Question

How many hours would it take for a $64g$ sample to decay one-eighth of its original concentration? Given that the half-life of radioactive sodium is $15.0$ hours.

Answer 1

A radioactive substance is one that is unstable and one which undergoes decay spontaneously to emit $\alpha$ -radiation or $\beta$ -radiation as a result. In order to find out the time taken for the sample to decay the formula for the time taken in terms of the decay constant and the sample amount at the starting and ending interval. The formula for the half-life is also required to be applied so that the value of the decay constant is calculated given the half-life.

Complete Step By Step Answer:
The above problem revolves around the concept of radioactive samples that decay to produce radiation and this process is known as disintegration. The radioactive sample reduces in its mass or the amount of the sample decreases periodically as the sample disintegrates.
It is given that initially the sample is $64g$ which is the original concentration of the same and the time taken for it to reduce to a concentration which is one-eighth of this original concentration is required to be found out. Hence, we have:
Given,
${N_0} = 64$
$N = \dfrac{1}{8} \times 64 = 8$
The decay constant or the disintegration constant in terms of the time taken for the sample to disintegrate and the initial and final concentrations of the sample is given as:
$\lambda = \dfrac{{2.303}}{t}\log \dfrac{{{N_0}}}{N}$
By rearranging the terms we get:
$t = \dfrac{{2.303}}{\lambda }\log \dfrac{{{N_0}}}{N}$ --------( $1$ )
Here,
${N_0}$ is the number of radioactive nuclei present initially at the start of the process of decay at time t=0 in a sample of radioactive substance
$N$ is the number of radioactive nuclei present in the sample at any instant $t$
$\lambda$ is the decay constant
$t$ is the time taken for the sample to reach one-eighth of its original concentration
The half-life of the substance is given and hence we now construct the equation relating the half-life and the decay constant in order to find the value of the decay constant. The equation is as follows:
${T_{\dfrac{1}{2}}} = \dfrac{{0.693}}{\lambda }$
By rearranging the terms we get:
$\lambda = \dfrac{{0.693}}{{{T_{\dfrac{1}{2}}}}}$
The half-life of the sample is given to be 15 and hence by substituting this in the above equation we get:
Since, ${T_{\dfrac{1}{2}}} = 15$
$\lambda = \dfrac{{0.693}}{{15}}$
On solving further we get:
$\lambda = 0.0462$
Now, that the value of the decay constant is obtained is substitute this value in the equation ( $1$ ) along with the known values of the initial and final concentrations of the sample and hence we get:
$t = \dfrac{{2.303}}{{0.0462}}\log \dfrac{{64}}{8}$
On solving further we get:
$t = 49.8 \times \log 8$
$\Rightarrow t = 49.8 \times \log {2^3}$
Since, $\log {a^b} = b\log a$
$\Rightarrow t = 50 \times 3 \times \log 2$
$\Rightarrow t = 45.15$
$\Rightarrow t \approx 45$ years
Thus the time taken for the sample to decay to one-eighth of its concentration is approximately.

Note:
The disintegrations that occur in any radioactive sample is purely probability, that is, the disintegrations occur at random since the atom or the nuclei which will disintegrate first is completely unknown. The disintegrations occur regardless of the physical conditions of its surroundings which makes this decay process completely independent.