection of a beam of particles in electrical and magnetic fields led to the focusing of particles with the same value the ratio of charge to mass (regardless of their speed) at the same point on the screen. Along with Aston, a mass spectrometer of a slightly different design was created in the same years by the American Dempster. As a result of the subsequent use and improvement of mass spectrometers through the efforts of many researchers, by 1935 almost full table isotopic compositions of all chemical elements known at that time.
Methods for isotope separation. To study the properties of isotopes and especially for their use for scientific and applied purposes, it is necessary to obtain them in more or less noticeable quantities. In conventional mass spectrometers, almost complete separation of isotopes is achieved, but their quantity is negligible. Therefore, the efforts of scientists and engineers were aimed at searching for other possible methods isotope separation. First of all, physicochemical methods of separation were mastered, based on differences in such properties of isotopes of the same element as evaporation rates, equilibrium constants, chemical reactions etc. The most effective among them were the methods of rectification and isotope exchange, which found wide application in the industrial production of isotopes of light elements: hydrogen, lithium, boron, carbon, oxygen and nitrogen.
Another group of methods consists of the so-called molecular kinetic methods: gas diffusion, thermal diffusion, mass diffusion (diffusion in a vapor flow), centrifugation. Gas diffusion methods, based on different rates of diffusion of isotopic components in highly dispersed porous media, were used during the Second World War to organize industrial production separation of uranium isotopes in the USA as part of the so-called Manhattan Project to create atomic bomb. To obtain the required quantities of uranium enriched to 90% with the light isotope 235 U, the main “combustible” component of the atomic bomb, plants were built, occupying an area of about four thousand hectares. More than 2 billion dollars were allocated for the creation of an atomic center with plants for the production of enriched uranium. After the war, plants for the production of enriched uranium for military purposes, also based on the diffusion method of separation, were developed and built in the USSR. IN recent years this method gave way to the more efficient and less expensive method of centrifugation. In this method, the effect of separating an isotope mixture is achieved due to the different effects of centrifugal forces on the components of the isotope mixture filling the centrifuge rotor, which is a thin-walled cylinder limited at the top and bottom, rotating at a very high speed in a vacuum chamber. Hundreds of thousands of centrifuges connected in cascades, the rotor of each of which makes more than a thousand revolutions per second, are currently used in modern separation plants both in Russia and in other developed countries of the world. Centrifuges are used not only to obtain enriched uranium, which is necessary to ensure the operation of nuclear reactors nuclear power plants, but also for the production of isotopes of about thirty chemical elements in the middle part of the periodic table. To separate various isotopes, electromagnetic separation units with powerful ion sources are also used; in recent years, laser methods divisions.
Application of isotopes. Various isotopes of chemical elements are widely used in scientific research, V various areas industry and agriculture, nuclear energy, modern biology and medicine, environmental research and other areas. In scientific research (for example, chemical analysis) are usually required small quantities rare isotopes of various elements, calculated in grams and even milligrams per year. At the same time, for a number of isotopes widely used in nuclear energy, medicine and other industries, the need for their production can amount to many kilograms and even tons. Thus, in connection with the use of heavy water D 2 O in nuclear reactors its global production by the early 1990s of the last century was about 5000 tons per year. The hydrogen isotope deuterium, which is part of heavy water, the concentration of which in the natural mixture of hydrogen is only 0.015%, along with tritium, will become in the future, according to scientists, the main component of energy fuels. thermonuclear reactors, working on the basis of reactions nuclear fusion. In this case, the need for the production of hydrogen isotopes will be enormous.
In scientific research, stable and radioactive isotopes are widely used as isotopic indicators (tracers) in the study of the most various processes occurring in nature.
IN agriculture isotopes (“labeled” atoms) are used, for example, to study the processes of photosynthesis, the digestibility of fertilizers, and to determine the efficiency of plants’ use of nitrogen, phosphorus, potassium, trace elements, and other substances.
