Vanadium has two naturally occurring isotopes, 50-V with an atomic mass of 49.9472 amu and 51-V with an atomic mass of 50.9440. The atomic weight of vanadium is 50.9415. What is the percent abundance of the vanadium isotopes?

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Vanadium has two naturally occurring isotopes, 50-V with an atomic mass of 49.9472 amu and 51-V with an atomic mass of 50.9440. The atomic weight of vanadium is 50.9415. What is the percent abundance of the vanadium isotopes?

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Md Ariful Islam · Answer 1 · Apr 02, 2024 09:48:30

The idea with that have two naturally occurring is that the percent abundances of those two must add up to give $100%$.

In calculations, it is often easier to work with decimal abundances, which are simply percent abundances divided by $100$.

So, if you're working with decimal abundances, you can say for sure that if $x$ is the decimal abundance of the first isotope, then the decimal abundance of the second isotope will have to be $(1-x)$, since

$x + (1-x) = 1$

The decimal abundances of the two isotopes must add up to give $1$.

Now, an element's relative is calculated by taking the weighted average of the atomic masses of its naturally occurring isotopes.

$color(blue)("relative atomic mass" = sum_i ("isotope"_i xx "abundance"_i))$

So, if we take $x$ to be the decimal abundance of $""^50"V"$ and $(1-x)$ as the decimal abundance of $""^51"V"$, you can say that

$49.9472 color(red)(cancel(color(black)("u"))) xx x + 50.9440 color(red)(cancel(color(black)("u"))) xx (1-x) = 50.9415 color(red)(cancel(color(black)("u")))$

Solve this equation for $x$ to get

$49.9472 * x + 50.9440 - 50.9440 * x = 50.9415$

$0.9968 * x = 0.0025 implies x = 0.0025/0.9968 = 0.00250803$

Since $x$ represents the decimal abundance of $""^50"V"$, it follows that the decimal abundance of $""^51"V"$ will be

$1 - x = 1 - 0.00250803 = 0.9974920$

The percent abundances of the two isotopes will be

$""^50"V: " 0.002508 xx 100 = color(green)(0.250803%)$

$""^51"V: " 0.9974920 xx 100 = color(green)(99.7492%)$

I'll leave the values rounded to six , the number of sig figs you have for the atomic masses of the isotopes and for the relative atomic mass of the element.

Vanadium has two naturally occurring isotopes, 50-V with an atomic mass of 49.9472 amu and 51-V with an atomic mass of 50.9440. The atomic weight of vanadium is 50.9415. What is the percent abundance of the vanadium isotopes?

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