Question

Two radioactive materials A and B have decay constants 10 $\lambda $ and $\lambda $ respectively. Initially they have the same number of nuclei, and then the ratio of the number of nuclei of A to that of B will be 1/e after a time:
A. 11/10 $\lambda $
B. 1/9 $\lambda $
C. 1/10 $\lambda $
D. 11/11 $\lambda $

Answer 1

Hint: Here, we use the formula of number of particles present in decay. The two radioactive materials will give two different equations and from these two equations the value of time is being calculated. Radioactive material is any material containing unstable atoms that emit ionizing radiation as it decays.

Formula used:
$ N = {N_0}{e^{ - \lambda t}}$

Complete step-by-step answer:
Radioactivity is the process by which certain naturally occurring or artificial nuclides undergo spontaneous decay releasing a new energy. So, this decay process is accompanied by the emission of one or more types of radiation, ionizing or non-ionizing particles.
Radioactive material is any material containing unstable atoms the emit ionizing radiation as it decays.
The number of nuclei having decay constant and life time is given by:
 $\eqalign{& {N_1} = {N_0}{e^{ - 10\lambda t}} \cr
  & {N_2} = {N_0}{e^{ - \lambda t}} \cr}$
Now, by taking the ratio of the above two equations and solving, we get:
$\eqalign{
  & \dfrac{1}{e} = \dfrac{{{N_1}}}{{{N_2}}} = {e^{ - 9\lambda t}} \cr
  & \Rightarrow 9\lambda t = 1 \cr
  & \therefore t = \dfrac{1}{{9\lambda }} \cr} $
Therefore, the correct option is B) i.e., the ratio of the number of nuclei of A to that of B will be 1/e after a time t is given by 1/9 $\lambda $.

Additional Information: A material which has unstable nuclei is considered as radioactive. There are three types of radioactive decay first is alpha decay, second is beta decay and the last is gamma decay. All of this decay involves emitting one or more particles or photons.
The ability of radiation to damage molecules is analyzed in terms of ionizing power. When a radiation particle interacts with atoms, the interaction can cause the atom, the interaction can cause the atom to lose electrons and thus become ionized.
The ability of each type of radiation to pass through matter is expressed as penetration power. The more material the radiation can pass through, the greater the penetration power. If the penetration [power is more than the radiation is more dangerous.
It can also be noticed that, the greater the mass present the greater the ionizing power and the lower the penetration power.

Note: Radioactive materials are dangerous. The mean lifetime of a radioactive material is very small. Mean life of radioactive elements is expected to be longer than the half-life of the element. Decay constant is given by the ratio of 0.693 to half-time of the element.