Advertisements
Advertisements
प्रश्न
How is the mean life of a given radioactive nucleus related to the decay constant?
Advertisements
उत्तर
To find the mean life t1, we need to use the equation of radioactive law.
The number of nuclei which decay in the time interval t to t + Δt is R(t)Δt (= λN0 e–λt Δt). Each of them has lived for time t.
Thus, the total life of all these nuclei would be t λN0 e–λt Δt. It is clear that some nuclei may live for a short time, while others may live longer. Therefore, to obtain the mean life, we have to integrate the above expression over all times from 0 to ∞ and divide it by the total number N0 of nuclei at t = 0.
Therefore, we get
`t=(lambdaN_0int_0^oote^(-lambdat)dt)/N_0=lambdaint_0^oote^(-lambdat)dt`
Solving by integration-by-parts we get
`t=lambdaxx1/lambda^2=1/lambda`
APPEARS IN
संबंधित प्रश्न
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.
Define 'activity' of a radioactive substance ?
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.)
When charcoal is prepared from a living tree, it shows a disintegration rate of 15.3 disintegrations of 14C per gram per minute. A sample from an ancient piece of charcoal shows 14C activity to be 12.3 disintegrations per gram per minute. How old is this sample? Half-life of 14C is 5730 y.
A radioactive isotope is being produced at a constant rate dN/dt = R in an experiment. The isotope has a half-life t1/2. Show that after a time t >> t1/2 the number of active nuclei will become constant. Find the value of this constant.
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?
Before the year 1900 the activity per unit mass of atmospheric carbon due to the presence of 14C averaged about 0.255 Bq per gram of carbon.
(a) What fraction of carbon atoms were 14C?
(b) An archaeological specimen containing 500 mg of carbon, shows 174 decays in one hour. What is the age of the specimen, assuming that its activity per unit mass of carbon when the specimen died was equal to the average value of the air? The half-life of 14C is 5730 years.
The half-life of the radioactive substance is 40 days. The substance will disintegrate completely in
