To determine the half-life of a radioactive element, a student plots a graph of ln d t d N ( t ) versust. Here d t d N ( t ) is the rate of radioactive decay at time t . If the number of radio

Question

To determine the half-life of a radioactive element, a student plots a graph of $ln \frac{d N ( t )}{d t}$ versust. Here $\frac{d N ( t )}{d t}$ is the rate of radioactive decay at time $t$ . If the number of radioactive nuclei of this element decreases by a factor of $p$ after $4.16 yr$ , the value of $p$ is

TestHub · Accepted Answer

​ d t d N ​ ​ =∣ Activity of radioactive substance| = λ N = λ N 0 ​ e − λ t Taking log both sides ln ​ d t d N ​ ​ = ln ( λ N 0 ​ ) − λ t Hence, ln ​ d t d N ​ ​ versus t graph is a straight line with slope − λ From the graph we can see that, λ = 2 1 ​ = 0.5 yr − 1 Now applying the equation, N ​ = N 0 ​ e − λ t = N 0 ​ e − 0.5 × 4.16 = N 0 ​ e − 2.08 = 0.125 N 0 ​ = 8 N 0 ​ ​ ​ ie, nuclei decreases by a factor of 8 . Hence the answer is 8 .

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