Challenge Level

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.*

What information does this give you about the possible initial composition of the sample?

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?

NOTES AND BACKGROUND

Data sourced from http://ie.lbl.gov/education/isotopes.htm.

The following Wikipedia page might also be of interest: http://en.wikipedia.org/wiki/Cobalt-59.