Question

A typical sample of vinegar has a pH of \(3.0 .\) Assuming that vinegar is only an aqueous solution of acetic acid \(\left(K_{\mathrm{a}}=1.8 \times\right.\) \(10^{-5}\) ), calculate the concentration of acetic acid in vinegar.

Short answer

The concentration of acetic acid in the vinegar solution is approximately \(0.056 M\).

Step-by-step solution

1. Convert pH to H+ concentration

First, we need to convert the pH value into the concentration of H+ ions in the solution using the formula: pH = -log[H+] where [H+] is the concentration of H+ ions in the solution. From the problem, we have pH = 3.0. 3.0 = -log[H+] Let us now find the [H+] value.

2. Calculate H+ concentration

To find the [H+] value, we will take the anti-logarithm of both sides: [H+] = 10^(-pH) [H+] = 10^(-3.0) [H+] = 0.001 M Now that we have the H+ concentration, we can use the Ka expression for the acetic acid dissociation.

3. Write the Ka expression

The dissociation reaction for acetic acid (\(CH_3COOH\)) in water can be written as: \(CH_3COOH \rightleftharpoons H+ + CH_3COO^-\) The Ka expression for this reaction is: \(K_a\) = \(\frac{[H+][CH_{3}COO^-]}{[CH_{3}COOH]}\) Since we are assuming the vinegar solution is only acetic acid, the concentration of acetic acid [CH_3COOH] will be almost the same as the initial concentration (we'll call this "C") before any dissociation occurs. After dissociation, we will have: [H+] = 0.001 M \([CH_{3}COO^-] = 0.001 M\) And: \([CH_{3}COOH] = C - 0.001\) Now let's substitute these values into the Ka expression and solve for the initial concentration C of acetic acid.

4. Substitute values into the Ka expression

\(K_a\) = \(\frac{[H+][CH_{3}COO^-]}{[CH_{3}COOH]}\) \(1.8 \times 10^{-5}\) = \(\frac{(0.001)(0.001)}{(C - 0.001)}\) Now we need to solve for C, the concentration of acetic acid.

5. Solve for the initial concentration of acetic acid

To find the value of C, we need to rearrange the equation and solve for C: \(C - 0.001 = \frac{(0.001)(0.001)}{1.8 \times 10^{-5}}\) Now, let's solve for C: \(C = 0.001 + \frac{(0.001)(0.001)}{1.8 \times 10^{-5}}\) \(C \approx 0.056 M\) So, the concentration of acetic acid in the vinegar solution is approximately 0.056 M.

Key concepts

pH and H+ concentration

The concept of pH is a simple yet essential part of understanding acids and bases. pH is a measure of how acidic or basic a solution is, determined by the concentration of hydrogen ions \([H^+]\) in the solution.
The pH scale ranges from 0 to 14, with lower values being more acidic and higher values being more basic. Neutral solutions, like pure water, have a pH of 7.

To find the concentration of hydrogen ions based on pH, we use the formula:

• \( \text{pH} = -\log[H^+] \)

With a given pH of 3.0, we can rearrange the formula to solve for \([H^+]\):

• \([H^+] = 10^{-\text{pH}} \)
• \([H^+] = 10^{-3} = 0.001 \, M \)

This conversion helps us understand the level of acidity in solutions of acids like acetic acid.

acid dissociation constant (Ka)

The acid dissociation constant, known as \(K_a\), is a crucial value when studying acids. It gives us insight into the degree to which an acid dissociates in a solution. A higher \(K_a\) value means more dissociation and, thus, a stronger acid.

In the case of acetic acid \(CH_3COOH\), the dissociation in water can be represented as:

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

The \(K_a\) expression is:

• \(K_a = \frac{[H^+][CH_3COO^-]}{[CH_3COOH]}\)

By knowing the \(K_a\) value and the concentrations of the dissociated species, we can better comprehend the behavior of acetic acid in different environments.

For acetic acid, \(K_a = 1.8 \times 10^{-5}\). Understanding this helps us determine the initial concentration of acetic acid in solutions like vinegar.

vinegar acidity

Vinegar's acidity primarily comes from acetic acid, which gives it that characteristic sour taste. Typically, household vinegar is a dilute solution of acetic acid in water.

Given its pH of around 3.0, vinegar is acidic enough for culinary and cleaning purposes without being harmful to the skin. In chemical terms, this pH level indicates a certain concentration of acetic acid that can be quantified using the steps we've discussed.

By calculating the concentration of \([H^+]\) and using the acid dissociation constant \(K_a\), we can determine that the concentration of acetic acid in vinegar is approximately 0.056 M.
This concentration aligns with vinegar's expected strength, making it effective for everyday uses like pickling or salad dressings.

chemical equilibrium in solutions

Chemical equilibrium is an essential concept in understanding reactions in solutions, particularly for weak acids like acetic acid. In an equilibrated system, the rate of the forward reaction equals the rate of the reverse reaction, leading to stable concentrations of reactants and products.

For acetic acid in vinegar, the establishment of equilibrium involves the dissociation of the acid into \(H^+\) ions and acetate ions \(CH_3COO^-\):

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

The equilibrium constant, or \(K_a\), helps us understand this balance.
In the vinegar exercise, we use the equilibrium expression to solve for the original concentration of the acetic acid before any dissociation. Reacting systems like this help in calculating concentrations in various scenarios, crucial for both theoretical studies and practical applications in chemistry.

Understanding equilibrium principles empowers us to predict how a system responds to changes, like dilution or temperature shifts, keeping chemistry in practical balance.