# Question 03711

Feb 4, 2016

$\text{0.214 mL}$

#### Explanation:

This one is a little tricky because it wants you to perform a dilution by adding solute to the solvent, and not the other way around.

Here's how you can approach thus type of problems.

When performing a regular dilution, the dilution factor is used to express the ratio between the concentration of the stock solution and the concentration of the target solution by using the ratio between their two volumes.

Starting from the equation for dilution calculations

$\textcolor{b l u e}{{c}_{1} \times {V}_{1} = {c}_{2} \times {V}_{2}} \text{ }$, where

${c}_{1}$, ${V}_{1}$ - the concentration and volume of the stock solution
${c}_{2}$, ${V}_{2}$ - the concentration and volume of the target solution

you can rearrange the terms to get

${c}_{1} / {c}_{2} = {V}_{2} / {V}_{1}$

This is the dilution factor, $\text{D.F}$, for your dilution

$\textcolor{b l u e}{\text{D.F.} = {V}_{2} / {V}_{1}}$

In your case, you want to perform a $1 : 38$ dilution of an antibiotic solution. This is equivalent to having a dilution factor equal to $38$

$\text{D.F.} = 38 = {V}_{2} / {V}_{1}$

Now, do not be confused by the fact that you're adding the antibiotic solution to the saline solution!

Let's say that you will end up using a volume of $x$ milliliters of antibiotic solution. This will be the initial volume of the solution.

${V}_{1} = x \textcolor{w h i t e}{a} \text{mL}$

Since you're adding this solution to $\text{7.91 mL}$ of sterile saline, the final volume of the solution will be

${V}_{2} = \text{7.91 mL" + xcolor(white)(a)"mL" = (7.91 + x)color(white)(a)"mL}$

Plug this into the equation for the dilution factor to get

$38 = {V}_{2} / {V}_{1}$

$38 = \left(\left(7.91 + x\right) \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{mL"))))/(xcolor(red)(cancel(color(black)("mL}}}}\right) = \frac{7.91 + x}{x}$

This means that you have

$38 \cdot x = 7.91 + x$

$37 \cdot x = 7.91 \implies x = \frac{7.91}{37} = 0.214$

Therefore, adding $\text{0.214 mL}$ of antibiotic solution to $\text{7.91 mL}$ of sterile saline solution will result in a $1 : 38$ dilution of the antibiotic solution.

V_"antibiotic" = color(green)("0.214 mL") -># rounded to thre sig figs

So, remember that dilutions are all about keeping the number of moles of solute (or the amount, if you will) constant while increasing the volume of the solution.

So always try to think

final volume $\to$ initial volume $\to$ dilution factor

or

initial concentration $\to$ final concentration $\to$ dilution factor