Calculation of the half-life of a radioactive element (8299)

, por F_y_Q

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A sample of 300 grams of a radioactive element remains 18.75 grams after 24 hours. Calculate the half-life.

P.-S.

You must use the decay law, but referring to the mass of the substance:

\color[RGB]{2,112,20}{\bm{m = m_0\cdot e^{- \lambda\cdot t}}}

Solving for the value of \lambda, you get:

- \lambda = \frac{ln \frac{18.75\ \cancel{g}}{300\ \cancel{g}}}{86\ 400\ s} = \color[RGB]{0,112,192}{\bm{3.2\cdot 10^{-5}\ s^{-1}}}

You know that radioactive activity is related to the half-life and this to the half-life period:

\left \lambda = \frac{1}{\tau} \atop \tau = \frac{t_{1/2}}{ln\ 2} \right \}

Substituting and solving:

t_{1/2} = \frac{ln\ 2}{\lambda} = \fbox{\color[RGB]{192,0,0}{\bf 21\ 661\ s}}


Expressed in hours, it will be:

t_{1/2} = 21\ 661\ \cancel{s}\cdot \frac{1\ h}{3\ 600\ s} = \fbox{\color[RGB]{192,0,0}{\bf 6\ h}}