# Question 12ca1

Mar 30, 2016

Here's what I got.

#### Explanation:

The underlying principle of a dilution is that you can decrease the concentration of the solution by

• keeping the number of moles of solute constant
• increasing the total volume of the solution

So, in order to dilute a solution you must increase its volume without changing the amount of solute it contains.

Let's assume that solution $\text{A}$ contains an unknown number of moles of solute, ${n}_{A}$. The initial concentration of the solution can be written as

${\left[\text{A}\right]}_{0} = {n}_{A} / {V}_{A 0}$

["A"]_0 = n_A/(0.3 * 10^(-3)"L") = (10/3 * 10^3 * n_A)color(white)(a)"mol L"^(-1)

After you mix this solution with solution $\text{B}$, the total volume of the resulting solution will be

${V}_{\text{total" = "0.3 mL" + "6.7 mL" = "7.0 mL}}$

Keeping in mind the fact tha the resulting solution must contain ${n}_{A}$ moles of solute $\text{A}$, you an say that its new concentration is equal to

["A"]_"dil" = n_A/(7.0 * 10^(-3)"L") = (1/7 * 10^3 * n_A)color(white)(a)"mol L"^(-1)

So, by what factor, $\text{D.F}$, was solution $\text{A}$ diluted? Divide the initial concentration by the final concentration to get

"D.F." = ( 10/3 * color(blue)(cancel(color(black)(10^3 * n_a))) color(red)(cancel(color(black)("mol L"^(-1)))))/(1/7 * color(blue)(cancel(color(black)(10^3 * n_a))) color(red)(cancel(color(black)("mol L"^(-1)))))

$\text{D.F.} = \frac{10}{3} \cdot 7 = \textcolor{g r e e n}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} 23.3 \textcolor{w h i t e}{\frac{a}{a}} |}}}$

This tells you that the resulting solution is $23.3$ times less concentrated than solution $\text{A}$ compared with the initial concentration of $\text{A}$.

An interesting thing to notice here is that you can get the dilution factor by dividing the final volume by the initial volume of the solution

"D.F." = (7.0 color(red)(cancel(color(black)("mL"))))/(0.3color(red)(cancel(color(black)("mL")))) = color(green)(bar(ul(|color(white)(a/a)23.3color(white)(a/a)|))) -> for solution $\text{A}$

The exact same approach can be used to find the dilution of $\text{B}$. This time, the initial volume will be equal to $\text{6.7 mL}$ and the final volume to $\text{7.0 mL}$

"D.F." = (7.0color(red)(cancel(color(black)("mL"))))/(6.7color(red)(cancel(color(black)("mL")))) = color(green)(bar(ul(|color(white)(a/a)1.04color(white)(a/a)|))) -># for solution $\text{B}$

So, to sum this, the dilution factor for a given dilution is equal to

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \text{D.F." = "initial concentration"/"final concentration" = "final volume"/"initial volume} \textcolor{w h i t e}{\frac{a}{a}} |}}}$