# Question 37461

Dec 27, 2017

See explanation.

#### Explanation:

For starters, it's worth mentioning that you won't find a volume by volume percent concentration, $\text{% v/v}$, solution of sodium chloride because this type of percent concentration is usually reserved for solutions that contain liquid solutes.

A much more plausible percent concentration to use for a solution of sodium chloride is the mass by volume percent concentration, $\text{% m/v}$, which is defined as the number of grams of solute present for every $\text{100 mL}$ of the solution.

"% m/v" = "grams of solute"/"100 mL solution" xx 100%

For, for example, if you have a $\text{25% m/v}$ solution of sodium chloride, you can say that this solution contains $\text{25 g}$ of sodium chloride, the solute, for every $\text{100 mL}$ of the solution.

By comparison, the mass by mass percent concentration, $\text{% m/m}$, tells you the number of grams of solute present for every $\text{100 g}$ of the solution.

"% m/m" = "grams of solute"/"100 g solution" xx 100%

In this example, a $\text{25% m/m}$ solution of sodium chloride contains $\text{25 g}$ of sodium chloride for every $\text{100 g}$ of the solution.

Notice that you can convert between the two percent concentrations by using the density of the solution. For example, if you have a solution of sodium chloride that has a density of $\rho$ ${\text{g mL}}^{- 1}$, which means that every $\text{1 mL}$ of this solution has a mass of $\rho$ $\text{g}$, you can say that $\text{100 mL}$ of the solution will have a mass of

100 color(red)(cancel(color(black)("mL solution"))) * (rho color(white)(.)"g solution")/(1color(red)(cancel(color(black)("mL solution")))) = (100 * rho)color(white)(.)"g"

So, you know that

$\text{100 mL solution " -> " "(100 * rho) color(white)(.)"g solution}$

This means that you have

"% m/v" = "grams of solute"/"100 mL solution" xx 100%

"% m/v" = "grams of solute"/((100 * rho)color(white)(.)"g solution") xx 100%

This is equivalent to

"% m/v" = overbrace("grams of solute"/"100 g solution" xx 100%)^(color(blue)("= %m/m")) * 1/rho#

which gets you

$\textcolor{b l u e}{\underline{\textcolor{b l a c k}{\text{% m/v" = 1/rho * "% m/m}}}}$

Notice that if the density of the solution is $> {\text{1 g mL}}^{- 1}$, the mass by volume percent concentration is lower than the mass by mass percent concentration, which happens because the mass of $\text{100 mL}$ of the solution is $>$ $\text{100 g}$.