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 $${N}_{X}$$ of nuclei of $$X$$ at time $$t={T}_{1/2}$$ is
A
$$\cfrac { K+\lambda { N }_{ 0 } }{ 2\lambda } $$
C
$$\left( 2\lambda { N }_{ 0 }-K \right) \cfrac { 1 }{ \lambda } $$
D
$$\left[ \lambda { N }_{ 0 }+\cfrac { K }{ 2 } \right] \cfrac { 1 }{ \lambda } $$
Correct option is A. $$\cfrac { K+\lambda { N }_{ 0 } }{ 2\lambda } $$
$$\cfrac { d{ N }_{ X } }{ dt } =K-\lambda { N }_{ X }$$
$$\dfrac{dN_X}{K-\lambda N_X}=dt$$
Integrating both sides
$$\Rightarrow { N }_{ X }=\cfrac { 1 }{ \lambda } \left[ K-(K-\lambda { N }_{ 0 }){ e }^{ -\lambda t } \right] $$
at $$t=T_{1/2}=\dfrac{0.693}\lambda$$
$$N_X=\dfrac{K+\lambda N_0}{2\lambda}$$