# "750. mL" of a "5.0 M" solution needs to be diluted to "1.0 M". How much water should be added to reach this concentration?

Mar 3, 2017

$3.0 \cdot {10}^{3} \text{mL}$

#### Explanation:

The trick here is to realize that because the amount of solute must remain constant in a dilution, the decrease in concentration must be equal to the increase in volume.

In other words, diluting a solution will decrease its concentration by a factor $\text{DF}$ and increase its volume by the same factor $\text{DF}$.

This factor is called the dilution factor and can be written as

color(blue)(ul(color(black)("DF" = overbrace(c_"concentrated"/c_"diluted")^(color(red)("decrease in concentration")) = overbrace(V_"diluted"/V_"concentrated")^(color(purple)("increase in volume")) )))

In your case, the concentration decreases by a factor of

"DF" = (5.0 color(red)(cancel(color(black)("M"))))/(1.0color(red)(cancel(color(black)("M")))) = color(blue)(5)

which means that the volume must increase by factor of $\textcolor{b l u e}{5}$

${V}_{\text{diluted" = color(blue)(5) * "750. mL" = "3750 mL}}$

This means that you must add

"volume of water" = "3750 mL" - "750. mL" = color(darkgreen)(ul(color(black)(3.0 * 10^(3)color(white)(.)"mL")))

of water to your concentrated solution in order to dilute it from $\text{5 M}$ to $\text{1 M}$.

The answer is rounded to two sig figs.