Question

how to calculate volume of concentrated solution needed to prepare a diluted solution
example: how many milliliters of a 0.20 m solution of copper(ii) sulfate solution must be used to prepare 250.00 ml of a 0.0040 m solution?

1. convert the volume of diluted solution from milliliters to liters.

$v_{diluted\\ solution}=250.00\\ ml\times\frac{1\\ l}{1000\\ ml}=0.25000\\ l$

1. calculate the number of moles of solute contained in the diluted solution.

$n_{solute}=0.25000\\ l\\ solution\times\frac{0.0040\\ mol\\ cuso_4}{1\\ l\\ solution}=0.0010\\ mol\\ cuso_4$
3 the number of moles of solute present in both the concentrated and diluted solutions remains the same. calculate the volume (in milliliters) of the concentrated solution needed to make the dilute solution.
$v_{concentrated\\ solution}=0.0010\\ mol\\ cuso_4\times\frac{1\\ l\\ solution}{0.20\\ mol\\ cuso_4}\times\frac{1000\\ ml}{1\\ l}=5.0\\ ml\\ solution$
an alternative way to solve for the volume is to use the following relationship:
$m_iv_i = m_fv_f$
where $m_i$ and $v_i$ are the initial molarity and volume of the concentrated solution and $m_f$ and $v_f$ are the final molarity and final volume of the dilute solution

Explanation:

Step1: Convert volume to liters

$V_{diluted\ solution}=250.00\ mL\times\frac{1\ L}{1000\ mL}=0.25000\ L$

Step2: Calculate moles in diluted solution

$n_{solute}=0.25000\ L\ solution\times\frac{0.0040\ mol\ CuSO_4}{1\ L\ solution}=0.0010\ mol\ CuSO_4$

Step3: Calculate volume of concentrated solution

$V_{concentrated\ solution}=0.0010\ mol\ CuSO_4\times\frac{1\ L\ solution}{0.20\ mol\ CuSO_4}\times\frac{1000\ mL}{1\ L}=5.0\ mL\ solution$

Answer:

$5.0\ mL$