Question

Potassium is mildly radioactive because one of its natural isotopes is radioactive: ⁴⁰K undergoes β^- decay with a half-life of t_1/2 = 1.248 × 10 ⁹ years. The isotopic abundances for potassium are:    ³⁹K 93.258 % (atom percent) stable   

   ⁴⁰K 0.0117 at % β- decay

⁴¹K 6.730 at % stable

Calculate the activity (measured in curies or becquerel) of 1 cubic centimeter of KCl.

Answer 1

For a given mass (in grams) of an isotope with atomic mass ${\displaystyle m_{\text{a}}}$ (in g/mol) and a half-life of $t_{1/2}$ (in s), the radioactivity can be calculated using:

${\displaystyle A_{\text{Bq}}={\frac {m}{m_{\text{a}}}}N_{\text{A}}{\frac {\ln 2}{t_{1/2}}}}$ ------Eq(1)

With $N_{{\text{A}}}$ = 6.02214076×1023 mol−1, the Avogadro constant.

Since ${\displaystyle m/m_{\text{a}}}$ is the number of moles , the amount of radioactivity can be calculated by:

${\displaystyle A_{\text{Bq}}=nN_{\text{A}}{\frac {\ln 2}{t_{1/2}}}}$ ------Eq(2)

Here given that, only 40K  undergoes Beta decay and it has 0.000117 atomic percent and the atomic mass of the potassium is 40 g/mol and the t1/2 given as 1.248 *109 years,

substuting all in the above eqaution 2,

ABq = 29.32 Bq.

Thank You