Question

At time $t=0$, $N_1$ nuclei of decay constant ${\lambda }_1$ and $N_2$ nuclei of decay constant ${\lambda }_1$ 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. $-(N_1{\lambda }_1{e}^{-{\lambda }_{1}t}+N_2{\lambda }_2{e}^{-{\lambda }_{2}t})$
• D. $-N_1{\lambda }_1{N}_2{\lambda }_2{e}^{-({\lambda }_1+{\lambda }_2)t}$

Answer 1

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

Decay rate of one particle only depends on its number od atoms but not on the other particles.So the total rate will be the addition of the rates of two compounds.

No. of atoms of first compound at any time $-N_1{e}^{-d_{1}t}$

No. of atoms of $2^{nd}$ compound at any time $=N_1{e}^{-{\lambda }_{2}t}$

We know that $\frac{dN}{dt}=\lambda N$

decay rate of $1^{st}$ compound $={\lambda }_1{N}_1{e}^{-{\lambda }_{1}t}$

decay rate of $2^{nd}$ compound $={\lambda }_2{N}_2{e}^{-{\lambda }_{2}t}$

Total rate of mixture $={\lambda }_1{N}_1{e}^{-{\lambda }_{1}t}+{\lambda }_2{N}_2{e}^{-{\lambda }_{2}t}$