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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%
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%
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