Advertisements
Advertisements
Question
State the law of radioactive decay.
Advertisements
Solution
Law of radioactive decay:
The number of nuclei undergoing the decay per unit time is proportional to the number of unchanged nuclei present at that instant.
If ‘N’ is the number of nuclei present at any instant ‘t’, ‘dN’ is the number of nuclei that disintegrated in short interval of time ‘dt’, then according to decay law,
`- (dN)/dt ∝ N`
`:.(dN)/dt = -lambda N`
where, `lambda` is known as decay constant or disintegration constant. The negative sign indicates disintegration of atoms.
APPEARS IN
RELATED QUESTIONS
(a) Write the basic nuclear process involved in the emission of β+ in a symbolic form, by a radioactive nucleus.
(b) In the reactions given below:
(i)`""_16^11C->_y^zB+x+v`
(ii)`""_6^12C+_6^12C->_a^20 Ne + _b^c He`
Find the values of x, y, and z and a, b and c.
Derive the mathematical expression for law of radioactive decay for a sample of a radioactive nucleus
How is the mean life of a given radioactive nucleus related to the decay constant?
A radioactive isotope has a half-life of T years. How long will it take the activity to reduce to a) 3.125%, b) 1% of its original value?
Obtain the amount of `""_27^60"Co"` necessary to provide a radioactive source of 8.0 mCi strength. The half-life of `""_27^60"Co"` is 5.3 years.
The half-life of `""_38^90 "Sr"` is 28 years. What is the disintegration rate of 15 mg of this isotope?
The Q value of a nuclear reaction \[\ce{A + b → C + d}\] is defined by
Q = [ mA+ mb− mC− md]c2 where the masses refer to the respective nuclei. Determine from the given data the Q-value of the following reactions and state whether the reactions are exothermic or endothermic.
\[\ce{^1_1H + ^3_1H -> ^2_1H + ^2_1H}\]
Atomic masses are given to be
`"m"(""_1^2"H")` = 2.014102 u
`"m"(""_1^3"H")` = 3.016049 u
`"m"(""_6^12"C")` = 12.000000 u
`"m"(""_10^20"Ne")` = 19.992439 u
The Q value of a nuclear reaction A + b → C + d is defined by
Q = [mA+ mb − mC − md]c2 where the masses refer to the respective nuclei. Determine from the given data the Q-value of the following reactions and state whether the reactions are exothermic or endothermic.
\[\ce{^12_6C + ^12_6C ->^20_10Ne + ^4_2He}\]
Atomic masses are given to be
`"m"(""_1^2"H")` = 2.014102 u
`"m"(""_1^3"H")` = 3.016049 u
`"m"(""_6^12C)` = 12.000000 u
`"m"(""_10^20"Ne")` = 19.992439 u
Under certain circumstances, a nucleus can decay by emitting a particle more massive than an α-particle. Consider the following decay processes:
\[\ce{^223_88Ra -> ^209_82Pb + ^14_6C}\]
\[\ce{^223_88 Ra -> ^219_86 Rn + ^4_2He}\]
Calculate the Q-values for these decays and determine that both are energetically allowed.
Represent Radioactive Decay curve using relation `N = N_o e^(-lambdat)` graphically
Define 'activity' of a radioactive substance ?
Why is it experimentally found difficult to detect neutrinos in this process ?
In a given sample, two radioisotopes, A and B, are initially present in the ration of 1 : 4. The half lives of A and B are respectively 100 years and 50 years. Find the time after which the amounts of A and B become equal.
A radioactive nucleus ‘A’ undergoes a series of decays according to the following scheme:
The mass number and atomic number of A are 180 and 72 respectively. What are these numbers for A4?
The decay constant of a radioactive sample is λ. The half-life and the average-life of the sample are respectively
Lithium (Z = 3) has two stable isotopes 6Li and 7Li. When neutrons are bombarded on lithium sample, electrons and α-particles are ejected. Write down the nuclear process taking place.
28Th emits an alpha particle to reduce to 224Ra. Calculate the kinetic energy of the alpha particle emitted in the following decay:
`""^228"Th" → ""^224"Ra"^(∗) + alpha`
`""^224"Ra"^(∗) → ""^224"Ra" + γ (217 "keV")`.
Atomic mass of 228Th is 228.028726 u, that of 224Ra is 224.020196 u and that of `""_2^4H` is 4.00260 u.
(Use Mass of proton mp = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron mn = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c2,1 u = 931 MeV/c2.)
Calculate the maximum kinetic energy of the beta particle emitted in the following decay scheme:
12N → 12C* + e+ + v
12C* → 12C + γ (4.43MeV).
