How can I calculate the relative abundance of three isotopes? The letters have nothing to do with the periodic table. 22y = 22.341 amu, 23y =...

Saturday, February 28, 2015

How can I calculate the relative abundance of three isotopes? The letters have nothing to do with the periodic table. 22y = 22.341 amu, 23y =...

Denote the abundances in numbers (not percent) as $\displaystyle{a}_{{22}},$ $\displaystyle{a}_{{23}}$ and $\displaystyle{a}_{{24}}.$ Then if we assume that there are no other isotopes found in nature,

$\displaystyle{a}_{{22}}+{a}_{{23}}+{a}_{{24}}={1}.$

Also, by the definition of relative atomic mass:

$\displaystyle{23.045}={22.341}\cdot{a}_{{22}}+{23.041}\cdot{a}_{{23}}+{24.941}\cdot{a}_{{24}}.$

There is no more information and we have only two linear equations for three unknowns. Usually this means infinitely many solutions.

From the first equation $\displaystyle{a}_{{22}}={1}-{a}_{{23}}-{a}_{{24}},$ substitute it into the second equation:

$\displaystyle{23.045}={22.341}\cdot{\left({1}-{a}_{{23}}-{a}_{{24}}\right)}+{23.041}\cdot{a}_{{23}}+{24.941}\cdot{a}_{{24}},$ or

$\displaystyle{23.045}-{22.341}={\left({23.041}-{22.341}\right)}\cdot{a}_{{23}}+{\left({24.941}-{22.341}\right)}\cdot{a}_{{24}},$ ...

$\displaystyle{a}_{{22}}+{a}_{{23}}+{a}_{{24}}={1}.$

Also, by the definition of relative atomic mass:

$\displaystyle{23.045}={22.341}\cdot{a}_{{22}}+{23.041}\cdot{a}_{{23}}+{24.941}\cdot{a}_{{24}}.$

There is no more information and we have only two linear equations for three unknowns. Usually this means infinitely many solutions.

From the first equation $\displaystyle{a}_{{22}}={1}-{a}_{{23}}-{a}_{{24}},$ substitute it into the second equation:

$\displaystyle{23.045}={22.341}\cdot{\left({1}-{a}_{{23}}-{a}_{{24}}\right)}+{23.041}\cdot{a}_{{23}}+{24.941}\cdot{a}_{{24}},$ or

$\displaystyle{23.045}-{22.341}={\left({23.041}-{22.341}\right)}\cdot{a}_{{23}}+{\left({24.941}-{22.341}\right)}\cdot{a}_{{24}},$ or

$\displaystyle{0.704}={0.7}\cdot{a}_{{23}}+{2.6}\cdot{a}_{{24}}.$

Therefore $\displaystyle{a}_{{23}}=\frac{{{0.704}-{2.6}\cdot{a}_{{24}}}}{{0.7}}\approx{1.006}-{3.714}\cdot{a}_{{24}}.$ So if we would know $\displaystyle{a}_{{24}}$ (or $\displaystyle{a}_{{23}},$ or $\displaystyle{a}_{{22}}$ ), we could find the other abundances.

To find the relative abundances in percent, we have to multiply $\displaystyle{a}_{{24}},$ $\displaystyle{a}_{{23}}$ and $\displaystyle{a}_{{22}}$ by $\displaystyle{100}.$ The formal answer for your question is "not enough data." Also note that the atomic masses of isotopes are almost always integers, while your numbers aren't.