To solve the problem of finding the volume of the new solution when diluting a chemist's magnesium nitrate solution, we'll use the principle of conservation of moles during dilution. This is encapsulated in the formula:
[tex]\[ C_1 V_1 = C_2 V_2 \][/tex]
where:
- [tex]\( C_1 \)[/tex] is the initial concentration
- [tex]\( V_1 \)[/tex] is the initial volume
- [tex]\( C_2 \)[/tex] is the final concentration
- [tex]\( V_2 \)[/tex] is the final volume
Let's break it down step-by-step:
1. Identify initial parameters:
- [tex]\( C_1 = 2.13 \)[/tex] M (initial concentration)
- [tex]\( V_1 = 1.24 \)[/tex] liters (initial volume)
2. Identify the final concentration:
- [tex]\( C_2 = 1.60 \)[/tex] M (final concentration)
3. Rearrange the formula to solve for [tex]\( V_2 \)[/tex]:
[tex]\[ V_2 = \frac{C_1 V_1}{C_2} \][/tex]
4. Plug in the known values:
[tex]\[ V_2 = \frac{(2.13 \, \text{M}) \cdot (1.24 \, \text{L})}{1.60 \, \text{M}} \][/tex]
5. Calculate the final volume:
[tex]\[ V_2 = \frac{2.6412 \, \text{M} \cdot \text{L}}{1.60 \, \text{M}} \][/tex]
[tex]\[ V_2 = 1.65075 \, \text{L} \][/tex]
6. Round to three significant figures:
[tex]\[ V_2 \approx 1.65 \, \text{L} \][/tex]
Thus, the volume of the new solution is [tex]\( \boxed{1.65} \)[/tex] liters.
