Question

Question #38939

Answer 1

`"pH" = 2.2`

Explanation:

Lactic acid will only partially ionize in aqueous solution to produce lactate anions and hydronium cations as described by the equilibrium equation

`"C"_ 2"H"_ 5"OCOOH"_ ((aq)) + "H"_ 2"O"_ ((l)) rightleftharpoons "C"_ 2"H"_ 5"OCOO"_ ((aq))^(-) + "H"_ 3"O"_ ((aq))^(+)`

Notice that for every mole of lactic acid that ionizes, you get `1` mole of lactate anions and `1` mole of hydronium cations.

This means that if you take `x` `"M"` to be the concentration of the acid that ionizes, you can say that, at equilibrium, the solution will contain

`["C"_ 2"H"_ 5"OCOO"^(-)] = ["H"_ 3"O"^(+)] = x quad "M"`

`["C"_ 2"H"_ 5"OCOOH"] = (0.06 - x) quad "M"`

This basically means that in order for the reaction to produce `x` `"M"` of lactate anions and `x` `"M"` of hydronium cations, the initial concentration of the acid must decrease by `x` `"M"`.

By definition, the acid dissociation constant for this reaction looks like this

`K_a = (["C"_ 2"H"_ 5"OCOO"^(-)] * ["H"_ 3"O"^(+)])/(["C"_ 2"H"_ 5"OCOOH"])`

This will be equivalent to

`8.4 * 10^(-4) = (x * x)/(0.06 - x)`

`8.4 * 10^(-4) = x^2/(0.06 - x)`

Rearrange to quadratic equation form

`x^2 + 8.4 * 10^(-4) * x - 0.06 * 8.4 * 10^(-4) = 0`

This quadratic equation will produce two solutions, one positive and one negative. Since `x` represents concentration, you can discard the negative value to get

`x = 0.0066917`

Since `x` `"M"` represents the equilibrium concentration of the hydronium cations, you can say that you have

`["H"_3"O"^(+)] = "0.0066917 M"`

Consequently, the `"pH"` of the solution, which is calculated using the equation

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

will be equal to

`"pH" = - log(0.0066917) = color(darkgreen)(ul(color(black)(2.2)))`

The answer must be rounded to one decimal place, the number of significant figure you have for the initial concentration of the acid.