To calculate the concentration of the working solution, we need to follow a series of steps to determine how the concentration changes when the stock solution is diluted. Here is a detailed, step-by-step solution:
1. Determine the amount of solute in the stock solution:
- The volume of the stock solution is [tex]\(0.100 \, \text{L}\)[/tex].
- The concentration of the stock solution is [tex]\(0.0000382 \, \frac{\text{mol}}{\text{L}}\)[/tex].
The amount of solute (in moles) in the stock solution can be calculated using the formula:
[tex]\[
\text{Amount of solute} = \text{Volume of stock solution} \times \text{Concentration of stock solution}
\][/tex]
Substituting in the given values:
[tex]\[
\text{Amount of solute} = 0.100 \, \text{L} \times 0.0000382 \, \frac{\text{mol}}{\text{L}} = 3.82 \times 10^{-6} \, \text{mol}
\][/tex]
2. Calculate the concentration of the working solution:
- The total volume of the working solution is [tex]\(0.910 \, \text{L}\)[/tex].
- The amount of solute remains constant during dilution (it is the same as calculated above).
The concentration of the working solution can be determined using the formula:
[tex]\[
\text{Concentration of working solution} = \frac{\text{Amount of solute}}{\text{Volume of working solution}}
\][/tex]
Substituting the values:
[tex]\[
\text{Concentration of working solution} = \frac{3.82 \times 10^{-6} \, \text{mol}}{0.910 \, \text{L}} = 4.197802197802198 \times 10^{-6} \, \frac{\text{mol}}{\text{L}}
\][/tex]
3. Express the final concentration with the correct number of significant digits:
- The initial concentration [tex]\(0.0000382 \, \frac{\text{mol}}{\text{L}}\)[/tex] has 3 significant digits.
- The volume measurements (0.100 L and 0.910 L) each have 3 significant digits.
Therefore, the final concentration should also be expressed with 3 significant digits:
[tex]\[
\text{Concentration of working solution} = 4.20 \times 10^{-6} \, \frac{\text{mol}}{\text{L}}
\][/tex]
Thus, the concentration of the chemist's working solution is [tex]\( \boxed{4.20 \times 10^{-6} \, \frac{\text{mol}}{\text{L}}} \)[/tex].
