# Question ff032

Jan 12, 2017

$\text{0.000625 M}$

#### Explanation:

The thing to remember about serial dilutions is that the total dilution factor is the product of the individual dilution factors that you use for each dilution.

For a serial dilution that consists of $n$ dilutions, you have

$\textcolor{b l u e}{\underline{\textcolor{b l a c k}{{\text{DF"_ "total" = "DF"_ 1 xx "DF"_ 2 xx ... xx "DF}}_{n}}}}$

Now, the dilution factor is simply the ratio that exists between

• the volume of the diluted solution and the volume of the concentrated solution
• the concentration of the concentrated solution and the concentration of the diluted solution

Mathematically, this is written as

$\text{DF" = V_"diluted"/V_"concentrated" = c_"concentrated"/c_"diluted}$

In your case, you're working with a $\text{5.00-mL}$ aliquot that is being diluted to a total volume of $\text{100.00 mL}$. You will have

$\left\{\begin{matrix}{V}_{\text{diluted" = "100.00 mL" \\ V_"concentrated" = "5.00 mL}}\end{matrix}\right.$

This means that the dilution factor for one dilution will be

"DF"_ 1 = (100.00 color(red)(cancel(color(black)("mL"))))/(5.00color(red)(cancel(color(black)("mL")))) = color(blue)(20)#

You can thus say that the concentration of the solution after the first dilution will be equal to

${c}_{\text{diluted 1" = c_"concentrated 1"/"DF}} _ 1$

${c}_{\text{diluted 1" = "5.00 M"/color(blue)(20) = "0.25 M}}$

Now, you're doing $3$ identical dilutions, so you can sue the first equation to calculate the total dilution factor for this serial dilution

${\text{DF"_ "total" = "DF"_ 1 xx "DF"_ 1 xx "DF}}_{1}$

$\text{DF"_ "total} = \textcolor{b l u e}{20} \times \textcolor{b l u e}{20} \times \textcolor{b l u e}{20} = \textcolor{b l u e}{8000}$

You can thus say that the concentration of the diluted solution after the third and final dilution will be

${c}_{\text{diluted 3" = c_"concentrated 1"/"DF"_ "total}}$

${c}_{\text{diluted 3" = "5.00 M"/color(blue)(8000) = "0.000625 M}} \to$ rounded to three sig figs

As you can see, the dilution factor for an individual dilution tells how concentrated the concentrated solution was compared with the diluted solution.

For a dilution factor of $\textcolor{b l u e}{20}$, we get that the concentrated solution was $\textcolor{b l u e}{20}$ times as concentrated as the diluted solution.

After $3$ such dilutions, we got to a point where the concentration of the initial solution, i.e. the concentrated sample, was $\textcolor{b l u e}{8000}$ times as concentrated as the final diluted solution.