To determine the half-life of a radioactive element, a student plots graph of ln d N ( t ) d t v e r s u s t. Here 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 e - 2 . 08 = 0 . 125

Step-by-step solution

d N d t = |Activity of radioactive substance| = λ N = λ N 0 e - λ t ( ∵ N = N 0 e - λ t ) Taking log both sides l n d N d t = l n ( λ N 0 ) - λ t Hence, ln d N d t versus t graph is a straight line with slope- λ . From the graph, we can see that, λ = 1 2 = 0.5 y r - 1 Now applying the equation N = N 0 e - λ t = N 0 e - 0.5 × 4.16 = N 0 e - 2.08 = N 0 e - 2 × 0 . 693 = 0.125 N 0 = N 0 8 i.e. nuclei decrease by a factor of 8 . Hence the answer is 8 .