# Question 03d29

Apr 11, 2016

Here's what I got.

#### Explanation:

The first thing to do here is use the volume by volume percent concentration, $\text{%v/v}$, of the target solution to determine how many liters of solute, which in your case is sulfuric acid, ${\text{H"_2"SO}}_{4}$, it must contain.

A $\text{29% v/v}$ sulfuric acid solution will contain $\text{29 L}$ of sulfuric acid for every $\text{100 L}$ of solution, which means that your solution must contain

600 color(red)(cancel(color(black)("L solution"))) * ("29 L H"_2"SO"_4)/(100color(red)(cancel(color(black)("L solution")))) = "174 L H"_2"SO"_4

Now, let's assume the $x$ represents the volume of the $\text{70% v/v}$ sulfuric acid solution and $y$ represents the volume of the $\text{25% v/v}$ sulfuric acid solution.

The first equation that you can write here will be

$x + y = \text{600 L"" " " } \textcolor{\mathmr{and} a n \ge}{\left(1\right)}$

This simply uses the fact that the two solutions must be mixed together to form a total volume of $\text{600 L}$.

Now, use the given percent concentrations to figure out how many liters of sulfuric acid you'd get in those two solutions

xcolor(white)(a)color(red)(cancel(color(black)("L solution"))) * ("70 L H"_2"SO"_4)/(100color(red)(cancel(color(black)("L solution")))) = 7/10xcolor(white)(a)"L H"_2"SO"_4

ycolor(white)(a)color(red)(cancel(color(black)("L solution"))) * ("25 L H"_2"SO"_4)/(100color(red)(cancel(color(black)("L solution")))) = 1/4ycolor(white)(a)"L H"_2"SO"_4

This means that you can write

$\frac{7}{10} x + \frac{1}{4} y = \text{174 L"" " " } \textcolor{\mathmr{and} a n \ge}{\left(2\right)}$

This equation describes the fact that the amount of sulfuric acid you get from the two solutions you're mixing must add up to give $\text{174 L}$.

Use equation $\textcolor{\mathmr{and} a n \ge}{\left(1\right)}$ to get

$x = 600 - y$

Plug this into the second equation to get

$\frac{7}{10} \left(600 - y\right) + \frac{y}{4} = 174$

$420 - \frac{7}{10} y + \frac{y}{4} = 174$

$- \frac{9}{20} y = - 246 \implies y = 547$

This means that you have

$x = 600 - 547 = 53$

"volume of 70% solution" = color(green)(|bar(ul(color(white)(a/a)"50 L"color(white)(a/a)|)))
"volume of 25% solution" = color(green)(|bar(ul(color(white)(a/a)"550 L"color(white)(a/a)|)))
Finally, the result makes sense because the concentration of the target solution is much closer to the concentration of the 25% solution, which implies that you'd need a lot more of this solution than of the 70%# one.