The isotopes $Ra-226$ undergoes $\alpha$ decay with a half-life of $1620$ years. What is the activity of $1\;g$ of $Ra-226$? Avogadro number = $6.023 * 10^{23}\; /mole$.

Given,
Half life $T_{1/2} = 1620$ years = $5.1 * 10^{10}$ Sec
Mass of $Ra-226$ = 1 gm
Activity of $Ra-226\; ; $ $(\frac{dN}{dt}) = \; ?$

Now,

$\lambda = $  $\frac{0.693}{5.1 \; * \; 10^{10}}$ $ = 1.36 * 10^{-11}\; Sec^{-1}$

Again,

           226 gm of $Ra-226$ contains $6.023 * 10^{23}$ atoms


           1 gm of $Ra-226$ contains $\frac{6.023 \; * \; 10^{23}}{226}$ $ = 2.66 * 10^{21}$ atoms


           $\therefore \;\;\; N= 2.66 * 10^{21}$ atoms

Now,

           $\frac{dN}{dt}$ $= \lambda \,N = 1.36 * 10^{-11} * 2.66 * 10^{10}\; dis/Sec.$

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