Question

A particular sample of vinegar has a pH of \(2.90 .\) If acetic acid is the only acid that vinegar contains \(\left(K_{a}=1.8 \times 10^{-5}\right)\) calculate the concentration of acetic acid in the vinegar.

Short answer

The concentration of acetic acid in the vinegar sample is approximately 0.0446 M.

Step-by-step solution

Finding the concentration of H+ ions from the pH

First, we need to find out the H+ ion concentration from the given pH using the pH formula: \(pH = -\log[H^+]\) where pH is 2.90. Now, solving for [H+]: \([H^+] = 10^{-pH}\) Plug in the value of pH: \([H^+] = 10^{-2.90}\)

Ka expression and setting up the ICE table

Next, we need to write the Ka expression for acetic acid with its Ka value: \( K_a = 1.8 x 10^{-5}\) The balanced equation for the dissociation of acetic acid is: \(CH_3COOH \rightleftharpoons H^+ + CH_3COO^-\) Now we establish an Initial, Change, and Equilibrium (ICE) table to help solve for the concentration of acetic acid.

Writing the equilibrium equation and solving for [CH3COOH]

We know the K_a expression for acetic acid dissociation will be: \(K_a = \cfrac{[H^+][CH_3COO^-]}{[CH_3COOH]}\) From the ICE table, we have: \([H^+] = 10^{-2.90}\) Since acetic acid is the only acid in the solution, the dissociation of acetic acid produces the H+ ions and every mole of acetic acid dissociating forms one mole of CH3COO-, hence [CH3COO-] is equal to [H+]. Thus, \([CH_3COO^-] = 10^{-2.90}\) Using [H+] and [CH3COO-], we can determine the concentration of acetic acid: \(K_a = \cfrac{(10^{-2.90})(10^{-2.90})}{[CH_3COOH]}\) Solving for [CH3COOH] using the provided Ka value: \(1.8 \times 10^{-5} = \cfrac{(10^{-2.90})(10^{-2.90})}{[CH_3COOH]}\)

Calculating the concentration of acetic acid

Finally, rearrange the equation to find [CH3COOH]: \([CH_3COOH] = \cfrac{(10^{-2.90})(10^{-2.90})}{1.8 \times 10^{-5}}\) Now, perform the calculation: \([CH_3COOH] \approx 0.0446 \, \text{M}\) Thus, the concentration of acetic acid in the vinegar sample is approximately 0.0446 M.

Key concepts

pH calculation

Understanding the basic calculation of pH is essential when dealing with acids and bases. The pH is a measure of the acidity or basicity of a solution. It's defined as the negative base-10 logarithm of the concentration of hydrogen ions ([H+]).

So, to find the pH, you use the formula:
\( pH = -\log[H^+] \)

For instance, if you're given a pH of 2.90, as in our vinegar sample problem, you can calculate the concentration of hydrogen ions by rearranging this formula:
\[ [H^+] = 10^{-pH} \]

Therefore,
\[ [H^+] = 10^{-2.90} \]

This gives us the [H+] we need to find the concentration of acetic acid in our next steps.

Ka expression for acetic acid

The equilibrium constant (Ka) for an acid is a quantitative measure of the acid's strength - the tendency of its molecules to dissociate into hydrogen ions (H+) and anions (A-). To find the Ka expression specifically for acetic acid, we consider its dissociation in an aqueous solution:

\( CH_3COOH \rightleftharpoons H^+ + CH_3COO^- \)

The Ka expression thus can be formulated as:
\( K_a = \frac{[H^+][CH_3COO^-]}{[CH_3COOH]} \)

Where \( K_a \) is the acid dissociation constant, [H+] is the concentration of hydrogen ions, [CH3COO-] is the concentration of acetate ions, and [CH3COOH] is the concentration of undissociated acetic acid. For our acetic acid example, the Ka value is given as \( 1.8 \times 10^{-5} \), which we will use to find the acetic acid concentration.

ICE table for equilibrium

An ICE table, which stands for Initial, Change, and Equilibrium, is a tool used to keep track of the concentrations of reactants and products during a chemical reaction that has reached equilibrium.

In the context of acetic acid dissociation, we start by setting the initial concentration of acetic acid and assuming initially there are zero products formed. During the change phase, a certain amount of acetic acid will dissociate into H+ and CH3COO- ions. At equilibrium, the concentration of reactants and products will not change with time.

The practicality of an ICE table relies on the stoichiometry of the balanced chemical equation and the assumption that the degree of dissociation is small compared to the initial concentration of the weak acid, allowing us to simplify the expressions involved in the equilibrium constant calculation.

Equilibrium concentration calculation

Calculating the equilibrium concentration of an acid requires the use of the equilibrium constant (Ka) and the concentrations of products and reactants at equilibrium. From our exercise, by knowing the Ka for acetic acid and the concentration of H+ ions (which is equal to the concentration of CH3COO- since one molecule of acetic acid dissociates into one ion of each), we can solve for the concentration of acetic acid ([CH3COOH]) at equilibrium.

This equation looks like:
\( [CH_3COOH] = \frac{[H^+][CH_3COO^-]}{K_a} \)

Substitute in the values we have for \( [H^+] \), \( [CH_3COO^-] \), and \( K_a \), and then solve for \( [CH_3COOH] \) to find the equilibrium concentration of acetic acid in the sample of vinegar, thereby giving us a clear picture of the acid's strength and concentration in the solution.