Question

Half-life of radium is 1620 years. How many radium nuclei decay in 5 hours in 5 gram radium? (Atomic weight of radium = 223)
(A) $ 9.1\times {{10}^{12}} $
(B) $ 3.15\times {{10}^{15}} $
(C) $ 1.72\times {{10}^{20}} $
(D) $ 3.3\times {{10}^{17}} $

Answer 1

Hint: Radioactive decay is the deterioration of an unstable particle with an emanation of radiation. As a radioisotope molecule decays to a more stable particle, it produces radiation just a single time. To transform from an unstable particle to a completely stable particle, one may require a few disintegration steps and radiation will be emitted at each progression. Nonetheless, when the molecule arrives at a steady state, no more radiation is emitted. The decay of radioactive components happens at a fixed rate. The half-life of a radioisotope is the time needed for one portion of the measure of unstable material to decay into a more stable material. For instance, a source will have an intensity of 100% when new. At one half-life, its power will be sliced to half of the first one. At two half-lives, it will have a power of 25% of formedr source. After ten half-lives, short of what one-thousandth of the first action will remain. Despite the fact that the half-life design is the equivalent for each radioisotope, the length of a half-life is extraordinary.
Number of nuclei decayed is equal to the present number of nuclei minus the number of nuclei left.
The formula for number of nuclei left in the system is given by:
 $ N={{N}_{0}}{{e}^{-\lambda t}} $
Where $ \lambda =\dfrac{0.693}{{{t}_{{}^{1}/{}_{2}}}} $ .

Complete step by step solution
We are given that
 $ {{t}_{{}^{1}/{}_{2}}} $ $ \text{of Radium = 1620 years} $
t = 5 hours
Mass of Radium = 5 gm
Atomic mass of Radium = 223 gm
So,
 $ \begin{align}
  & \lambda =\dfrac{0.693}{{{t}_{{}^{1}/{}_{2}}}} \\
 & \lambda =\dfrac{0.693}{1620\times 365\times 24}h{{r}^{-1}} \\
\end{align} $
Also,
 $ \begin{align}
  & {{N}_{0}}=n{{N}_{A}} \\
 & {{N}_{0}}=\dfrac{m}{M}{{N}_{A}} \\
 & {{N}_{0}}=\dfrac{5}{223}\times 6.022\times {{10}^{23}} \\
 & {{N}_{0}}=1.292\times {{10}^{22}} \\
\end{align} $
Therefore, number of nuclei decayed = $ N-{{N}_{0}} $
 $ \begin{align}
  & N-{{N}_{0}}={{N}_{0}}-{{N}_{0}}{{e}^{-\lambda t}} \\
 & ={{N}_{0}}\left( 1-{{e}^{-\lambda t}} \right) \\
 & =1.292\times {{10}^{22}}\left[ 1-\exp \left( -\dfrac{0.693}{1620\times 365\times 24}\times 5 \right) \right] \\
 & =3.152\times {{10}^{-7}}\times {{10}^{22}} \\
 & =3.152\times {{10}^{15}} \\
\end{align} $
Therefore, the number of nuclei of Radium decayed is $ 3.152\times {{10}^{15}} $ and option (B) is the correct answer.

Note
Remember 5gm is not the present number of nuclei but is the given mass for radium for which number of nuclei present need to be calculated first before calculating the number of nuclei decayed.
ORIGEN is a PC code framework for figuring the development, rot, and preparing of radioactive materials. ORIGEN utilizes a grid dramatic strategy to settle an enormous arrangement of coupled, straight, first-order ordinary differential conditions with steady coefficients.