Question

A 625 mL sample of an aqueous solution containing 0.275 mol propionic acid, \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CO}_{2} \mathrm{H},\) has \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=0.00239 \mathrm{M} .\) What is the value of \(K_{\mathrm{a}}\) for propionic acid? $$\begin{aligned} &\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CO}_{2} \mathrm{H}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}+\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CO}_{2}\\\ &&K_{\mathrm{a}}=? \end{aligned}$$

Short answer

\(K_a\) for propionic acid is 1.3 × 10-5.

Step-by-step solution

Find processing concentration

The first thing to do is to find the concentration of propionic acid by dividing the number of moles by the volume in liters. To convert mL to L, divide by 1000. Thus, concentration of the acid = \(\frac{0.275 mol}{0.625 L}\) = 0.44 M

Find equilibrium values

The propionic acid will dissociate according the reaction \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CO}_{2} \mathrm{H}=\mathrm{H}_{2} \mathrm{O}+\mathrm{H}_{3} \mathrm{O}^{+}+\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CO}_{2}^{─}\). The equilibrium concentration of \([H_{3}O^{+}]\) is 0.00239 M. At equilibrium, the acid and conjugate base have equal concentrations \(x\) M where \(x\) = \([H_{3}O^{+}]\).

Calculate \(K_a\)

The expression for \(K_a\) is \(\frac{[H_{3}O^{+}][CH3CH2CO2^-]}{[CH3CH2CO2H]}\). Substituting the equilibrium concentrations, we get \(K_a\) = \(\frac{(0.00239 M)^2}{0.44 M - 0.00239 M}\). Solving this equation using a calculator, we get \(K_a\) = 1.3 × 10-5.

Key concepts

Propionic Acid

Propionic acid, with the chemical formula \(\text{CH}_3\text{CH}_2\text{CO}_2\text{H}\), is a carboxylic acid, meaning it contains a carboxyl group \(-\text{COOH}\) attached to an alkyl group. It is naturally occurring and can be found in various fermented foods and dairy products. In chemistry, propionic acid is studied due to its weak acidic properties in aqueous solutions. The weak acidity means it partially dissociates into ions when dissolved in water. Understanding how this dissociation occurs is key to calculating its acid dissociation constant \(K_a\), which measures how readily the acid donates protons (\(\text{H}^+\)) in a solution.

Knowing the \(K_a\) value of propionic acid helps predict its behavior in chemical reactions and its effectiveness as a preservative. For students, being able to calculate \(K_a\) is a fundamental skill in understanding the nature of weak acids in aqueous solutions.

Equilibrium Concentration

The term 'equilibrium concentration' refers to the concentration of all species in a reaction at equilibrium. This is the point in a chemical reaction where the rates of the forward and backward reactions are equal, leading to a stable concentration of reactants and products. In the case of propionic acid dissociation, it involves the concentration of \(\text{H}_3\text{O}^+\) (hydronium ions), \(\text{CH}_3\text{CH}_2\text{CO}_2^-\) (propionate ions), and undissociated propionic acid.

Students often determine equilibrium concentrations by setting up an "ICE" (Initial, Change, Equilibrium) table where they track changes in concentration over the course of reaching equilibrium. By finding the concentrations at equilibrium, it's possible to understand how much of the acid dissociates and how much remains intact, which provides insight into the effectiveness of the acidic solution.

Chemical Equilibrium

Chemical equilibrium is a central concept in chemistry that describes the state of a reversible reaction where the rate of the forward reaction equals the rate of the reverse reaction. This balance does not imply that the concentrations of reactants and products are equal but that their ratios remain constant.

In the context of propionic acid dissociation, equilibrium is achieved when the rate at which the acid molecules donate a proton equals the rate at which the conjugate base recombines with a proton. Studying this equilibrium provides insights into the strength and behavior of an acid in a solution.

Understanding chemical equilibrium involves concepts such as Le Chatelier’s Principle, which predicts how a change in conditions (like concentration or temperature) will shift the equilibrium position to restore balance. Knowledge of these shifts is crucial for predicting how changes affect the dissociation of weak acids like propionic acid in various conditions.

Aqueous Solution Chemistry

Aqueous solution chemistry focuses on reactions and interactions that occur in water. Water is a universal solvent, meaning many substances dissolve in it to form solutions. This study is fundamental in understanding how acids and bases behave in water, including how they dissociate and interact with solvents.

Propionic acid, when dissolved in water, partially dissociates into hydronium ions and propionate ions. This partial dissociation is a characteristic of weak acids and is an important aspect of aqueous solution chemistry. In contrast, strong acids dissociate completely, and this difference is highlighted when calculating the acid dissociation constant \(K_a\).

Aqueous solution chemistry also forms the basis for many real-world applications, such as pharmaceuticals, cleaning products, and biological systems, where understanding the behavior of substances in water is essential. For students, grasping these principles is key to solving complex chemistry problems related to acid-base reactions and equilibria.