Isotope technologies are widely used in medicine. So in the USA, according to statistics, more than 36 thousand are carried out. medical procedures per day and about 100 million. laboratory tests using isotopes. The most common procedures associated with computed tomography. The carbon isotope C13, enriched to 99% (natural content about 1%), is actively used in the so-called " diagnostic control breathing." The essence of the test is very simple. The enriched isotope is introduced into the patient’s food and, after participating in the metabolic process in various organs of the body, is released as exhaled by the patient carbon dioxide CO 2 which is collected and analyzed using a spectrometer. The differences in the rates of processes associated with the release of different amounts of carbon dioxide, labeled with the C 13 isotope, make it possible to judge the condition of the patient’s various organs. In the US, the number of patients who will undergo this test is estimated at 5 million per year. Now laser separation methods are used to produce highly enriched C13 isotope on an industrial scale.
Related information.
>> The law of radioactive decay. Half life
§ 101 LAW OF RADIOACTIVE DECAY. HALF LIFE
Radioactive decay obeys a statistical law. Rutherford, studying the transformations of radioactive substances, established experimentally that their activity decreases over time. This was discussed in the previous paragraph. Thus, radon activity decreases by 2 times after 1 minute. The activity of elements such as uranium, thorium and radium also decreases with time, but much more slowly. For each radioactive substance there is a certain time interval during which the activity decreases by 2 times. This interval is called the half-life. Half-life T is the time it takes for half of the initial number radioactive atoms.
The decline in activity, i.e., the number of decays per second, depending on time for one of the radioactive drugs is shown in Figure 13.8. The half-life of this substance is 5 days.
Let us now derive mathematical form law of radioactive decay. Let the number of radioactive atoms at the initial moment of time (t= 0) be equal to N 0. Then, after the half-life, this number will be equal to
After another similar time interval, this number will become equal to:
§ 15-g. Law of Radioactive Decay
The advent of “manual” scintillation counters and, mainly, Geiger-Muller counters, which helped automate particle counts (see § 15), led physicists to an important conclusion. Any radioactive isotope is characterized by a spontaneous weakening of radioactivity, expressed in a decrease in the number of decaying nuclei per unit time.
Plotting graphs of the activity of various radioactive isotopes led scientists to the same dependence, expressed exponential function(see chart). The horizontal axis shows the observation time, and the vertical axis shows the number of undecayed nuclei. The curvature of the lines could be different, but the function itself, which expressed the dependencies described by the graphs, remained the same:
This formula expresses law of radioactive decay: the number of nuclei that have not decayed over time is determined as the product of the initial number of nuclei by 2 to the power equal to the ratio of the observation time to the half-life, taken with a negative sign.
As it turned out during the experiments, various radioactive substances can be characterized by different half-life – the time during which the number of still undecayed nuclei is halved(see table).
Half-lives of some isotopes of some chemical elements. Values are given for both natural and artificial isotopes.
| Iodine-129 | 15 Ma | Carbon-14 | 5.7 thousand years | |
| Iodine-131 | 8 days | Uran-235 | 0.7 billion years | |
| Iodine-135 | 7 o'clock | Uran-238 | 4.5 billion years |
Half-life is a generally accepted physical quantity characterizing the rate of radioactive decay. Numerous experiments show that even with a very long observation of a radioactive substance, its half-life is constant, that is, it does not depend on the number of atoms that have already decayed. Therefore, the law of radioactive decay has found application in the method of determining the age of archaeological and geological finds.
Radiocarbon dating method. Carbon is a very common chemical element on Earth, which includes the stable isotopes carbon-12, carbon-13 and the radioactive isotope carbon-14, which has a half-life of 5.7 thousand years (see table). Living organisms, consuming food, accumulate all three isotopes in their tissues. After the end of the life of the organism, the supply of carbon stops, and over time its content decreases naturally, due to radioactive decay. Since only carbon-14 decays, the ratio of carbon isotopes in the fossil remains of living organisms changes over centuries and millennia. By measuring this “carbon proportion,” we can judge the age of an archaeological find.
