# Question 9329f

Jan 14, 2017

Here's why that is the case.

#### Explanation:

The key here is the definition of a solution's $\text{pH}$, which as you know is aid to be equal to the negative log base $10$ of the concentration of hydronium cations, ${\text{H"_3"O}}^{+}$.

color(blue)(ul(color(black)("pH" = - log(["H"_3"O"^(+)]))))

Now, in order to see why hydrochloric acid is $10$ times stronger than citric acid, you need to look at the relative $\text{pH}$ values of the two acids.

You know that you have a difference of $1$ unit between the $\text{pH}$ of hydrochloric acid and the $\text{pH}$ of citric acid

"pH"_ "HCl" = "pH"_ "citric acid" + 1" " " "color(orange)("(*)")

Use the $\text{pH}$ equation to write

log(["H"_3"O"^(+)]) = - "pH"

10^log(["H"_3"O"^(+)]) = 10^(-"pH")

$\left[\text{H"_3"O"^(+)] = 10^(-"pH}\right)$

You can thus say that the concentration of hydronium cations in a solution of hydrochloric acid will be

$\left[\text{H"_ 3"O"^(+)] _ "HCl" = 10^(-"pH"_ "HCl}\right)$

Use equation $\textcolor{\mathmr{and} a n \ge}{\text{(*)}}$ to write

$\left[\text{H"_ 3"O"^(+)]_ "HCl" = 10^(-"pH"_ "citric acid} + 1\right)$

This is equivalent to

$\left[\text{H"_ 3"O"^(+)]_ "HCl" = 10^(-"pH"_ "citric acid}\right) \cdot {10}^{1}$

But since

$\left[\text{H"_ 3"O"^(+)]_ "citric acid" = 10^(-"pH"_ "citric acid}\right)$

it follows that you will have

color(darkgreen)(ul(color(black)(["H"_ 3"O"^(+)]_ "HCl" = 10 * ["H"_ 3"O"^(+)]_ "citric acid")))#

This shows that a solution of hydrochloric acid is $10$ times stronger than an equimolar solution of citric acid because it contains $10$ times more hydronium cations.

This is, of course, equivalent to saying that the $\text{pH}$ of the hydrochloric acid solution if $1$ unit lower than the $\text{pH}$ of the citric acid solution.