Sure, let's solve this step by step.
Given:
- The average atomic mass of natural copper is 63.5.
- Natural copper consists of two isotopes: [tex]\( ^{63}\text{Cu} \)[/tex] and [tex]\( ^{65}\text{Cu} \)[/tex].
- We need to find the percentage abundance of [tex]\( ^{63}\text{Cu} \)[/tex] and [tex]\( ^{65}\text{Cu} \)[/tex].
Let:
- [tex]\( x \)[/tex] be the percentage abundance of [tex]\( ^{63}\text{Cu} \)[/tex].
- [tex]\( 100 - x \)[/tex] be the percentage abundance of [tex]\( ^{65}\text{Cu} \)[/tex].
The average atomic mass can be represented as a weighted average of the masses of the two isotopes:
[tex]\[
63.5 = \left(\frac{x}{100}\right) \times 63 + \left(\frac{100 - x}{100}\right) \times 65
\][/tex]
First, set up the equation:
[tex]\[
63.5 = \left(\frac{x}{100}\right) \times 63 + \left(\frac{100 - x}{100}\right) \times 65
\][/tex]
To eliminate the fractions, multiply through by 100:
[tex]\[
63.5 \times 100 = x \times 63 + (100 - x) \times 65
\][/tex]
This simplifies to:
[tex]\[
6350 = 63x + 6500 - 65x
\][/tex]
Combine like terms:
[tex]\[
6350 = 6500 - 2x
\][/tex]
To isolate [tex]\( x \)[/tex], subtract 6350 from 6500:
[tex]\[
6500 - 6350 = 2x
\][/tex]
Simplify the left side:
[tex]\[
150 = 2x
\][/tex]
Finally, solve for [tex]\( x \)[/tex] by dividing both sides by 2:
[tex]\[
x = \frac{150}{2}
\][/tex]
[tex]\[
x = 75
\][/tex]
Therefore, the percentage abundance of [tex]\( ^{63}\text{Cu} \)[/tex] is 75%.
Since the total percentage abundance must add up to 100%, the percentage abundance of [tex]\( ^{65}\text{Cu} \)[/tex] is:
[tex]\[
100 - 75 = 25
\][/tex]
So, the percentage abundances are:
- [tex]\( ^{63}\text{Cu} \)[/tex]: 75%
- [tex]\( ^{65}\text{Cu} \)[/tex]: 25%