The atomic mass of 12N is 12.018613 u.
(Use Mass of proton mp = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron mn = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c2,1 u = 931 MeV/c2.)
Consider the situation of the previous problem. Suppose the production of the radioactive isotope starts at t = 0. Find the number of active nuclei at time t.
Define the term 'decay constant' of a radioactive sample. The rate of disintegration of a given radioactive nucleus is 10000 disintegrations/s and 5,000 disintegrations/s after 20 hr. and 30 hr. respectively from start. Calculate the half-life and the initial number of nuclei at t= 0.
Identify the nature of the radioactive radiations emitted in each step of the decay process given below.
`""_Z^A X -> _Z^A _-1^-4 Y ->_Z^A _-1^-4 W`
Define one Becquerel.
A radioactive substance disintegrates into two types of daughter nuclei, one type with disintegration constant λ1 and the other type with disintegration constant λ2 . Determine the half-life of the radioactive substance.
What is the amount of \[\ce{_27^60Co}\] necessary to provide a radioactive source of strength 10.0 mCi, its half-life being 5.3 years?
The isotope \[\ce{^57Co}\] decays by electron capture to \[\ce{^57Fe}\] with a half-life of 272 d. The \[\ce{^57Fe}\] nucleus is produced in an excited state, and it almost instantaneously emits gamma rays.
(a) Find the mean lifetime and decay constant for 57Co.
(b) If the activity of a radiation source 57Co is 2.0 µCi now, how many 57Co nuclei does the source contain?
c) What will be the activity after one year?
A source contains two species of phosphorous nuclei, \[\ce{_15^32P}\] (T1/2 = 14.3 d) and \[\ce{_15^33P}\] (T1/2 = 25.3 d). At time t = 0, 90% of the decays are from \[\ce{_15^32P}\]. How much time has to elapse for only 15% of the decays to be from \[\ce{_15^32P}\]?
Obtain an expression for the decay law of radioactivity. Hence show that the activity A(t) =λNO e-λt.
Two radioactive materials X1 and X2 have decay constants 10λ and λ respectively. If initially, they have the same number of nuclei, then the ratio of the number of nuclei of X1 to that of X2 will belie after a time.
A radioactive element disintegrates for an interval of time equal to its mean lifetime. The fraction that has disintegrated is ______
Which one of the following nuclei has shorter meant life?
'Half-life' of a radioactive substance accounts for ______.
Two radioactive materials Y1 and Y2 have decay constants '5`lambda`' and `lambda` respectively. Initially they have same number of nuclei. After time 't', the ratio of number of nuclei of Y1 to that of Y2 is `1/"e"`, then 't' is equal to ______.
What percentage of radioactive substance is left after five half-lives?
The half-life of a radioactive nuclide is 20 hrs. The fraction of the original activity that will remain after 40 hrs is ______.
The half-life of the radioactive substance is 40 days. The substance will disintegrate completely in
Suppose we consider a large number of containers each containing initially 10000 atoms of a radioactive material with a half life of 1 year. After 1 year ______.
Samples of two radioactive nuclides A and B are taken. λA and λB are the disintegration constants of A and B respectively. In which of the following cases, the two samples can simultaneously have the same decay rate at any time?
- Initial rate of decay of A is twice the initial rate of decay of B and λA = λB.
- Initial rate of decay of A is twice the initial rate of decay of B and λA > λB.
- Initial rate of decay of B is twice the initial rate of decay of A and λA > λB.
- Initial rate of decay of B is the same as the rate of decay of A at t = 2h and λB < λA.
The variation of decay rate of two radioactive samples A and B with time is shown in figure.
Which of the following statements are true?
- Decay constant of A is greater than that of B, hence A always decays faster than B.
- Decay constant of B is greater than that of A but its decay rate is always smaller than that of A.
- Decay constant of A is greater than that of B but it does not always decay faster than B.
- Decay constant of B is smaller than that of A but still its decay rate becomes equal to that of A at a later instant.
Which sample, A or B shown in figure has shorter mean-life?
Consider a radioactive nucleus A which decays to a stable nucleus C through the following sequence:
A→B→C
Here B is an intermediate nuclei which is also radioactive. Considering that there are N0 atoms of A initially, plot the graph showing the variation of number of atoms of A and B versus time.
A piece of wood from the ruins of an ancient building was found to have a 14C activity of 12 disintegrations per minute per gram of its carbon content. The 14C activity of the living wood is 16 disintegrations per minute per gram. How long ago did the tree, from which the wooden sample came, die? Given half-life of 14C is 5760 years.
What is the half-life period of a radioactive material if its activity drops to 1/16th of its initial value of 30 years?
