Question

The half-life of ${I^{131}}$ is 8 days. Given a sample of ${I^{131}}$ at time t=0, we can assert that
(1) No nucleus will decay before t=4 days
(2) No nucleus will decay before t=8 days
(3) All nuclei will decay before t=16 days
(4) A given nucleus will decay at t=0

Answer 1

Hint: We know that the half-life relates the stability and the disintegration of atomic particles. It tells us how quickly the unstable particle would disintegrate and the stable particle would survive when the nuclei become half the initial value.

Complete step by step answer:
We know that the relation to calculate the half-life is given by:
$N = {N_0}{e^{ - \lambda t}}$
Here, N is the remaining quantity that is left after decay in time t, ${N_0}$ is the initial quantity present that is going to be decayed, $\lambda $ is the decay constant and t is the time.We know that all nuclei will start to decay immediately after time t. At time t=0, the number of particles that are present are ${N_0}$ we can be proved by the above equation by substituting t=0. Therefore, we get the result as,
$\begin{array}{l}
N = {N_0}{e^{ - \lambda \times 0}}\\
 = {N_0}{e^0}\\
 = {N_0}
\end{array}$
Here in the above equation, just after the time t=0, the value of $N$ is always less than ${N_0}$.Hence at the time will increase that is as $t > 0$ then the amount of nuclei that are initially present will start to decay exponentially.Therefore, a nucleus will immediately start to decay after t=0 and the correct option is (4).

Note:We must know that some biological process also decays at a particular rate. But the nature of decay of the biological process is not exponential. Therefore, in this case we determine the half-life of the process in two particular halves. First half is the amount of time required to come from initial state to 50% decay.