To determine the amount of copper that can be recovered after the final step given a series of chemical transformations, we must account for the percentage yield at each step. Here's the step-by-step solution:
1. Understanding the Initial Conditions:
- We start with 0.90 grams of copper.
2. Percentage Yield per Step:
- The percentage yield of copper in each step is [tex]\( 99\% \)[/tex]. This means we retain [tex]\( 99\% \)[/tex] of the copper after each transformation.
3. Convert the Percentage to a Decimal:
- [tex]\( 99\% \)[/tex] can be written as [tex]\( 0.99 \)[/tex] in decimal form.
4. Number of Steps:
- There are 4 steps in the chemical transformation series.
5. Calculate the Final Amount of Copper:
- In each step, the amount of copper is multiplied by [tex]\( 0.99 \)[/tex] to account for the yield.
- Since there are 4 steps, the final amount of copper [tex]\( A \)[/tex] can be calculated using the formula:
[tex]\[
A = \text{initial amount of copper} \times (\text{yield percentage})^n
\][/tex]
where [tex]\( n \)[/tex] is the number of steps.
6. Substitute the Values:
- Initial amount of copper = [tex]\( 0.90 \)[/tex] grams.
- Yield percentage per step = [tex]\( 0.99 \)[/tex].
- Number of steps ([tex]\( n \)[/tex]) = 4.
- Therefore:
[tex]\[
A = 0.90 \times (0.99)^4
\][/tex]
7. Perform the Calculation:
- [tex]\( 0.99^4 \approx 0.96059601 \)[/tex]
- So, [tex]\( A = 0.90 \times 0.96059601 \approx 0.864536409 \)[/tex]
Therefore, the amount of copper that can be recovered at the end of the last step is approximately [tex]\( 0.8645 \)[/tex] grams.
