Question

The radioactive decay rate of a radioactive element is found to be \[{10^3}\] disintegrations/sec., at a certain time. If the half-life of the element is one second, the decay rate after one second is x disintegrations per sec and after three seconds is y disintegrations per sec. Then \[y/x = \]

Answer 1

Hint: Use the relation between the number of half lives and half-life to calculate the number of half lives for a given time. Use the decay equation relation number of nuclei presents at a certain time relating the number of half lives.

Formula used:
The number of half lives in time t is given by,
\[n = \dfrac{t}{T}\]

Here, T is the half-life.

\[N = {N_0}{\left( {\dfrac{1}{2}} \right)^n}\]

Here, \[{N_0}\] is the actual number of nuclei present in the sample at \[t = 0\] and n is the number of half lives.

Complete step by step answer:We know that the rate of decay is proportional to the number of nuclei left in the sample.

The number of half lives in time t is given by,
\[n = \dfrac{t}{T}\]

Here, T is the half-life.

In one second, the number of half-lives is,
\[n = \dfrac{1}{1}\]
\[ \Rightarrow n = 1\]

We can calculate the number of disintegrations per sec after one second using the equation,
\[N = {N_0}{\left( {\dfrac{1}{2}} \right)^n}\]

Here, \[{N_0}\] is the actual number of nuclei present in the sample at \[t = 0\] and n is the number of half lives.

Substitute \[n = 1\] in the above equation.
 \[N = {N_0}{\left( {\dfrac{1}{2}} \right)^1}\]
\[ \Rightarrow N = \dfrac{{{N_0}}}{2}\]
\[ \Rightarrow N = \dfrac{{1000}}{2}\]
\[ \Rightarrow N = 500\]

Therefore, the number of disintegrations/sec after one second is \[x = 500\].

In three seconds, the number of half lives becomes,
\[n = \dfrac{{3\,s}}{{1\,s}}\]
\[n = 3\]

We calculate the number of disintegrations per sec after three seconds using the equation,
\[N = {N_0}{\left( {\dfrac{1}{2}} \right)^n}\]

Here, \[{N_0}\] is the actual number of nuclei present in the sample at \[t = 0\] and n is the time.

Substitute \[n = 3\] in the above equation.
 \[N = {N_0}{\left( {\dfrac{1}{2}} \right)^3}\]
\[ \Rightarrow N = \dfrac{{{N_0}}}{8}\]
\[ \Rightarrow N = \dfrac{{1000}}{8}\]
\[ \Rightarrow N = 125\]

Therefore, the number of disintegrations/sec after one second is \[y = 125\].

Now, \[\dfrac{y}{x} = \dfrac{{125}}{{500}}\]
 \[\dfrac{y}{x} = \dfrac{1}{4}\]

Note:To solve such types of questions students must recognize which formula from the decay equation fits our given criteria. In this question, we were asked to determine the number of disintegrations per sec for different times. Therefore, we have used the decay equation to relate the number of half lives. Number of half lives is the ratio of the same quantities, therefore, it is unitless.