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How would you calculate the percent relative abundance of \[Cu - 63\] with the mass \[62.9296{\text{ }}g\] and \[Cu - 65\] with the mass \[64.9278{\text{ }}g\] , when the average mass of Cu is \[63.546\] ?

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Last updated date: 25th Feb 2024
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IVSAT 2024
Answer
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Hint: The relative abundance definition in science is the percentage of a specific isotope that happens in nature. The nuclear mass listed for a component on the periodic table is an average mass of all known isotopes of that component.

Complete step by step answer:
As you most likely are aware, the average nuclear mass of a component is determined by taking the weighted average of the nuclear masses of its normally occurring isotopes.
Step 1: Find the Average Atomic Mass
Basically, a component's normally occurring isotopes will contribute to the average nuclear mass of the component relative to their abundance.
\[avg.{\text{ }}atomic{\text{ }}mass\] =$\sum \left( {{\text{isotope}} \times {\text{abundance}}} \right)$
Step 2: Set Up the Relative Abundance Problem
With regards to the genuine count, it's simpler to use decimal abundances, which are basically percent abundances divided by \[100\] .
Thus, you realize that copper has two naturally occurring isotopes, \[copper - 63\] and \[copper - 65\] . This implies that their respective decimal abundance should amount to give\[1\] .
In the event that you take x to be the decimal bounty of \[copper - 63\] , you can say that the decimal abundance of \[copper - 65\] will be equivalent to \[1 - x\] .
So we can say that:
\[x \cdot 62.9296u + (1 - x) \cdot 64.9278u = 63.546u\]
Step 3: Solve for x to Get the Relative Abundance of the Unknown Isotope.
  To finding the value of x we get
\[62.9296 \cdot x - 64.9278 \cdot x = 63.546 - 64.9278\] \[1.9982 \cdot x = 1.3818\]
\[x = \] $\dfrac{{1.38181}}{{0.9982}}$
\[x = {\text{ }}0.69152\]
Step 4: Find percent abundance
This implies that the percent abundances of the two isotopes will be
\[69.152\% \]---->\[^{63}Cu\]
\[30.848\% \]------.\[^{65}Cu\]

Note:
 If a mass spectrum of the component was given, the relative rate isotope abundances are generally introduced as a vertical bar graph. The all-out may look as though it exceeds \[100\% ,\] however, that is because the mass spectrum works with relative rate isotope abundances.
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