# How many grams of acetic acid should be dissolved in 100.0 g of water to make a 4.0 percent (by mass) solution of vinegar?

Dec 25, 2015

$\text{4.2 g}$

#### Explanation:

As you know, percent concentration by mass is defined as mass of solute, which in your case will be acetic acid, divided by mass of solution, and multiplied by $100$.

$\textcolor{b l u e}{\text{% m/m" = "mass of solute"/"mass of solution} \times 100}$

Now, the trick here is to realize that adding the acetic acid will Increase the mass of the solution.

If the mass of glacial acetic acid, which is pure acetic acid, is equal to $x$ grams, then you can say that adding this amount to your sample of water will bring the mass of the solution to

${m}_{\text{sol" = (100.0 + x)" g}}$

You need this solution to be $\text{4.0% m/m}$ acetic acid, so you can say that

"4.0%" = (xcolor(red)(cancel(color(black)("g"))))/((100.0+x)color(red)(cancel(color(black)("g")))) xx 100

This is equivalent to

$4.0 \cdot \left(100.0 + x\right) = 100 x$

$400 = 96 x \implies x = \frac{400}{96} = 4.1667$

Rounded to two sig figs, the number of sig figs you have for the percent concentration of the target solution, the answer will be

m_"acetic acid" = color(green)("4.2 g")

So, if you add $\text{4.2 g}$ of glacial acetic acid to $\text{100.0 g}$ of water, you will get a $\text{4.0% m/m}$ acetic acid solution, also known as vinegar.