What units to use to compute the mass of a radioactive material?

Question

So, I was practising some radioactivity physics problems for a test and I got on an easy but quite tricky question :

Given the element $\rm Co^{60}_{27}$ with a half life of $t_{1/2} = 5.2 $ years and its radioactivity is equal to $A = 0.5~\mathrm{ uBq}$ Compute its mass $m$ .

After using some formulas I got to this :

$$ m = \frac{M\cdot t_{1/2}\cdot A}{\ln2} $$

The way I got to this is as follows :

$$ A = N_0 \cdot \lambda$$ $$ A = \frac{m \cdot N_a}{M} \cdot \frac{\ln2}{t_{1/2}}$$

Where :

1. $M$ is the molar mass of $Co^{60}$
2. $\lambda$ is the radioactivity decay constant
3. $N_a$ is the avogadro num

My question is that whether I should convert the $t_1/2$ to seconds or just use years ? Also should I convert $A$ to $\rm Bq$ instead of $\rm uBq$ ? Or is it possible to work directly without converting the units ?

Answer 1

Dimensional analysis is your friend here. A Becquerel (Bq) is decays per second; working in seconds and becquerels (rather than years and µBq) is the only way your numbers are going to come out correctly.

Now a year is of course just $3.154\cdot 10^{7}$ seconds - so whether you think you are "converting" when you write $5.2 ~(3.154\cdot 10^{7}~\rm{s})$ is a matter of opinion. Same with $\rm{1~\mu Bq = 10^{-6} ~Bq}$

And the same applies to the units of mass, of course. If you use molar mass in kg, then your answer will be in kg; if you use g, the answer will be g.