The method of radiocarbon analysis is applicable for geological rocks, as well as for fossil human objects, but provided that the ratio of isotopes in the sample has not been disturbed during its existence, for example, by a fire or the influence of a strong source of radiation. Non-accounting similar reasons immediately after the discovery of this method, it led to errors for several centuries and millennia. Today, “secular calibration scales” are used for the carbon-14 isotope, based on its distribution in long-lived trees (for example, the American millennial redwood). Their age can be calculated very accurately - by the annual rings of wood.
The limit of application of the radiocarbon dating method at the beginning of the 21st century was 60,000 years. To measure the age of older specimens, e.g. rocks or meteorites, use similar method, but instead of carbon, isotopes of uranium or other elements are observed, depending on the origin of the sample being studied.
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Lecture 2. The basic law of radioactive decay and the activity of radionuclides
The rate of decay of radionuclides is different - some decay faster, others slower. An indicator of the rate of radioactive decay is radioactive decay constant, λ [sec-1], which characterizes the probability of the decay of one atom in one second. For each radionuclide, the decay constant has its own value; the larger it is, the faster the nuclei of the substance decay.
The number of decays recorded in a radioactive sample per unit time is called activity (a ), or the radioactivity of the sample. The activity value is directly proportional to the number of atoms N radioactive substance:
a =λ· N , (3.2.1)
Where λ – radioactive decay constant, [sec-1].
Currently, according to the current International System of Units SI, the unit of measurement of radioactivity is becquerel [Bk]. This unit received its name in honor of the French scientist Henri Becquerel, who discovered the phenomenon of natural radioactivity of uranium in 1856. One becquerel equals one decay per second 1 Bk = 1 .
However, the non-system unit of activity is still often used – curie [Ki], introduced by the Curies as a measure of the decay rate of one gram of radium (in which ~3.7 1010 decays occur per second), therefore
1 Ki= 3.7·1010 Bk.
This unit is convenient for assessing activity large quantities radionuclides.
The decrease in radionuclide concentration over time as a result of decay obeys an exponential relationship:
, (3.2.2)
Where N t– the number of atoms of a radioactive element remaining after time t after the start of observation; N 0 – number of atoms at the initial moment of time ( t =0 ); λ – radioactive decay constant.
The described dependence is called basic law of radioactive decay .
The time it takes for half of the total number radionuclides is called half-life T½ . After one half-life, out of 100 radionuclide atoms, only 50 remain (Fig. 2.1). Over the next similar period, only 25 of these 50 atoms remain, and so on.
The relationship between half-life and decay constant is derived from the equation of the fundamental law of radioactive decay:
at t=T½ And
we get https://pandia.ru/text/80/150/images/image006_47.gif" width="67" height="41 src="> Þ ;
https://pandia.ru/text/80/150/images/image009_37.gif" width="76" height="21">;
i.e..gif" width="81" height="41 src=">.
Therefore, the law of radioactive decay can be written as follows:
https://pandia.ru/text/80/150/images/image013_21.gif" width="89" height="39 src=">, (3.2.4)
Where at – drug activity over time t ; a0 – activity of the drug at the initial moment of observation.
It is often necessary to determine the activity of a given amount of any radioactive substance.
Remember that the unit of quantity of a substance is the mole. A mole is the amount of a substance containing the same number of atoms as are contained in 0.012 kg = 12 g of the carbon isotope 12C.
One mole of any substance contains Avogadro's number N.A. atoms:
N.A. = 6.02·1023 atoms.
For simple substances(elements) the mass of one mole corresponds numerically to atomic mass A element
1mol = A G.
For example: For magnesium: 1 mol 24Mg = 24 g.
For 226Ra: 1 mol 226Ra = 226 g, etc.
Taking into account what has been said in m grams of the substance will be N atoms:
https://pandia.ru/text/80/150/images/image015_20.gif" width="156" height="43 src="> (3.2.6)
Example: Let's calculate the activity of 1 gram of 226Ra, which λ = 1.38·10-11 sec-1.
a= 1.38·10-11·1/226·6.02·1023 = 3.66·1010 Bq.
