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Chapter 16: Problem 55

A particular sample of vinegar has a \(\mathrm{pH}\) 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

Expert verified
The concentration of acetic acid in the vinegar is approximately \(6.9 \times 10^{-2}\) M.

Step by step solution

01

Write down the given information

From the exercise, we have: pH = 2.90 \(K_a = 1.8 \times 10^{-5}\) (Acid dissociation constant for acetic acid) Our goal is to find the concentration of acetic acid in the vinegar.
02

Write down the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is given by: \[pH = pK_a + \log_{10} \frac{[A^-]}{[HA]}\] Where: pH is the pH of the solution, \(pK_a\) is the negative logarithm of the acid dissociation constant \(K_a\), \([A^-]\) is the concentration of the conjugate base of the weak acid, and \([HA]\) is the concentration of the weak acid.
03

Calculate pKa

In order to use the Henderson-Hasselbalch equation, we need to find \(pK_a\) first. We can calculate pKa using the equation: \[pK_a = -\log_{10}K_{a}\] Substitute the given \(K_a\) value: \[pK_a = -\log_{10}(1.8 \times 10^{-5})\] Calculate the value of pKa: \[pK_a \approx 4.74\]
04

Use the Henderson-Hasselbalch equation to find acetic acid concentration

Now that we have pH and pKa, we can use the Henderson-Hasselbalch equation to find the concentration of acetic acid in the vinegar. Since acetic acid is the only acid present, the term \(\frac{[A^-]}{[HA]}\) will represent the ratio of conjugate base concentration to acetic acid concentration. Rearrange the Henderson-Hasselbalch equation to find the ratio \(\frac{[A^-]}{[HA]}\): \[\frac{[A^-]}{[HA]} = 10^{(pH - pK_a)}\] Substitute the values of pH and pKa: \[\frac{[A^-]}{[HA]} = 10^{(2.90 - 4.74)}\] \[ \frac{[A^-]}{[HA]} \approx 6.9 \times 10^{-2}\] For dilute solutions as assumed, the concentration of the conjugate base \([A^-]\) will be approximately equal to that of the acetic acid \([HA]\). Let \(x\) be the concentration of acetic acid, \[ \frac{x}{x} \approx 6.9 \times 10^{-2}\] \[ x \approx 6.9 \times 10^{-2} \; \text{M}\]
05

State the final answer

The concentration of acetic acid in the vinegar is approximately \(6.9 \times 10^{-2}\) M.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Henderson-Hasselbalch equation
The Henderson-Hasselbalch equation is a critical tool in acid-base chemistry. This equation is derived from the acid dissociation equilibrium and is used to relate the pH of a solution to the pKa (the negative logarithm of the acid dissociation constant) and the ratio of the concentrations of the conjugate base ([A^-]) to the weak acid ([HA]). The formula is:
  • \[ pH = pK_a + \log_{10} \left(\frac{[A^-]}{[HA]}\right) \]
This equation helps predict how the pH of a solution will change with different amounts of weak acid or its conjugate base added. In the exercise, the Henderson-Hasselbalch equation is used to find the concentration of acetic acid in the vinegar by considering the given pH and calculating the pKa from the known association constant (K_a) of acetic acid. Understanding this equation allows students to grasp the delicate equilibrium between acids and bases in solutions, especially in buffer systems.
Acetic Acid Concentration
Acetic acid (CH鈧僀OOH) is a common weak acid found in vinegar. In the exercise, we aim to determine its concentration in a vinegar sample with a known pH. The acetic acid dissociates slightly in water, forming hydrogen ions and acetate ions (CH鈧僀OO^-).To find the concentration of acetic acid, we use the calculated \(\frac{[A^-]}{[HA]}\) ratio from the Henderson-Hasselbalch equation:
  • Calculated as \(6.9 \times 10^{-2}\) from the given pH and the pKa of acetic acid.
Assuming that this ratio approximately represents the changes in molar concentrations, we find that the concentration of acetic acid can be closely estimated to be the same as this ratio when dealing with dilute solutions. Hence, the acetic acid concentration in the vinegar is about \(6.9 \times 10^{-2}\) M.
pH Calculation
Calculating pH is an essential skill in understanding acid-base equilibria. pH is defined as the negative log of the hydrogen ion concentration ([H^+]) in the solution. Given the initial information of pH = 2.90 for the vinegar, we back-calculate to derive other characteristics of the solution.The steps involve:
  • Recognizing that \(pH = -\log_{10} [H^+]\), which leads to \([H^+] = 10^{-pH}\).
  • Using pH to calculate the \(pK_a\), by relating it to the K_a of acetic acid (\[pK_a = -\log_{10} K_a\]), providing us a bridge to further apply the Henderson-Hasselbalch equation.
Understanding how to manipulate these equations and the significance of pH in assessing acidity or alkalinity allows students to predict how changes in concentration impact the system's balance. It's a practical demonstration of the interaction between chemical species in a solution and the importance of equilibrium constants in chemistry.

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Most popular questions from this chapter

Consider the base hydroxylamine, \(\mathrm{NH}_{2} \mathrm{OH}\). (a) What is the conjugate acid of hydroxylamine? (b) When it acts as a base, which atom in hydroxylamine accepts a proton? (c) There are two atoms in hydroxylamine that have nonbonding electron pairs that could act as proton acceptors. Use Lewis structures and formal charges ers (Section 8.5) to rationalize why one of these two atoms is a much better proton acceptor than the other.

Calculate the number of \(\mathrm{H}^{+}(a q)\) ions in \(1.0 \mathrm{~mL}\) of pure water at \(25^{\circ} \mathrm{C}\).

Arrange the following \(0.10 \mathrm{M}\) solutions in order of increasing acidity (decreasing \(\mathrm{pH}\) ): (i) \(\mathrm{NH}_{4} \mathrm{NO}_{3}\), (ii) \(\mathrm{NaNO}_{3}\), (iii) \(\mathrm{CH}_{3} \mathrm{COONH}_{4}\), (iv) \(\mathrm{NaF}\), (v) \(\mathrm{CH}_{3} \mathrm{COONa}_{\text {. }}\)

Addition of phenolphthalein to an unknown colorless solution does not cause a color change. The addition of bromthymol blue to the same solution leads to a yellow color. (a) Is the solution acidic, neutral, or basic? (b) Which of the following can you establish about the solution: (i) A minimum \(\mathrm{pH}\), (ii) A maximum \(\mathrm{pH}\), or (iii) A specific range of \(\mathrm{pH}\) values? (c) What other indicator or indicators would you want to use to determine the \(\mathrm{pH}\) of the solution more precisely?

Predict the products of the following acid-base reactions, and predict whether the equilibrium lies to the left or to the right of the equation: (a) \(\mathrm{O}^{2-}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons\) (b) \(\mathrm{CH}_{3} \mathrm{COOH}(a q)+\mathrm{HS}^{-}(a q) \rightleftharpoons\) (c) \(\mathrm{NO}_{2}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons\)
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