To find the concentration of the new solution, we can follow these steps:
1. Calculate the moles of solute in the initial solution:
Given:
- The mass of solute (ammonium sulfate, [tex]\((NH_4)_2SO_4\)[/tex]) is [tex]\( 66.05 \)[/tex] grams.
- The molar mass of [tex]\((NH_4)_2SO_4\)[/tex] is [tex]\( 132.1 \)[/tex] [tex]\(\text{g/mol}\)[/tex].
The number of moles is calculated using the formula:
[tex]\[
\text{moles of solute} = \frac{\text{mass of solute}}{\text{molar mass}}
\][/tex]
[tex]\[
\text{moles of solute} = \frac{66.05 \text{ g}}{132.1 \text{ g/mol}} = 0.500 \text{ mol}
\][/tex]
2. Convert the initial volume of the solution from mL to L:
Given:
- The initial volume of the solution is [tex]\( 250 \)[/tex] mL.
To convert mL to L:
[tex]\[
\text{Volume in liters} = \frac{\text{Volume in mL}}{1000}
\][/tex]
[tex]\[
\text{Volume in liters} = \frac{250 \text{ mL}}{1000} = 0.250 \text{ L}
\][/tex]
3. Calculate the molarity of the initial solution:
Molarity (M) is given by:
[tex]\[
M = \frac{\text{moles of solute}}{\text{liters of solution}}
\][/tex]
[tex]\[
M = \frac{0.500 \text{ mol}}{0.250 \text{ L}} = 2.00 \text{ M}
\][/tex]
4. Determine the volume of the sample taken from the initial solution and its molarity:
Given:
- The volume of the sample taken is [tex]\( 10.0 \)[/tex] mL.
- The volume of the diluted solution is [tex]\( 50.0 \)[/tex] mL.
First, convert the sample volume and the diluted solution volume to liters:
[tex]\[
\text{Volume of the sample in liters} = \frac{10.0 \text{ mL}}{1000} = 0.010 \text{ L}
\][/tex]
[tex]\[
\text{Volume of the diluted solution in liters} = \frac{50.0 \text{ mL}}{1000} = 0.050 \text{ L}
\][/tex]
5. Use the dilution formula [tex]\( M_i V_i = M_1 V_f \)[/tex] to calculate the molarity of the new solution:
Where:
- [tex]\( M_i \)[/tex] = Initial molarity of the solution = [tex]\( 2.00 \text{ M} \)[/tex]
- [tex]\( V_i \)[/tex] = Initial volume of the sample = [tex]\( 0.010 \text{ L} \)[/tex]
- [tex]\( M_1 \)[/tex] = Final molarity of the new solution = ?
- [tex]\( V_f \)[/tex] = Final volume of the new (diluted) solution = [tex]\( 0.050 \text{ L} \)[/tex]
Rearrange the dilution formula to solve for [tex]\( M_1 \)[/tex]:
[tex]\[
M_1 = \frac{M_i V_i}{V_f}
\][/tex]
Substitute the given values:
[tex]\[
M_1 = \frac{2.00 \text{ M} \times 0.010 \text{ L}}{0.050 \text{ L}} = 0.40 \text{ M}
\][/tex]
Therefore, the concentration of the new solution is [tex]\( 0.400 \text{ M} \)[/tex]. The correct answer is [tex]\( 0.400 \text{ M} \)[/tex].
