Question

Two radioactive nuclei P and Q, in a given sample decay into a stable nucleous R. At time t =0, number of P species are 4N_o and that of Q are N_o.Half -life of P (for conversion to R) is 1 min where asthat of Q is 12 min. Initially there are no nuclei of R present in the sample. When number of nuclei of R presnent in the sample would be

3N_o

9N_o / 2

5N_o/2

2N_o

Solution

9N_o / 2

Initially P → 4N_o
Q → N_oHalf life T_P →1 min
T_Q = 2 min
Let after time t number of nuclei of P and Q are equal i.e,
$fraction numerator 4 straight N subscript straight o over denominator 2 to the power of begin display style bevelled 1 over straight t end style end exponent end fraction space equals space N subscript o over 2 to the power of 1 divided by t end exponent 4 space equals space 2 to the power of t divided by 2 end exponent 2 squared space equals space 2 to the power of t divided by 2 end exponent t over 2 space space equals space 2 t equals space 4 space m i n D i s a c t i v e space n u c l e u s space o r space N u c l e i space o f space R equals space open parentheses 4 N subscript o minus fraction numerator 4 N subscript o over denominator 2 to the power of 4 end fraction close parentheses space plus open parentheses N subscript o space minus N subscript o over 2 squared close parentheses equals space 4 N subscript o space minus space N subscript o over 4 plus N subscript o space minus space N subscript o over 4 equals space 5 N subscript o space minus space N subscript o over 2 equals space 9 over 2 N subscript o$

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