Question

Figure 42-16 gives the activities of three radioactive samples versus time. Rank the samples according to their (a) half-life and (b) disintegration constant, greatest first. (Hint:For (a), use a straightedge on the graph.)

Short answer

1. The rank of the samples according to their half-life is C > B > A.
2. The rank of the samples according to their disintegration constant is A > B > C.

Step-by-step solution

Given data

Figure 42-16 gives the activities of three radioactive samples versus time is given.

Understanding the concept of decay  

Half-life is the time required for a quantity to reduce to half of its initial value during the radioactive decay process.

The disintegration constant of a radioactive substance is defined as the ratio of its instantaneous rate of disintegration to the number of nuclides present at that time.

The half-life and disintegration constant of a nuclide are inversely proportional to each other.

Formulae:

The disintegration constant,$\lambda =\frac{\ln 2}{T_{v2}}............\left(1\right)$

Where,$T_{v2}$is the half-life of the substance,

The rate of decay, $R=\lambda N........\left(2\right)$

a) Calculation for the rank of the samples according to their half-lives

From the graph, we can see that the rate of.decay of the samples at half of their time decay is in the order of: A > B > C

Thus, from equations (1) and (2), we can see that the rate of decay is inversely proportional to their half-lives,$R=\left(\frac{\ln @}{T_{v2}}\right)N$.

Hence, the rank of the samples according to their half-lives is given by C > B > A.

b) Calculation of the rank of the samples according to their disintegration constants

Now, from part (a) calculations, the rank according to half-lives is given as: C > B > A

Thus, from equation (1), we see that disintegration constant is inversely proportional to the half-life.

Hence, the ranking of the samples according to disintegration constant is A > B > C.