question_answer

The activity of a radioactive sample is measured as 9750 counts per minute at t = 0 and as 975 counts per minute at T = 5 minutes. The decay constant is approximately:

A)  0.922 per minute            

B)  0.691 per minute

C)  0.461 per minute            

D)  0.230 per minute

Correct Answer: C

Solution :

                According to law of radioactivity \[\frac{N}{{{N}_{0}}}={{e}^{-\lambda t}}\]           ??.(i) Taking logarithm on both sides of Eq. (i), we have                 \[{{\log }_{e}}\left( \frac{N}{{{N}_{0}}} \right)={{\log }_{e}}({{e}^{-\lambda t}})\]                 \[=-\lambda t\,{{\log }_{e}}\,e=-\lambda t\] As we know that \[{{\log }_{e}}\,x=2.3026\,\,{{\log }_{10}}x\] Making substitution, we get                 \[\lambda =\frac{2.3026\,\,{{\log }_{10}}\,\,\left( \frac{9750}{975} \right)}{5}\]                 \[=\frac{2.3026}{e}\,{{\log }_{10}}10\]                 \[=\frac{2.3026}{5}\,{{\min }^{-1}}=0.461\,{{\min }^{-1}}\] NOTE: 1. The graph between   number  of   nuclei decayed            \[({{N}_{0}}-N)\] with time (t) is as shown              The graph between probability of a nucleus     for survival \[\left( \frac{N}{{{N}_{0}}} \right)\] with time (f) is as shown    below.