Question

Activities of three radioactive substances A, B and C are represented by the curves A, B and C in the figure. Then their half-lives\[{T_{1/2}}(A):{T_{1/2}}(B):{T_{1/2}}(C)\] are in the ratio:

A. 3:2:1
B. 2:1:1
C. 4:3:1
D. 2:1:3

Answer 1

Hint:Radioactive decay defines the process of losing energy by an unstable atomic by radiation. A material that possesses unstable nuclei is a radioactive element. Alpha decay, gamma decay and beta decay are different types of decay.

Formula used:
The relation between decay constant and activity is,
\[\lambda t = \ln \dfrac{{{A_0}}}{{{A_t}}}\]
Where, \[\lambda \] indicates decay constant, \[{A_0}\] is initial activity and \[{A_0}\]is activity at particular time t and t is time.

Complete step by step solution:
In the provided graph, y axis represents time and the x axis represents activity of three substances A, B and C. In the equation,
\[\lambda t = \ln \dfrac{{{A_0}}}{{{A_t}}} \\ \]
\[\Rightarrow \lambda t = \ln {A_0} - \ln {A_t} \\ \]
Rearranging the equation, we get,
\[\ln {A_t} = - \lambda t + \ln {A_0}\]
If we compare the above equation with the equation of straight line, that is, \[y = mx + C\]
Then, \[y = - \ln {A_t}\] and \[m = - \lambda = \tan \theta \\ \]
Taking mod of \[m = - \lambda = \tan \theta \], we get,
\[m = \lambda = \tan \theta \\ \]
Again,
\[\tan \theta = {m_A} = \dfrac{6}{{10}} = \dfrac{3}{5} \\ \]
\[\Rightarrow \tan \theta = {m_B} = \dfrac{6}{5} \\ \]
\[\Rightarrow \tan \theta = {m_C} = \dfrac{2}{5}\]
Therefore,
\[{\lambda _A} = \dfrac{3}{5}\] \[{\lambda _B} = \dfrac{6}{5}\] and \[{\lambda _c} = \dfrac{2}{5}\]

Now, the relation between Half life and \[\lambda \] is,
\[{t_{1/2}} = \dfrac{{\ln 2}}{\lambda }\]
The above equation says that half life is inversely proportional to \[\lambda \] . Therefore, half lives of A, B and C are,
\[{t_{1/2}}(A):{t_{1/2}}(B):{t_{1/2}}(C) = \dfrac{5}{3}:\dfrac{5}{6}:\dfrac{5}{2}\]
Canceling 5 and multiplying 6 in Right Hand side of the equation gives the ratio,
\[{t_{1/2}}(A):{t_{1/2}}(B):{t_{1/2}}(C) = \dfrac{6}{3}:\dfrac{6}{6}:\dfrac{6}{2} = 2:1:3\]
Therefore, the ratio of Half lives of A, B and C is 2:1:3.

Hence, option D is the correct answer.

Note: Half-life defines the requirement of time by a quantity in reducing its quantity into half. This term is used commonly in the study of nuclear physics to understand how unstable atoms quickly undergo radioactive decay.