Rate Of Radioactive Decay
Category : JEE Main & Advanced
“According to the law of radioactive decay, the quantity of a radio-element which disappears in unit time (rate of disintegration) is directly proportional to the amount present.”
The law of radioactive decay may also be expressed mathematically.
Suppose N0 be the number of atoms of the radioactive element present at the commencement of observation, \[t=0\] and after time t, the number of atoms remaining unchanged is \[{{N}_{t}}\]. The rate of disintegration \[\left( -\frac{d{{N}_{t}}}{dt} \right)\]at any time t is directly proportional to N. Then,\[-\frac{d{{N}_{t}}}{dt}\]= lN
where l is a radioactive constant or decay constant.
Various forms of equation for radioactive decay are,
\[{{N}_{t}}={{N}_{0}}{{e}^{-\lambda t}}\]; \[\log {{N}_{0}}-\log {{N}_{t}}=0.4343\,\lambda t\]
\[\log \frac{{{N}_{0}}}{{{N}_{t}}}=\frac{\lambda t}{2.303}\]; \[\lambda =\frac{2.303}{t}\log \frac{{{N}_{0}}}{{{N}_{t}}}\]
This equation is similar to that of first order reaction, hence we can say that radioactive disintegration are examples of first order reactions. However, unlike first order rate constant (K), the decay constant (l) is independent of temperature.
Rate of decay of nuclide is independent of temperature, so its energy of activation is zero.
(1) Half-life period (T1/2 or t1/2) : The half-life period of a radioelement is defined, as the time required by a given amount of the element to decay to one-half of its initial value.
\[{{t}_{1/2}}=\frac{0.693}{\lambda }\]
Now since l is a constant, we can conclude that half-life period of a particular radioelement is independent of the amount of the radioelement. In other words, whatever might be the amount of the radioactive element present at a time, it will always decompose to its half at the end of one half-life period.
Let the initial amount of a radioactive substance be \[{{N}_{0}}\]
Amount of radioactive substance left after n half-life periods
\[N={{\left( \frac{1}{2} \right)}^{n}}{{N}_{0}}\]
Total time T \[=n\times {{t}_{1/2}}\] where n is a whole number.
(2) Average-life period (T) : Since total decay period of any element is infinity, it is meaningless to use the term total decay period (total life period) for radioelements. Thus the term average life is used.
Average life (T)\[=\frac{\text{Sum of lives of the nuclei}}{\text{Total number of nuclei}}\]
Average life (T) of an element is the inverse of its decay constant, i.e., \[T=\frac{1}{\lambda }\], Substituting the value of l in the above equation,
\[T=\frac{{{t}_{1/2}}}{0.693}=1.44\,{{t}_{1/2}}\]
Thus, Average life (T) \[=1.44\times \text{Half life}({{T}_{1/2}})=\sqrt{2}\times {{t}_{1/2}}\]
Thus, the average life period of a radioisotope is approximately under-root two times of its half life period.
(3) Activity of population or specific activity : It is the measure of radioactivity of a radioactive substance. It is defined as ' the number of radioactive nuclei, which decay per second per gram of radioactive isotope.' Mathematically, if 'm' is the mass of radioactive isotope, then
\[\text{Specific activity}=\frac{\text{Rate of decay}}{m}=\frac{\lambda N}{m}=\lambda \times \frac{\text{Avogadro number}}{\text{Atomic mass in }g}\]
where N is the number of radioactive nuclei which undergoes disintegration.
(4) Radioactive equilibrium : Suppose a radioactive element A disintegrates to form another radioactive element B which in turn disintegrates to still another element C.
\[A\xrightarrow{{}}B\xrightarrow{{}}C\]
B is said to be in radioactive equilibrium with A if its rate of formation from A is equal to its rate of decay into C.
It is important to note that the term equilibrium is used for reversible reactions but the radioactive reactions are irreversible, hence it is preferred to say that B is in a steady state rather than in equilibrium state.
At a steady state, \[\frac{{{N}_{A}}}{{{N}_{B}}}=\frac{{{\lambda }_{B}}}{{{\lambda }_{A}}}=\frac{{{T}_{A}}}{{{T}_{B}}}=\frac{{{\left( {{t}_{1/2}} \right)}_{A}}}{{{\left( {{t}_{1/2}} \right)}_{B}}}\]
Thus at a steady state (at radioactive equilibrium), the amounts (number of atoms) of the different radioelements present in the reaction series are inversely proportional to their radioactive constants or directly proportional to their half-life and also average life periods.
(5) Units of radioactivity : The standard unit in radioactivity is curie (c) which is defined as that amount of any radioactive material which gives \[3.7\times {{10}^{10}}\] disintegration’s per second (dps), i.e.,1c = Activity of 1g of \[R{{a}^{226}}=3.7\times {{10}^{10}}dps\]
The millicurie (mc) and microcurie (mc) are equal to \[{{10}^{-3}}\] and \[{{10}^{-6}}\] curies i.e. \[3.7\times {{10}^{7}}\] and \[3.7\times {{10}^{4}}\]dps respectively.
\[1c={{10}^{3}}mc={{10}^{6}}\mu c\];\[1c=3.7\times {{10}^{10}}dps\]
\[1mc=3.7\times {{10}^{7}}dps\] ; \[1\mu c=3.7\times {{10}^{4}}dps\]
But now a day, the unit curie is replaced by rutherford (rd) which is defined as the amount of a radioactive substance which undergoes \[{{10}^{6}}\,dps.\] i.e., \[1\,rd={{10}^{6}}\,dps\]. The millicurie and microcurie correspondingly rutherford units are millirutherford (mrd) and microrutherford (mrd) respectively.
\[1\,c=3.7\times {{10}^{10}}\,dps=37\times {{10}^{3}}\,rd\]
\[1\,mc=3.7\times {{10}^{7}}\,dps=37\,rd\]
\[1\,\mu c=3.7\times {{10}^{4}}\,dps=37\,mrd\]
However, in SI system the unit of radioactivity is Becquerel (Bq)
1 Bq = 1 disintegration per second = 1 dps = 1mrd, \[{{10}^{6}}\,Bq=1\,rd\], \[3.7\times {{10}^{10}}\,Bq=1\,c\]
(6) The Geiger-Nuttal relationship : It gives the relationship between decay constant of an a- radioactive substance and the range of the a-particle emitted.
\[\log \lambda =A+B\log R\]
Where R is the range or the distance which an a-particle travels from source before it ceases to have ionizing power. A is a constant which varies from one series to another and B is a constant for all series. It is obvious that the greater the value of l the greater the range of the a-particle.
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