If a radioactive element is included in the composition chemical compound, then when determining the activity of a drug, it is necessary to take into account its formula. Taking into account the composition of the substance, it is determined mass fraction χ radionuclide in a substance, which is determined by the ratio:
https://pandia.ru/text/80/150/images/image017_17.gif" width="118" height="41 src=">
Example of problem solution
Condition:
Activity A0 radioactive element 32P per day of observation is 1000 Bk. Determine the activity and number of atoms of this element after a week. Half life T½ 32P = 14.3 days.
Solution:
a) Let’s find the activity of phosphorus-32 after 7 days:
https://pandia.ru/text/80/150/images/image019_16.gif" width="57" height="41 src=">
Answer: after a week, the activity of the drug 32P will be 712 Bk, and the number of atoms of the radioactive isotope 32P is 127.14·106 atoms.
Security questions
1) What is the activity of a radionuclide?
2) Name the units of radioactivity and the relationship between them.
3) What is the radioactive decay constant?
4) Define the basic law of radioactive decay.
5) What is half-life?
6) What is the relationship between activity and mass of a radionuclide? Write the formula.
Tasks
1. Calculate activity 1 G 226Ra. T½ = 1602 years.
2. Calculate activity 1 G 60Co. T½ = 5.3 years.
3. One M-47 tank shell contains 4.3 kg 238U. Т½ = 2.5·109 years. Determine the activity of the projectile.
4. Calculate the activity of 137Cs after 10 years, if at the initial moment of observation it is equal to 1000 Bk. T½ = 30 years.
5. Calculate the activity of 90Sr a year ago if it is currently equal to 500 Bk. T½ = 29 years.
6. What kind of activity will 1 create? kg radioisotope 131I, T½ = 8.1 days?
7. Using reference data, determine activity 1 G 238U. Т½ = 2.5·109 years.
Using reference data, determine activity 1 G 232Th, Т½ = 1.4·1010 years.
8. Calculate the activity of the compound: 239Pu316O8.
9. Calculate the mass of a radionuclide with an activity of 1 Ki:
9.1. 131I, T1/2=8.1 days;
9.2. 90Sr, T1/2=29 years;
9.3. 137Cs, Т1/2=30 years;
9.4. 239Pu, Т1/2=2.4·104 years.
10. Determine mass 1 mCi radioactive carbon isotope 14C, T½ = 5560 years.
11. It is necessary to prepare a radioactive preparation of phosphorus 32P. After what period of time will 3% of the drug remain? Т½ = 14.29 days.
12. The natural potassium mixture contains 0.012% of the 40K radioactive isotope.
1) Determine the mass of natural potassium, which contains 1 Ki 40K. Т½ = 1.39·109 years = 4.4·1018 sec.
2) Calculate the radioactivity of the soil using 40K, if it is known that the potassium content in the soil sample is 14 kg/t.
13. How many half-lives are required for the initial activity of a radioisotope to decrease to 0.001%?
14. To determine the effect of 238U on plants, seeds were soaked in 100 ml solution UO2(NO3)2 6H2O, in which the mass of radioactive salt was 6 G. Determine the activity and specific activity of 238U in solution. Т½ = 4.5·109 years.
15. Identify Activity 1 grams 232Th, Т½ = 1.4·1010 years.
16. Determine mass 1 Ki 137Cs, Т1/2=30 years.
17. The ratio between the content of stable and radioactive isotopes of potassium in nature is a constant value. The 40K content is 0.01%. Calculate the radioactivity of the soil using 40K, if it is known that the potassium content in the soil sample is 14 kg/t.
18. Lithogenic radioactivity of the environment is formed mainly due to three main natural radionuclides: 40K, 238U, 232Th. The proportion of radioactive isotopes in the natural sum of isotopes is 0.01, 99.3, ~100, respectively. Calculate radioactivity 1 T soil, if it is known that the relative content of potassium in the soil sample is 13600 g/t, uranium – 1·10-4 g/t, thorium – 6·10-4 g/t.
