# Question #a9870

##### 1 Answer

#### Explanation:

All you have to do here is keep track of the **number of moles** of solute as you progress towards the target solution.

For instance, the initial solution contains

#51.0 color(red)(cancel(color(black)("mL"))) * (1color(red)(cancel(color(black)("L"))))/(10^3color(red)(cancel(color(black)("mL")))) * "1.30 moles solute"/(1color(red)(cancel(color(black)("L")))) = "0.0663 moles solute"#

When you **dilute** this solution to a *total volume* of **constant**. This means that after the first dilution, the solution will contain

#"0.0663 moles solute " -># in atotal volumeof#"288 mL"#

Next, you take *second dilution*. Notice that

#"144 mL" = "288 mL"/2#

This tells you that the **half as many moles of solute** as the

#"0.0663 moles solute"/2 = "0.03315 moles solute " -># in a volume of#"144 mL"#

When you're diluting this sample by adding **total volume** of the resulting solution will be

#"144 mL " + " 161 mL" = "305 mL"#

This solution will contain **the same number of moles of solute** as you had in the **final concentration**, i.e. the concentration of the target solution, is equal to

#"0.03315 moles solute"/(305 * 10^(-3)"L") = color(green)(bar(ul(|color(white)(a/a)color(black)("0.109 M")color(white)(a/a)|)))#

The answer is rounded to three **sig figs**.

**ALTERNATIVE APPROACH**

You can solve this problem by using the **dilution factor**

#color(blue)(bar(ul(|color(white)(a/a)"DF" = V_"final"/V_"initial"color(white)(a/a)|)))#

Here

**diluted solution**

**concentrated solution**

In essence, the dilution factor tells you how concentrated the initial solution was compared with the *diluted solution*.

So, for the **first dilution**, you have

#"DF"_1 = (288 color(red)(cancel(color(black)("mL"))))/(51.0color(red)(cancel(color(black)("mL")))) = 5.647#

This means that the **times more concentrated** than the solution that resulted from the *first dilution*.

Now look at the **second dilution** -- this time, the starting volume is

#"DF"_2 = (305color(red)(cancel(color(black)("mL"))))/(144color(red)(cancel(color(black)("mL")))) = 2.118#

This means that the sample we took from the **times more concentrated** than the target solution.

We've thus performed a **serial dilution**. The **overall dilution factor**, **product** of the two dilution factors

#color(blue)(bar(ul(|color(white)(a/a)"DF"_ "serial" = "DF"_ 1 xx "DF"_ 2color(white)(a/a)|)))#

You will have

#"DF"_"serial" = 5.647 * 2.118 = 11.96#

This means that the initial **times more concentrated** than the *target solution*, i.e. the solution that resulted from the **second dilution**.

Therefore, the concentration of the target solution is

#c_"target" = c_"initial"/"DF"_"serial"#

#c_"target" = "1.30 M"/11.96 = color(green)(bar(ul(|color(white)(a/a)color(black)("0.109 M")color(white)(a/a)|)))#