Problem 109 - Chemistry The Central Science

Question

Caproic acid \(\left(\mathrm{C}_{5} \mathrm{H}_{11} \mathrm{COOH}\right)\) is found in small amounts in coconut and palm oils and is used in making artificial flavors. A saturated solution of the acid contains \(11 \mathrm{~g} / \mathrm{L}\) and has a \(\mathrm{pH}\) of 2.94. Calculate \(K_{a}\) for the acid.

Step-by-Step Solution

Answer

The acid dissociation constant (\(K_{a}\)) for caproic acid is approximately \(1.35 \times 10^{-5}\).

Step 1: Determine the concentration of H+ ions using the pH value

To determine the concentration of H+ ions, we can use the formula: pH = -log10[H+] Given pH = 2.94, we can calculate the H+ concentration: 2.94 = -log10[H+] Rearranging the equation to solve for [H+]: [H+] = 10^(-2.94)

Step 2: Write the balanced dissociation equation of caproic acid

The balanced dissociation equation for caproic acid (HA) would be: HA <-> H+ + A-

Step 3: Set up an ICE table to determine the equilibrium concentrations

Set up the ICE (Initial, Change, Equilibrium) table for concentration changes: | [HA] | [H+] | [A-] ------+--------+--------+-------- I | Co | 0 | 0 C | -x | +x | +x E | Co - x | x | x We are given that the concentration of caproic acid is \(\frac{11 g}{L}\) in the saturated solution. Let's find the moles of the acid: \(Co = \frac{11 g}{(C_{5}H_{11}COOH)g/mol}\) \(Co = \frac{11}{116.15}\) mol/L \(Co ≈ 0.095 M\) Since it's a saturated solution, most of the caproic acid will remain undissociated. Hence, we can assume \(Co - x ≈ Co = 0.095M\). Also, the concentration of H+ is calculated from Step 1, which is our x value: [H+] = x = \(10^{-2.94}\)

Step 4: Calculate the Ka using the equilibrium concentrations

Now, let's calculate the Ka: \(K_{a} = \frac{[H+][A^-]}{[HA]}\) Substitute the equilibrium concentration values: \(K_{a} = \frac{x \cdot x}{Co - x}\) \(K_{a} = \frac{(10^{-2.94})^2}{0.095}\) Calculate Ka: \(K_{a} ≈ 1.35 \times 10^{-5}\) So, the calculated \(K_{a}\) for caproic acid is \(1.35 \times 10^{-5}\).

Key Concepts

Acid Dissociation Constant

To understand the acid dissociation constant, or \( K_a \), let's first look at what it tells us. \( K_a \) is a measure of the strength of an acid in solution. It describes the extent to which an acid will dissociate into its ions.
For a weak acid, which does not completely dissociate, the \( K_a \) value is usually a small number.
When an acid like caproic acid is dissolved in water, it separates into hydrogen ions \( (H^+) \) and its conjugate base \( (A^-) \). The equation is as follows:

• HA \( \rightleftharpoons H^+ + A^- \)

The smaller the \( K_a \), the fewer ions are produced, indicating a weaker acid.
The calculation of \( K_a \) requires the equilibrium concentrations of all species in the solution. If you only have the initial concentration and the equilibrium concentration of hydrogen ions, you can plug these into the expression for \( K_a \) to find it:\[ K_a = \frac{[H^+][A^-]}{[HA]} \]This relationship helps us understand the relative amounts of each species present at equilibrium. Knowing \( K_a \) can help predict how an acid will behave in different chemical reactions.

pH Calculation

Calculating the pH of a solution is a fundamental concept in chemistry. The pH is a measure of the acidity or basicity of a solution. It is calculated by taking the negative logarithm (base 10) of the hydrogen ion concentration \([H^+]\).
Thus, the pH is calculated using:

• \( \text{pH} = -\log_{10}[H^+] \)

This scale runs from 0 to 14, with lower values indicating acidic solutions and higher values suggesting basic solutions. Neutral solutions have a pH around 7.
In the exercise, the pH of the saturated caproic acid solution is given as 2.94.
To find the concentration of hydrogen ions, we rearrange the formula:

• \([H^+] = 10^{-2.94}\)

This calculation is crucial because it provides the "x" value, or the concentration of \(H^+\), that we will use in the ICE table to solve for \( K_a \).
pH helps us gauge the strength of acids and bases and predict the direction of acid-base reactions. Understanding pH calculations can be very useful in a variety of fields, from chemistry and biology to environmental science and medicine.

ICE Table

An ICE table is a handy tool used to simplify calculations involving equilibrium in chemical reactions. ICE stands for Initial, Change, and Equilibrium.
Setting up an ICE table helps to keep track of the concentrations of species at these three stages.

• Initial: The concentrations of reactants and products before any reaction takes place.
• Change: The adjustments made to each concentration as the reaction proceeds towards equilibrium.
• Equilibrium: The concentrations when the reaction has reached equilibrium.

When working with an ICE table, the change in concentration is often represented as "x," which signifies the amount that dissociates from the reactant, forming the products.
In this exercise, the table helps us maintain an organized way of seeing how much caproic acid dissociates in solution:

• \([HA]_{initial} = C_0\)
• \(\text{Change:}- x\)
• \([HA]_{equilibrium} = C_0 - x\)
• \([H^+]_{equilibrium} = x\)
• \( [A^-]_{equilibrium} = x \)

The ICE table serves as the foundation for solving the equilibrium concentrations, making it a crucial step in determining the \( K_a \) value. By understanding how to effectively use an ICE table, you can tackle a wide range of equilibrium problems in chemistry.
It showcases all necessary steps in both algebraic and conceptual forms, ensuring you grasp the full picture of the process.