19. In shells bivalves discovered 23200 Bq/kg 90Sr. Determine the activity of samples after 10, 30, 50, 100 years.
20. The main pollution of closed reservoirs in the Chernobyl zone took place in the first year after the accident at the nuclear power plant. In the bottom sediments of the lake. Azbuchin in 1999 discovered 137Cs with a specific activity of 1.1·10 Bq/m2. Determine the concentration (activity) of fallen 137Cs per m2 of bottom sediments as of 1986-1987. (12 years ago).
21. 241Am (T½ = 4.32·102 years) is formed from 241Pu (T½ = 14.4 years) and is an active geochemical migrant. Taking advantage reference materials, calculate with an accuracy of 1% the decrease in the activity of plutonium-241 over time, in which year after Chernobyl disaster 241Am's formation environment will be maximum.
22. Calculate the activity of 241Am in the emissions of the Chernobyl reactor as of April
2015, provided that in April 1986 the activity of 241Am was 3.82 1012 Bk,Т½ = 4.32·102 years.
23. 390 were found in soil samples nCi/kg 137Cs. Calculate the activity of samples after 10, 30, 50, 100 years.
24. Average concentration of lake bed pollution. Glubokoye, located in the Chernobyl exclusion zone, is 6.3 104 Bk 241Am and 7.4·104 238+239+240Pu per 1 m2. Calculate in what year these data were obtained.
Radioactivity
Ionizing radiation
Radiation effects
The Earth is under constant influence of a flow of fast particles and hard quanta electromagnetic radiation coming from space. This stream is called cosmic rays. Cosmic rays come from the depths of the universe and from the Sun. Part of the cosmic ray flux reaches the Earth's surface, and part is absorbed by the atmosphere, generating secondary radiation and leading to the formation of various radionuclides. The interaction of cosmic rays with matter leads to its ionization.
The flow of particles or electromagnetic quanta, the interaction of which with a medium leads to the ionization of its atoms, is called ionizing radiation.
Ionizing radiation can also be of terrestrial origin. For example, occur during radioactive decay.
The phenomenon of radioactivity was discovered in 1896 by A. Becquerel.
Radioactivity - ability of some atomic nuclei spontaneously (spontaneously) transform into other nuclei with the emission of particles.
There are two types of radioactivity:
Natural, which is found in natural unstable nuclei;
Artificial, which is found in radioactive nuclei formed as a result of various nuclear reactions.
Both types of radioactivity have common patterns.
Radioactive decay is a statistical phenomenon. Can be installed probability decay of one nucleus over a certain period of time. Over equal periods of time, equal shares of existing (i.e., not yet decayed at the beginning of a given period of time) nuclei of a radioactive element decay.
Let in a short time dt disintegrates dN cores. This number is proportional to the time interval dt and the total number of radioactive nuclei N:
where λ - decay constant, proportional to the probability of decay of a radioactive nucleus and depending on the nature of the element; the "-" sign indicates decreasing number of radioactive nuclei.
The solution to the differential equation (12.23) is exponential function:
Where N 0- the number of radioactive nuclei at the moment t = 0, a N- the number of undecayed nuclei at the current time t.
Formula (12.24) expresses the law of radioactive decay.
Number of radioactive nuclei decreases with time according to an exponential law.
In practice, instead of the decay constant A, another value is often used, called half-life.
Half-life (T)- this is the time during which it decays half radioactive nuclei.
The half-life can be very long or very short. For example, for uranium T = 4.5 10 9 years, and for lithium T Li = 0.89 s.
Decay characteristics T and λ are related by:
The law of radioactive decay using half-life is written as follows:
In Fig. 12.7 shows the processes of radioactive decay for two substances with different periods half-life
Rice. 12.7. Decrease in number of cores starting material during radioactive decay
