Nuclei of a radioactive element $$X$$ are being produced at a constant rate $$K$$ and this element decays to a stable nucleus $$Y$$ with a decay constant $$\lambda$$ and half-life $${T}_{1/2}$$. At time $$t=0$$, there are $${N}_{0}$$ nuclei of the element $$X$$
The number $${N}_{Y}$$ of nuclei of $$Y$$ at time $$t$$ is
A
$$\quad Kt+\left( \cfrac { K-\lambda { N }_{ 0 } }{ \lambda } \right) { e }^{ -\lambda t }$$
B
$$Kt-\left( \cfrac { K-\lambda { N }_{ 0 } }{ \lambda } \right) { e }^{ -\lambda t }+\cfrac { K-\lambda { N }_{ 0 } }{ \lambda } $$
C
$$Kt+\left( \cfrac { K-\lambda { N }_{ 0 } }{ \lambda } \right) { e }^{ -\lambda t }-\cfrac { K-\lambda { N }_{ 0 } }{ \lambda } $$
D
$$\quad Kt-\left( \cfrac { K-\lambda { N }_{ 0 } }{ \lambda } \right) { e }^{ -\lambda t }$$
Correct option is B. $$Kt+\left( \cfrac { K-\lambda { N }_{ 0 } }{ \lambda } \right) { e }^{ -\lambda t }-\cfrac { K-\lambda { N }_{ 0 } }{ \lambda } $$
$${ N }_{ X }=\cfrac { 1 }{ \lambda } \left[ K-(K-\lambda { N }_{ 0 }){ e }^{ -\lambda t } \right]$$
$$\cfrac { d{ N }_{ Y } }{ dt } =\lambda { N }_{ X }$$
$$dN_Y=\left[K-(K-\lambda N_0)e^{-\lambda t}\right]dt$$
Integrating both sides
$$\Rightarrow { N }_{ Y }=Kt+\left( \cfrac { K-\lambda { N }_{ 0 } }{ \lambda } \right) { e }^{ -\lambda t }-\cfrac { K-\lambda { N }_{ 0 } }{ \lambda } $$