Question

Mean life of a radioactive sample is 100 seconds. Then its half-life (in minutes) is
a) 0.693
b) 1
c) \[{10^{ - 4}}\]
d) 1.155

Answer 1

Hint:The process of spontaneous emission of radiations such as \[\alpha ,\beta ,\gamma \] rays by an element is called radioactivity and the elements which emit these radiations are known as radioactive elements. Once, the elements emit these radioactive radiations, they start to decompose and the time taken for an element to decay is calculated using its half-life.

Formula Used :
The half-life of a radioactive element is,
\[{T_{\frac{1}{2}}} = \dfrac{{0.6931}}{\lambda }\]
Where,
\[{T_{\frac{1}{2}}}\] is the half-life of a radioactive element
\[\lambda \] is the decay constant

The mean life of a radioactive element is,
\[\tau = \dfrac{1}{\lambda }\]
Where,
\[\tau \] is the mean-life

Complete step by step solution:
Given, \[\tau \]= 100 s
To find, \[{T_{\frac{1}{2}}}\] = ? min

The time taken for a given sample of N atoms to decay is difficult to calculate, hence we can calculate its half-life. The half-life \[\left( {{T_{\frac{1}{2}}}} \right)\]of the atoms can be defined as the time required for the number of atoms initially present to decay to its half of the amount present initially.

Since the time taken for complete decay varies from zero to infinite, so we can calculate the average or mean life\[\left( \tau \right)\], which is defined as the sum or integration of life time of all nuclei to the total number of nuclei present in the initial time.

We know the half-life is \[{T_{\frac{1}{2}}} = \dfrac{{0.6931}}{\lambda }\] and mean-life is \[\tau = \dfrac{1}{\lambda }\].

By rearranging the above formulae we get,
\[{T_{\frac{1}{2}}} = 0.6931\left( {\dfrac{1}{\lambda }} \right)\]
\[{T_{\frac{1}{2}}} = 0.6931(\tau )\]
\[{T_{\frac{1}{2}}} = 0.6931{\rm{x}}100\]
\[{T_{\frac{1}{2}}} = 69.3{\rm{ s}}\]

We need to find \[{T_{\frac{1}{2}}}\] in minutes,
\[{T_{\frac{1}{2}}} = \dfrac{{69.3}}{{60}} = 1.155{\rm{ min}}\]
\[{T_{\frac{1}{2}}} = 1.155{\rm{ min}}\]

So, option (d) is correct.

Note: Remember to convert the half-life to minutes as asked in the question. If you are not asked to convert it into minutes, then skip the last two steps and your answer will be in seconds. The decay constant (\[\lambda \]) is different for different elements.