Problem 50 - Chemistry The Science in Context

Question

Rancid Butter The odor of spoiled butter is due in part to butanoic acid, which results from the chemical breakdown of butterfat. A \(0.100 M\) solution of butanoic acid is \(1.23 \%\) ionized. Calculate the value of \(K_{\mathrm{a}}\) for butanoic acid.

Step-by-Step Solution

Answer

Answer: The acid dissociation constant (Ka) for butanoic acid is approximately 1.53 × 10^(-5).

Step 1: Write the balanced chemical equation for the ionization of butanoic acid

The ionization of butanoic acid (C\(_{3}\)H\(_{7}\)COOH) can be represented by the following chemical equation: C\(_{3}\)H\(_{7}\)COOH (aq) \(\rightleftharpoons\) C\(_{3}\)H\(_{7}\)COO\(^{-}\) (aq) + H\(^{+}\) (aq)

Step 2: Calculate the equilibrium concentration of the ions

Given the initial concentration of butanoic acid and its percentage of ionization, we can find the equilibrium concentration of the ions. Let's denote butanoic acid concentration as [C\(_{3}\)H\(_{7}\)COOH]\(_{0}\) = 0.100 M, and the percentage of ionization as 1.23%. Then, the concentrations of ions at equilibrium are: [C\(_{3}\)H\(_{7}\)COO\(^{-}\)]\(_{eq}\) = [H\(^{+}\)]\(_{eq}\) = 0.100 M × 1.23% = 0.100 M × 0.0123 = 0.00123 M

Step 3: Write the expression for Ka and plug in the equilibrium concentrations

Now, we can write the expression for the acid dissociation constant (\(K_{\mathrm{a}}\)) for butanoic acid: \(K_{\mathrm{a}}\) = \(\frac{[\mathrm{C_{3}H_{7}COO^{-}][H^{+}]}{[\mathrm{C_{3}H_{7}COOH}]}\) Plugging the equilibrium concentrations into the equation, we have: \(K_{\mathrm{a}}\) = \(\frac{(0.00123\,\mathrm{M})(0.00123\,\mathrm{M})}{(0.100\,\mathrm{M} - 0.00123\,\mathrm{M})}\)

Step 4: Calculate the value of Ka

Calculating the \(K_{\mathrm{a}}\) using the given values, we get: \(K_{\mathrm{a}}\) = \(\frac{(0.00123\,\mathrm{M})(0.00123\,\mathrm{M})}{(0.09877\,\mathrm{M})}\) ≈ 1.53 × 10\(^{-5}\) Therefore, the value of \(K_{\mathrm{a}}\) for butanoic acid is approximately 1.53 × 10\(^{-5}\).

Key Concepts

butanoic acid

Butanoic acid, also known as butyric acid, is an organic compound with the chemical formula C\(_{3}\)H\(_{7}\)COOH. It's commonly associated with the unpleasant smell that emanates from rancid butter. This aroma occurs due to the presence of butanoic acid as a breakdown product of butterfat. As a carboxylic acid, butanoic acid can release protons (H\(^{+}\)), reacting in aqueous solutions.
This behavior in water means butanoic acid can be ionized, forming butyrate ions (C\(_{3}\)H\(_{7}\)COO\(^{-}\)) and protons. This ionization characteristic is crucial for understanding its acid dissociation constant (\(K_{\mathrm{a}}\)), which measures the strength of the acid in solution. Knowing \(K_{\mathrm{a}}\) helps predict how butanoic acid behaves in chemical processes and environmental systems.
In applications, butanoic acid is used in various sectors, including the food industry, for its flavoring properties, and in manufacturing, where it's a key ingredient for producing different chemicals.

percentage ionization

Percentage ionization refers to the fraction of acid that dissociates into ions in solution, expressed as a percentage of the initial concentration. For butanoic acid, a \(0.100\,M\) solution is known to be \(1.23\%\) ionized. To calculate this, multiply the initial concentration by the percentage ionization factor converted to decimal \((1.23\% = 0.0123)\).

• The calculation is \(0.100 \times 0.0123 = 0.00123\,M\), indicating that \(0.00123\,M\) of the butanoic acid is ionized in solution.

This percentage tells us how much of the acid is effective in contributing protons to the solution, influencing the solution's acidity and various equilibrium reactions.
Understanding the percentage ionization helps chemists gauge an acid's potency. Even if an acid is weak (not completely ionized) like butanoic acid, knowing this measure allows planning for its use in buffering solutions and processes where specific pH levels are needed.

equilibrium concentration

Equilibrium concentration refers to the concentration of reactants and products present in a solution when a chemical reaction reaches a state of balance. For butanoic acid in the context of our problem, after ionization, we estimate this concentration using its initial amount and ionization percentage.
Initially, a \(0.100\,M\) solution begins, with some portion converting to ions and reaching equilibrium with both butyrate ions (C\(_{3}\)H\(_{7}\)COO\(^{-}\)) and protons at \(0.00123\,M.\)

• The remaining concentration of un-ionized butanoic acid is approximately \(0.100\,M - 0.00123\,M = 0.09877\,M.\)

This gives essential parameters for calculating the acid dissociation constant \(K_{\mathrm{a}}\), which is defined by the equation:\[K_{\mathrm{a}} = \frac{[\mathrm{C_{3}H_{7}COO^{-}}][\mathrm{H^{+}}]}{[\mathrm{C_{3}H_{7}COOH}]}\]Substituting in the equilibrium concentrations allows us to find \(K_{\mathrm{a}}\), determining how readily butanoic acid donate protons and how the solution's acidity is established under different conditions.