The question requires us to calculate the amount of a 0.250 M sucrose solution necessary to prepare 400.0 mL of a 0.0310 M solution.
The question provided the following information:
molarity of more concentrated solution = C(A) = 0.250 M = 0.250 mol/L
molarity of less concentrated solution = C(B) = 0.0310 M = 0.0310 mol/L
volume of less concentrated solution = V(B) = 400.0 mL
The question refers to a simple dilution process, where we use a more concentrated solution (A) to preare a less concentrated solution (B). The amount of sucrose in both solutions doesn't change, thus we can say that the number of moles of sucrose in A is the same as it is in B:
From the definition of molar concentration (also called molarity), we can say that the number of moles of a solution corresponds to its concentration (in mol/L) multiplied by its volume (in L):
Considering the two equations above, we can write:
We can also rearrange this equation in order to calculate the volume of more concentrated solution, V(A), necessary:
Now, we can use the values given by the question in the equation to calculate V(A):
Therefore, 49.6 mL of a 0.250 M sucrose solution are necessary to prepare 400.0 mL of a 0.0310 M solution. Note that, as 400.0 mL is the total volume of the solution B, this solution should be prepared with 49.6 mL of solution A + 350.4 mL of water.