Question

A solution is made by adding \(50.0 \mathrm{~mL}\) of \(0.200 M\) acetic acid \(\left(K_{\mathrm{a}}=1.8 \times 10^{-5}\right)\) to \(50.0 \mathrm{~mL}\) of \(1.00 \times 10^{-3} \mathrm{M} \mathrm{HCl}\) a. Calculate the \(\mathrm{pH}\) of the solution. b. Calculate the acetate ion concentration.

Short answer

a. The pH of the solution is approximately 4.06. b. The acetate ion concentration is \(3.60 \times 10^{-5}\) mol/L.

Step-by-step solution

Calculate initial moles of each acid

First, we will find the moles of acetic acid and HCl in the solution using the given concentrations and volumes. For acetic acid (CH3COOH), Moles = (Concentration) x (Volume) = (0.200 mol/L) x (50.0 mL) x (1 L/1000 mL) = 0.0100 mol For HCl, Moles = (Concentration) x (Volume) = (1.00 x 10^{-3} mol/L) x (50.0 mL) x (1 L/1000 mL) = 5.00 x 10^{-5} mol

Set up an ICE table

Let's set up an ICE (Initial, Change, Equilibrium) table for the reaction. CH3COOH (aq) + H2O (l) <-> CH3COO- (aq) + H3O+ (aq) Initial (moles): 0.0100 -- 0 5.00 x 10^{-5} Change (moles): -x -- +x +x Equilibrium: 0.0100-x -- x 5.00 x 10^{-5}+x Since the volume of the solution remains constant, we can work with concentrations directly.

Write the expression for Ka and make necessary approximations

The Ka expression for acetic acid is given by: Ka = ([CH3COO-] [H3O+])/([CH3COOH]) Where, Ka = 1.8 x 10^{-5} Since we know that x will be small compared to the initial concentration of acetic acid, we can approximate: 0.0100 - x ≈ 0.0100 Now, our Ka expression becomes: 1.8 x 10^{-5} = x(5.00 x 10^{-5}+x)/0.0100

Solve for x

Since x is small relative to 5.00 x 10^{-5}, we can also approximate: 5.00 x 10^{-5} + x ≈ 5.00 x 10^{-5} So, 1.8 x 10^{-5} = x(5.00 x 10^{-5})/0.0100 Solve for x: x = 3.60 x 10^{-5} Now, x represents the change in the concentration of H3O+ ions. a.

Calculate the pH of the solution

Recall the pH definition: pH = -log10 ([H3O+]) The final concentration of H3O+ ions is 5.00 x 10^{-5} + 3.60 x 10^{-5} = 8.60 x 10^{-5} mol/L Now, we can calculate the pH: pH = -log10(8.60 x 10^{-5}) ≈ 4.06 b.

Calculate the acetate ion concentration

The acetate ion concentration is equal to the value of x since the change in concentration of CH3COO- is +x. Therefore, the final acetate ion concentration is: [CH3COO-] = 3.60 x 10^{-5} mol/L

Key concepts

Acid-Base Equilibrium

Understanding acid-base equilibrium is essential when delving into pH calculations in chemistry. It's a state where the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of the acid, base, conjugate acid, and conjugate base. This equilibrium is crucial because it allows us to use the constant, \(K_a\), which represents the acid dissociation constant, a measure of the strength of an acid in solution. The smaller the \(K_a\) value, the weaker the acid and the less it will dissociate. When you add acids to water, they tend to donate protons (\(H^+\)) to water molecules, forming hydronium ions (\(H_3O^+\)). This donation and the subsequent collection of \(H^+\) ions by the base, determines the overall pH of the solution.

For a weak acid like acetic acid (\(CH_3COOH\)), which does not fully dissociate, an equilibrium is established between the undissociated acid and its ions. The equilibrium formula \(K_a = \frac{[CH_3COO^-] [H_3O^+]}{[CH_3COOH]}\), quantifies this balance and is pivotal for calculating the pH. To determine the pH, we use the known \(K_a\) value and start with the initial concentrations of the acids, moving towards the equilibrium concentrations.

Acetic Acid Dissociation

Acetic acid dissociation involves its separation into acetate ions (\(CH_3COO^-\)) and hydrogen ions (\(H^+\)). In an aqueous solution, these hydrogen ions associate with water molecules to form hydronium ions (\(H_3O^+\)). The extent of this dissociation is not complete for acetic acid as it's a weak acid. Its dissociation constant (\(K_a\)) reflects this property. The small \(K_a\) value of acetic acid (\(1.8 \times 10^{-5}\)) suggests that in water, most of the acetic acid remains undissociated. Only a tiny fraction breaks down to contribute to \(H_3O^+\) ions, and in turn, affects the pH of the solution. Through this careful balance, acetic acid establishes a specific acid-base equilibrium in the solution.

The dissociation process is reversible and reaches a state of dynamic equilibrium in water, crucial in predicting the concentrations of the products and reactants at equilibrium state. When another acid like HCl is added to the acetic acid solution, the interactions between these two acids need to be considered to accurately calculate pH and ion concentrations.

ICE Table Method

To navigate the complex world of reactions going to equilibrium, chemists employ the Initial, Change, Equilibrium (ICE) table method. This approach offers a structured way to calculate the concentration of reactants and products at equilibrium. Let's start by understanding each part of the ICE acronym:

• Initial: The starting concentrations or moles of reactants and products before the reaction begins.
• Change: The amount the concentrations or moles will change as the reactants proceed to products.
• Equilibrium: The final concentrations or moles when the reaction reaches equilibrium.

The ICE table incorporates the stoichiometry of the balanced equation and uses the initial concentrations along with the \(K_a\) to find the change needed to reach equilibrium. This method is paramount when dealing with weak acids like acetic acid where the dissociation does not go to completion, and an understanding of the equilibrium state is necessary to calculate pH. It is through setting up a meticulous ICE table that you can proceed to make the necessary approximations regarding the concentrations, enabling straightforward equilibrium calculations.

Acetic Acid and HCl Titration

Titration of acetic acid with a strong acid like hydrochloric acid (HCl) may not follow the traditional titration curves seen in acid-base titrations. Usually, in a titration, the addition of a titrant changes the pH of the solution significantly, allowing us to determine the equivalence point. However, in the case when mixtures of weak acids (like acetic acid) and strong acids (like HCl) are present, the strong acid reacts completely, and the weak acid only partially dissociates.

Determining the pH in these situations requires careful consideration of both acids' contribution to the \(H_3O^+\) concentration. Here, the strong acid HCl completely dissociates, immediately releasing \(H^+\) ions, which notably affects the pH. Acetic acid's dissociation also impacts the pH but to a lesser extent due to its weak nature. Overall, the pH calculation hinges on the cumulative effect of both acids' dissociation, with the strong acid typically governing the initial pH before any titration commences. The ICE table then comes into play to reflect the weak acid's dissociation, allowing for an equilibrium-based calculation of pH. Titrating acetic acid with HCl also changes the concentration of acetate ions (\( CH_3COO^-\)), so both the \(pH\) and acetate concentration calculations need to address the presence of both acids in the solution.