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 mixture is

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

Answer 1

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

Note the decay rate of single nuclei is:

$\frac{dN}{dt}=-\lambda N{e}^{-\lambda t}$

$\therefore$ The combination of two nuclei $N_1\&N_2$ the decay rate is:

$\frac{d{N}_1}{dt}+\frac{d{N}_2}{dt}=-({\lambda }_1{N}_1{e}^{-{\lambda }_{1}t}+{\lambda }_2{N}_2{e}^{-{\lambda }_{2}t})$