Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let the number of atoms present at t = 0 be N0.
Let N be the number of radio-active isotopes present at time t.
Then,
N = N0e−λt
Here, `lambda` = Disintegration constant
∴ Number of radioactive isotopes decayed = `N_0 - N = N_0 - N_0e^(-lambdat)`
= `N_0(1-e^(-lambdat))` ...(1)
Rate of decay (R) is given by
`R = lambdaN_0` ...(2)
Substituting the value of `N_0` from equation (2) to equation (1), we get
`N = N_0(1-e^(-lambdat))`
= `R/lambda (1 - e^(-lambdat))`
APPEARS IN
संबंधित प्रश्न
State the law of radioactive decay.
The half-life of `""_38^90 "Sr"` is 28 years. What is the disintegration rate of 15 mg of this isotope?
Define 'activity' of a radioactive substance ?
Two different radioactive elements with half lives T1 and T2 have N1 and N2 undecayed atoms respectively present at a given instant. Derive an expression for the ratio of their activities at this instant in terms of N1 and N2 ?
A freshly prepared radioactive source of half-life 2 h emits radiation of intensity which is 64 times the permissible safe level. The minimum time after which it would be possible to work safely with this source is
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.)
57Co decays to 57Fe by β+- emission. The resulting 57Fe is in its excited state and comes to the ground state by emitting γ-rays. The half-life of β+- decay is 270 days and that of the γ-emissions is 10−8 s. A sample of 57Co gives 5.0 × 109 gamma rays per second. How much time will elapse before the emission rate of gamma rays drops to 2.5 × 109per second?
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.
Obtain a relation between the half-life of a radioactive substance and decay constant (λ).
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?
Disintegration rate of a sample is 1010 per hour at 20 hours from the start. It reduces to 6.3 x 109 per hour after 30 hours. Calculate its half-life and the initial number of radioactive atoms in the sample.
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.
Which one of the following nuclei has shorter meant life?
The half-life of a radioactive sample undergoing `alpha` - decay is 1.4 x 1017 s. If the number of nuclei in the sample is 2.0 x 1021, the activity of the sample is nearly ____________.
The half-life of the radioactive substance is 40 days. The substance will disintegrate completely in
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.
Sometimes a radioactive nucleus decays into a nucleus which itself is radioactive. An example is :
\[\ce{^38Sulphur ->[half-life][= 2.48h] ^{38}Cl ->[half-life][= 0.62h] ^38Air (stable)}\]
Assume that we start with 1000 38S nuclei at time t = 0. The number of 38Cl is of count zero at t = 0 and will again be zero at t = ∞ . At what value of t, would the number of counts be a maximum?
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.
