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The element cobalt has one stable isotope: $^{59}$Co.

Chemists can construct several different isotopes of this element with varying numbers of neutrons. All are unstable; the five most stable are given in the following table (y=years, d=days, m=minutes).

Isotope Half life
$^{60}$Co 5.2714 y
$^{57}$Co 271.79 d
$^{56}$Co 77.27 d
$^{58}$Co 70.86 d
$^{55}$Co 17.53 h

I have a lump of pure cobalt, called sample $X$. I know that sample X is uniformly composed of differing quantities of the various isotopes of cobalt (including, possibly, some of the more unstable isotopes not listed in the table above).

I take a piece of sample $X$ of weight $100\textrm{ g}$ and leave it to decay.

After exactly $10$ days $68.402\textrm{ g}$ of cobalt remains in the sample.
What information does this give you about the possible initial composition of the sample?

For example, are there any isotopes which you can work out must be present in the sample? Use your common sense and simple calculations before attempting a numerically detailed analysis! You can assume that each isotope is present in a whole number of grams to start with.

After $90$ days in total $58.283\textrm{ g}$ of cobalt remains. After $360$ days $48.359\textrm{ g}$ of cobalt remains.

How accurately can you determine the composition of the original sample?


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