# Question 9932a

Jul 13, 2016

The solution is $\text{56% v/v}$ alcohol.

#### Explanation:

A solution's volume by volume percent concentration, $\text{% v/v}$, is calculated as the volume of solute, which in your case is the alcohol, expressed in milliliters, present in $\text{100 mL}$ of solution.

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \text{% v/v" = "mL of solute/100 mL solution} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

In your case, you know that you get $\text{225 mL}$ of solute in $\text{400 mL}$ of solution. To find the solution's volume by volume percent concentration, calculate how many milliliters of solute you get in $\text{100 mL}$ of solution

100 color(red)(cancel(color(black)("mL solution"))) * overbrace("225 mL alcohol"/(400color(red)(cancel(color(black)("mL solution")))))^(color(purple)("information given")) = "56.25 mL alcohol"

Since this is how many milliliters of solute you get in $\text{100 mL}$ of solution, it follows that the solution's volume by volume percent concentration is

"% v/v" = color(green)(|bar(ul(color(white)(a/a)color(black)("56% alcohol")color(white)(a/a)|)))

I'll leave the answer rounded to two sig figs, but keep in mind that you only have one sig fig for the volume of the solution.

Now, you know that your solution contains $\text{56 mL}$ of alcohol for every $\text{100 mL}$ of solution. This means that the volume of the solvent, which in this case is water, will be

${V}_{\text{solvent" = "100 mL" - "56 mL" = "44 mL}}$

This tells you that every $\text{100 mL}$ of solution contains $\text{44 mL}$ of water. This means that the solution is 44%"v/v" water.

For your solution, this is equivalent to

400 color(red)(cancel(color(black)("mL solution"))) * "44 mL water"/(100color(red)(cancel(color(black)("mL solution")))) = "176 mL water"#

Keep in mind that we've rounded the volume of alcohol from $\text{56.25 mL}$ to $\text{56 mL}$, which is why we didn't get

${V}_{\text{water" = V_"solution" - V_"alcohol}}$

${V}_{\text{water" = "400 mL" - "225 mL" = "175 mL water}}$