मराठी
महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता १२ वी

State the Law of Radioactive Decay. Hence Derive The Expression N = Noe^-λT Where Symbols Have Their Usual Meanings.

Advertisements
Advertisements

प्रश्न

State the law of radioactive decay.

दीर्घउत्तर
Advertisements

उत्तर

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.

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
2012-2013 (March)

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

How is the mean life of a given radioactive nucleus related to the decay constant?


Obtain the relation between the decay constant and half life of a radioactive sample.


Write symbolically the process expressing the β+ decay of `""_11^22Na`. Also write the basic nuclear process underlying this decay.


Why is it found experimentally difficult to detect neutrinos in nuclear β-decay?


The normal activity of living carbon-containing matter is found to be about 15 decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive `""_6^14"C"` present with the stable carbon isotope `""_6^12"C"`. When the organism is dead, its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life (5730 years) of `""_6^14"C"` and the measured activity, the age of the specimen can be approximately estimated. This is the principle of `""_6^14"C"` dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of 9 decays per minute per gram of carbon. Estimate the approximate age of the Indus-Valley civilisation.


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 radionuclide 11C decays according to 

\[\ce{^11_6C -> ^11_5B + e+ + \text{v}}\] : T1/2 = 20.3 min

The maximum energy of the emitted positron is 0.960 MeV.

Given the mass values: `"m"(""_6^11"C") = 11.011434 u and "m"(""_6^11"B") = 11.009305 "u"`

Calculate Q and compare it with the maximum energy of the positron emitted.


A source contains two phosphorous radio nuclides `""_15^32"P"` (T1/2 = 14.3d) and `""_15^33"P"` (T1/2 = 25.3d). Initially, 10% of the decays come from `""_15^33"P"`. How long one must wait until 90% do so?


Represent Radioactive Decay curve using relation `N = N_o e^(-lambdat)` graphically


Using the equation `N = N_0e^(-lambdat)` obtain the relation between half-life (T) and decay constant (`lambda`) of a radioactive substance.


(a) Derive the relation between the decay constant and half life of a radioactive substance. 
(b) A radioactive element reduces to 25% of its initial mass in 1000 years. Find its half life.


Define 'activity' of a radioactive substance ?


Define the activity of a given radioactive substance. Write its S.I. unit.


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.


The radioactive isotope D decays according to the sequence

If the mass number and atomic number of D2 are 176 and 71 respectively, what is (i) the mass number (ii) atomic number of D?


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.


The masses of 11C and 11B are respectively 11.0114 u and 11.0093 u. Find the maximum energy a positron can have in the β*-decay of 11C to 11B.

(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.)


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.)


The decay constant of `""_80^197`Hg (electron capture to `""_79^197`Au) is 1.8 × 10−4 S−1. (a) What is the half-life? (b) What is the average-life? (c) How much time will it take to convert 25% of this isotope of mercury into gold?


The decay constant of 238U is 4.9 × 10−18 S−1. (a) What is the average-life of 238U? (b) What is the half-life of 238U? (c) By what factor does the activity of a 238U sample decrease in 9 × 109 years?


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.


The half-life of 40K is 1.30 × 109 y. A sample of 1.00 g of pure KCI gives 160 counts s−1. Calculate the relative abundance of 40K (fraction of 40K present) in natural potassium.


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.


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?


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 ______.


After 1 hour, `(1/8)^"th"` of the initial mass of a certain radioactive isotope remains undecayed. The half-life of the isotopes is ______.


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 ______.


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?

  1. Initial rate of decay of A is twice the initial rate of decay of B and λA = λB.
  2. Initial rate of decay of A is twice the initial rate of decay of B and λA > λB.
  3. Initial rate of decay of B is twice the initial rate of decay of A and λA > λB.
  4. Initial rate of decay of B is the same as the rate of decay of A at t = 2h and λB < λA.

Draw a graph showing the variation of decay rate with number of active nuclei.


Which sample, A or B shown in figure has shorter mean-life?


The activity R of an unknown radioactive nuclide is measured at hourly intervals. The results found are tabulated as follows:

t (h) 0 1 2 3 4
R (MBq) 100 35.36 12.51 4.42 1.56
  1. Plot the graph of R versus t and calculate the half-life from the graph.
  2. Plot the graph of ln `(R/R_0)` versus t and obtain the value of half-life from the graph.

The radioactivity of an old sample of whisky due to tritium (half-life 12.5 years) was found to be only about 4% of that measured in a recently purchased bottle marked 10 years old. The age of a sample is ______ 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?


The half-life of `""_82^210Pb` is 22.3 y. How long will it take for its activity 0 30% of the initial activity?


In the uranium radioactive series, the initial nucleus is \[\ce{_92U^238}\] and that the final nucleus is \[\ce{_82U^206}\]. When uranium nucleus decays to lead, the number of α particles and β particles emitted are ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×