Atoms and activity in one gram of radium

Problem

How many atoms are contained in $1{,}0\ \text{g}$ of radium-226 ($A = 226$)? Knowing that its half-life is $\tau_{1/2} = 1600\ \text{years}$, calculate the activity of the sample in becquerel. Use $N_A = 6{,}022 \cdot 10^{23}$, $1\ \text{year} \approx 3{,}156 \cdot 10^{7}\ \text{s}$.

Solution

Number of atoms. With molar mass $A = 226\ \text{g/mol}$:

$N = \frac{m}{A}\,N_A = \frac{1{,}0}{226} \cdot 6{,}022 \cdot 10^{23} \approx 2{,}66 \cdot 10^{21}$

Half-life in seconds.

$\tau_{1/2} = 1600 \cdot 3{,}156 \cdot 10^{7}\ \text{s} \approx 5{,}05 \cdot 10^{10}\ \text{s}$

Decay constant.

$\lambda = \frac{\ln 2}{\tau_{1/2}} = \frac{0{,}693}{5{,}05 \cdot 10^{10}} \approx 1{,}37 \cdot 10^{-11}\ \text{s}^{-1}$

Activity. The activity in becquerel is $A = \lambda N$:

$A = 1{,}37 \cdot 10^{-11} \cdot 2{,}66 \cdot 10^{21} \approx 3{,}65 \cdot 10^{10}\ \text{Bq}$

$\ev{N \approx 2{,}66 \cdot 10^{21}; \quad A \approx 3{,}65 \cdot 10^{10}\ \text{Bq}}$

This value ($\approx 1\ \text{Ci}$) is no coincidence: the activity of one gram of radium-226 is the historical definition of the curie, the unit introduced precisely from this element.

Connections

Topics: Nuclear physics Concepts: Radioactive decay