Question

At time $t=0$, $N_1$ nuclei of decay constant ${\lambda }_1$ and $N_2$ nuclei of decay constant ${\lambda }_2$ are mixed. The decay rate of the mixture is :

• A. $N_1{N}_2{e}^{-({\lambda }_1+{\lambda }_2)t}$
• B. $N_1{\lambda }_1{e}^{-{\lambda }_{1}t}+N_2{\lambda }_2{e}^{-{\lambda }_{2}t}$
• C. $-\left(\frac{N_1}{N_2}\right)e^{-({\lambda }_1-{\lambda }_2)t}$
• D. $-N_1{\lambda }_1{N}_2{\lambda }_2{e}^{-({\lambda }_1+{\lambda }_2)}$

Answer 1

The correct option is B $N_1{\lambda }_1{e}^{-{\lambda }_{1}t}+N_2{\lambda }_2{e}^{-{\lambda }_{2}t}$

No. of atoms of $1^{st}$ compound at time $t$ $=N_1{e}^{-{\lambda }_{1}t}$
No. of atoms of $2^{nd}$ compound at time $t$ $=N_2{e}^{-{\lambda }_{2}t}$
At time $t$, the mixture will have the number of nuclei
$N=N_1{e}^{-{\lambda }_{1}t}+N_2{e}^{-{\lambda }_{2}t}$
The decay rate is $-\frac{dN}{dt}$
$-\frac{dN}{dt}={\lambda }_1{N}_1{e}^{-{\lambda }_{1}t}+{\lambda }_2{N}_2{e}^{-{\lambda }_{2}t}$