Advertisements
Advertisements
Question
How is the mean life of a given radioactive nucleus related to the decay constant?
Advertisements
Solution
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
RELATED QUESTIONS
The decay constant of radioactive substance is 4.33 x 10-4 per year. Calculate its half life period.
Represent Radioactive Decay curve using relation `N = N_o e^(-lambdat)` graphically
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.
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.
Define one Becquerel.
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 nuclide is 20 hrs. The fraction of the original activity that will remain after 40 hrs is ______.
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 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 |
- Plot the graph of R versus t and calculate the half-life from the graph.
- Plot the graph of ln `(R/R_0)` versus t and obtain the value of half-life from the graph.
