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 pH of 2.94. Calculate \(K_{a}\) for the acid.

Short answer

The acid dissociation constant, \(K_a\), for caproic acid is approximately \(1.4 \times 10^{-5}\).

Step-by-step solution

Convert the pH value to hydrogen ion concentration

To convert the pH value to the hydrogen ion concentration, we use the formula: \[ [\mathrm{H}^{+}] = 10^{-\mathrm{pH}} \] Plug in the given pH value of 2.94: \[ [\mathrm{H}^{+}] = 10^{-2.94} \approx 1.15 \times 10^{-3} \, \mathrm{M} \]

Determine the concentrations of caproic acid and its conjugate base

The initial concentration of caproic acid in the saturated solution is given as 11 g/L. To convert it to molarity, we need to divide by its molar mass. The molar mass of \(\mathrm{C}_{5} \mathrm{H}_{11} \mathrm{COOH}\) is roughly \(12 \times 5 + 1 \times 11 + 12 + 16 + 16 + 1 = 116 \, \mathrm{g/mol}\). Calculating the initial concentration of caproic acid: \[ [\mathrm{C}_{5} \mathrm{H}_{11} \mathrm{COOH}]_{\mathrm{initial}} = \frac{11\,\mathrm{g/L}}{116\,\mathrm{g/mol}} \approx 0.095\,\mathrm{M} \] Since we have the hydrogen ion concentration, \([\mathrm{H}^{+}]\), we can assume that an equal amount of caproic acid has dissociated and produced the conjugate base. Therefore, we can write the concentrations at equilibrium as: \[ [\mathrm{C}_{5} \mathrm{H}_{11} \mathrm{COOH}] = 0.095 - 1.15 \times 10^{-3} \approx 0.094\,\mathrm{M} \] \[ [\mathrm{C}_{5} \mathrm{H}_{11} \mathrm{COO}^{-}] = 1.15 \times 10^{-3}\,\mathrm{M} \]

Calculate \(K_a\) using the ionization equation and the equilibrium expression

The ionization equation for caproic acid is: \[ \mathrm{C}_{5} \mathrm{H}_{11} \mathrm{COOH} + \mathrm{H}_{2}\mathrm{O} \rightleftharpoons \mathrm{C}_{5} \mathrm{H}_{11} \mathrm{COO}^{-} + \mathrm{H}^{+} \] The equilibrium expression for \(K_a\) is: \[ K_a = \frac{[\mathrm{C}_{5} \mathrm{H}_{11} \mathrm{COO}^{-}][\mathrm{H}^{+}]}{[\mathrm{C}_{5} \mathrm{H}_{11} \mathrm{COOH}]} \] Plugging in the equilibrium concentrations we derived in Step 2: \[ K_a = \frac{(1.15 \times 10^{-3})(1.15 \times 10^{-3})}{0.094} \approx 1.4 \times 10^{-5} \] Therefore, the acid dissociation constant, \(K_a\), for caproic acid is approximately \(1.4 \times 10^{-5}\).

Key concepts

pH Calculation

Understanding how to calculate pH is essential to chemistry, particularly when working with acids and bases. The pH of a solution is a measure of its acidity or alkalinity, expressed on a scale where 7 is neutral, values less than 7 are acidic, and values greater than 7 are basic (alkaline).

The pH is calculated using the concentration of hydrogen ions (\text{H}^{+}) in the solution, with the formula:
\[\text{pH} = -\text{log}([\text{H}^{+}])\].

For example, if the concentration of \text{H}^{+} is known to be 1.15 x 10^{-3} M, as in the case of the saturated caproic acid solution, the pH calculation would be:\[\text{pH} = -\text{log}(1.15 \times 10^{-3})\].

This process involves logarithms which are a part of the mathematical field concerned with exponents and their inverses. This calculation is vital for chemists because it allows one to infer how acidic or basic a solution is, which can in turn affect reaction rates, solubility, and chemical equilibria. pH meters and indicators commonly provide the pH value, but knowing how to calculate it manually is crucial for a deeper understanding of chemical processes.

Molar Mass

Molar mass, the mass of one mole of a substance (usually expressed in grams per mole), is a pivotal concept in chemistry that allows chemists to convert between the mass of a substance and the amount of substance (number of moles).

To calculate the molar mass of a compound, such as caproic acid (\text{C}_{5} \text{H}_{11} \text{COOH}), sum the atomic masses of each element multiplied by their respective number of atoms in the formula:
\[\text{Molar mass of } \text{C}_{5} \text{H}_{11} \text{COOH} = (12 \times 5) + (1 \times 11) + 12 + (16 \times 2) + 1 = 116 \text{ g/mol}\].

Molar mass plays a critical role in stoichiometry, which is the study of quantitative relationships in chemical reactions. It enables the calculation of how much of a reactant is needed to produce a desired amount of product or how much of a product can be made from available reactants. This makes molar mass calculations crucial for laboratory work and industrial applications.

Chemical Equilibrium

Chemical equilibrium is a dynamic state where the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time. It is an essential concept in chemical reactions, representing a balance that allows the determination of reaction conditions and behavior.

To describe chemical equilibria quantitatively, chemists use equilibrium constants, such as the acid dissociation constant (\text{K}_{a}). This constant provides a measure of the strength of an acid in solution, defined for the generic acid dissociation reaction
\[\text{HA} \rightleftharpoons \text{A}^{-} + \text{H}^{+}\].as
\[\text{K}_{a} = \frac{[\text{A}^{-}][\text{H}^{+}]}{[\text{HA}]}\].

For caproic acid, the \text{K}_{a} calculation based on the equilibrium concentrations of \text{H}^{+}, \text{C}_{5} \text{H}_{11} \text{COO}^{-} (the conjugate base), and \text{C}_{5} \text{H}_{11} \text{COOH} demonstrates the acid's tendency to donate protons to the solution. A small \text{K}_{a} value suggests a weak acid, which is partially dissociated in solution. Understanding chemical equilibrium is key in predicting the extent of chemical reactions in various conditions and is widely applied in chemical, pharmaceutical, and environmental fields.