Question

Show that the mass of radium with an activity of 1 curie is almost I gram.
The half life is 1600 years and I curie=$3.7 \times {10^{10}}$ disintegrations per second.

Answer 1

Hint: In the above question, the activity of the radium sample is given. As the half life of the sample is given and the disintegration per second is also given, we can find the amount of sample integrated and amount of sample left for us. Hence, we can find the mass easily.

Complete answer:
Firstly, let us write down the given terms and quantities,
$\eqalign{
  & m = 1gm,{T_{\dfrac{1}{2}}} = 1600years \cr } $
Secondly, we must determine the number of atoms in radium. We can use molar mass to find the number of atoms,
$\eqalign{
  & N = 1gm \times \dfrac{{mol}}{{226gm}} \times \dfrac{{6.022 \times {{10}^{23}}atoms}}{{mol}} \cr
  & \Rightarrow N = 2.6646 \times {10^{21}}atoms \cr} $
Now, the half life of the radium sample is given and N is found out earlier, so,
$\eqalign{
  & R = \dfrac{{0.693N}}{{{T_{\dfrac{1}{2}}}}} \cr
  & \Rightarrow R = \dfrac{{0.693 \times 2.6646 \times {{10}^{21}}}}{{1.6 \times {{10}^3}y}} \times \dfrac{{1y}}{{3.156 \times {{10}^7}s}} \cr
  & \therefore R = 3.66 \times {10^{10}}Bq \cr} $
Hence, we can find the value of R in this way.

Additional information:
As per radioactive decay law, the radioactive decays per unit time are directly proportional to the total number of nuclei of radioactive compounds in the sample. Radioactive decay is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered radioactive. Three of the most common types of decay are alpha decay, beta decay and gamma decay. In all these decays, one or more particles or photons are emitted. The weak force is the mechanism that is responsible for beta decay. The radioactive process is a random process at a level of single atoms. In quantum theory, it is impossible to predict when an atom will decay, regardless of how long the atom has existed.

Note:
The probability of when an atom will decay in the universe is impossible to know, because of which the term half life is introduced. This half life expresses the decay rate of the group of atoms, particularly, identical atoms. The half life of radioactive atoms has a huge range.