To solve the problem where you need to prepare [tex]\(2.00 \, \text{L}\)[/tex] of a [tex]\(0.100 \, \text{M} \, \text{NH}_4\text{NO}_3\)[/tex] solution from a [tex]\(1.75 \, \text{M}\)[/tex] stock solution, you can use the dilution equation given by:
[tex]\[ M_i V_i = M_f V_f \][/tex]
Where:
- [tex]\( M_i \)[/tex] is the initial molarity ([tex]\(1.75 \, \text{M}\)[/tex]).
- [tex]\( V_i \)[/tex] is the initial volume (the volume of the stock solution we need to find).
- [tex]\( M_f \)[/tex] is the final molarity ([tex]\(0.100 \, \text{M}\)[/tex]).
- [tex]\( V_f \)[/tex] is the final volume ([tex]\(2.00 \, \text{L}\)[/tex]).
### Step-by-Step Solution
1. Identify and List Known Values:
- [tex]\( M_i = 1.75 \, \text{M} \)[/tex]
- [tex]\( M_f = 0.100 \, \text{M} \)[/tex]
- [tex]\( V_f = 2.00 \, \text{L} \)[/tex]
2. Set Up the Dilution Equation:
[tex]\[ 1.75 \times V_i = 0.100 \times 2.00 \][/tex]
3. Solve for [tex]\( V_i \)[/tex]:
[tex]\[ V_i = \frac{0.100 \times 2.00}{1.75} \][/tex]
[tex]\[ V_i = \frac{0.200}{1.75} \][/tex]
[tex]\[ V_i \approx 0.1142857142857143 \, \text{L} \][/tex]
4. Convert Volume to Milliliters:
Because [tex]\( V_i \)[/tex] is usually easier to measure in milliliters, we convert liters to milliliters:
[tex]\[ V_i \approx 0.1143 \, \text{L} \times 1000 \, \text{mL/L} \approx 114 \, \text{mL} \][/tex]
### Conclusion
You need to measure approximately [tex]\(114 \, \text{mL}\)[/tex] of the [tex]\(1.75 \, \text{M}\)[/tex] stock solution and dilute it with water to a total volume of [tex]\(2.00 \, \text{L}\)[/tex] to obtain a [tex]\(0.100 \, \text{M}\)[/tex] solution.
Therefore, the correct instruction to follow is:
- Measure [tex]\(114 \, \text{mL}\)[/tex] of the [tex]\(1.75 \, \text{M}\)[/tex] solution, and dilute it to [tex]\(2.00 \, \text{L}\)[/tex].
