To find the final volume \( V_f \) required to dilute the solution, we can use the dilution equation, which is given by:
\[ C_1 V_1 = C_2 V_2 \]
where:
- \( C_1 \) = initial concentration (2.19 M)
- \( V_1 \) = initial volume (97.9 mL)
- \( C_2 \) = final concentration (1.00 M)
- \( V_2 \) = final volume (unknown)
First, we need to convert the initial volume from mL to liters:
\[ V_1 = 97.9 , \text{mL} = 0.0979 , \text{L} \]
Now we can use the dilution equation. We have:
\[ C_1 V_1 = C_2 V_2 \]
Substituting in the known values:
\[ (2.19 , \text{M})(0.0979 , \text{L}) = (1.00 , \text{M})(V_2) \]
Calculating the left-hand side:
\[ 2.19 \times 0.0979 = 0.2147 , \text{mol} \]
Then we can set this equal to the expression for \( V_2 \):
\[ 0.2147 = 1.00 \times V_2 \]
This simplifies to:
\[ V_2 = 0.2147 , \text{L} \]
Now, rounding this result to three significant digits gives:
\[ V_2 \approx 0.215 , \text{L} \]
Thus, the final volume required to achieve a concentration of 1.00 M is
\[ \boxed{0.215 , \text{L}} \]