question_answer

A radioactive substance X decays into another radioactive substance Y. Initially only X was present. \[{{\lambda }_{x}}\text{ }and\text{ }{{\lambda }_{y}}\] are the disintegration constant of X and Y. \[{{N}_{x}}\text{ }and\text{ }{{N}_{y}}\] are the number of nuclei of X and Y at any time t. Number of nuclei \[{{N}_{y}}\] will be maximum when-

A) \[\frac{{{N}_{y}}}{{{N}_{x}}-{{N}_{y}}}\,\,=\,\,\frac{{{\lambda }_{y}}}{{{\lambda }_{x}}-{{\lambda }_{y}}}\]

B)   \[\frac{{{N}_{y}}}{{{N}_{x}}-{{N}_{y}}}\,\,=\,\,\frac{{{\lambda }_{y}}}{{{\lambda }_{x}}-{{\lambda }_{y}}}\]

C) \[{{\lambda }_{y}}{{N}_{y}}\,=\,\,{{\lambda }_{x}}{{N}_{x}}\]                     

D) \[{{\lambda }_{y}}{{N}_{x}}\,=\,\,{{\lambda }_{x}}{{N}_{y}}\]

Correct Answer: C

Solution :

  Net rate of formation of Y at any time t is \[\frac{d{{N}_{y}}}{dt}\,\,=\,\,{{\lambda }_{x}}{{N}_{x}}-\,{{\lambda }_{y}}{{N}_{y}}\] \[{{N}_{y}}\] is maximum, when \[\frac{d{{N}_{y}}}{dt}\,\,=\,\,0\]