Problem 45 - Chemistry: Principles and Reactions

Question

Caproic acid, \(\mathrm{HC}_{6} \mathrm{H}_{11} \mathrm{O}_{2}\), is found in coconut oil and is used in making artificial flavors. A solution is made by dissolving \(0.450 \mathrm{~mol}\) of caproic acid in enough water to make \(2.0 \mathrm{~L}\) of solution. The solution has \(\left[\mathrm{H}^{+}\right]=1.7 \times\) \(10^{-3} M\). What is \(K_{\mathrm{a}}\) for caproic acid?

Step-by-Step Solution

Answer

Question: Determine the acid dissociation constant (Ka) of caproic acid, given that its moles are 0.450 mol, the volume of the solution is 2.0 L, and the concentration of hydronium ions (H+) is 1.7 x 10^-3 M. Answer: The acid dissociation constant (Ka) of caproic acid is 1.29 x 10^-5.

Step 1: Write the chemical equation for the dissociation of caproic acid

Caproic acid, denoted as \(\mathrm{HC}_{6} \mathrm{H}_{11} \mathrm{O}_{2}\), dissociates in water as follows: $$\mathrm{HC}_{6} \mathrm{H}_{11} \mathrm{O}_{2} \rightleftharpoons \mathrm{H}^{+} + \mathrm{C}_{6} \mathrm{H}_{11} \mathrm{O}_{2}^{-}$$

Step 2: Write the expression for the dissociation constant, \(K_a\)

The equation for \(K_a\) is: $$K_a = \frac{[\mathrm{H}^+][\mathrm{C}_{6} \mathrm{H}_{11} \mathrm{O}_{2}^{-}]}{[\mathrm{HC}_{6} \mathrm{H}_{11} \mathrm{O}_{2}]}$$

Step 3: Calculate the initial concentration of caproic acid

To get the initial concentration of caproic acid, divide the moles by the volume of the solution: $$[\mathrm{HC}_{6} \mathrm{H}_{11} \mathrm{O}_{2}]_0 = \frac{0.450 \,\mathrm{mol}}{2.0 \,\mathrm{L}} = 0.225 \,\mathrm{M}$$

Step 4: Create an ICE (Initial, Change, Equilibrium) table

Construct an ICE table for the concentrations of each species involved in the dissociation of caproic acid: | | \(\mathrm{HC}_{6} \mathrm{H}_{11} \mathrm{O}_{2}\) | \(\mathrm{H}^+\) | \(\mathrm{C}_{6} \mathrm{H}_{11} \mathrm{O}_{2}^{-}\) | |------|-----------|-------|-------------| | I | 0.225 M | 0 M | 0 M | | C | -x M | +x M | +x M | | E | 0.225-x M | x M | x M | Since we know the final concentration of \(\mathrm{H}^+\) to be \(1.7 \times 10^{-3}\,\mathrm{M}\), we can replace x in the last row with the given concentration: | | \(\mathrm{HC}_{6} \mathrm{H}_{11} \mathrm{O}_{2}\) | \(\mathrm{H}^+\) | \(\mathrm{C}_{6} \mathrm{H}_{11} \mathrm{O}_{2}^{-}\) | |------|-----------|-------|-------------| | I | 0.225 M | 0 M | 0 M | | C | -x M | +x M | +x M | | E | 0.225-x M | \(1.7 \times 10^{-3}\,\mathrm{M}\) | \(1.7 \times 10^{-3}\,\mathrm{M}\) |

Step 5: Calculate the concentration of caproic acid at equilibrium

From the ICE table, the concentration of caproic acid at equilibrium is: $$[\mathrm{HC}_{6} \mathrm{H}_{11} \mathrm{O}_{2}]_{eq} = 0.225 - 1.7 \times 10^{-3} = 0.2233\,\mathrm{M}$$

Step 6: Calculate the \(K_a\) of caproic acid

Using the equilibrium concentrations found in the ICE table, solve for the \(K_a\) using the equilibrium constant expression: $$K_a = \frac{(1.7 \times 10^{-3})(1.7 \times 10^{-3})}{0.2233} = 1.29 \times 10^{-5} $$ The value of \(K_a\) for caproic acid is \(1.29 \times 10^{-5}\).

Key Concepts

Chemical Equilibrium

When a chemical reaction reaches a state where the rate of the forward reaction equals the rate of the reverse reaction, it has achieved chemical equilibrium. At this point, the concentrations of all reactants and products remain constant over time, although both reactions continue to occur. This dynamic process is critical in understanding many aspects of chemistry, including acid-base reactions. Occasionally, certain external factors such as temperature or pressure changes can shift the equilibrium, altering the concentrations of the involved substances. It's important to note that equilibrium does not imply equal concentrations of reactants and products but rather a constant ratio that depends on the specific reaction conditions and the inherent properties of the substances involved.

Acid Dissociation Constant (Ka)

The strength of an acid in a solution is quantified by its acid dissociation constant (Ka). Ka represents the equilibrium constant for the dissociation of an acid into its corresponding ions. The formula for the dissociation constant expresses the ratio of the product of the concentrations of the ions to the concentration of the undissociated acid. For a generic acid, denoted by HA, dissociating into H+ and A-, the expression for Ka would be:
\[\frac{[H^{+}][A^{-}]}{[HA]}\]
Acids with higher Ka values are stronger because they dissociate more completely into their ions in solution. Calculating Ka is crucial for understanding the extent of acidity in solutions, allowing chemists and students to predict how acids will react in different scenarios.

ICE Table Method

The ICE table method is a systematic tool used to organize and calculate changes in concentrations for a reaction that's attained equilibrium. ICE stands for Initial, Change, and Equilibrium. This approach starts with writing down the initial concentrations of reactants and products, the changes that occur as reactants form products, and finally the concentrations at equilibrium. This method simplifies the calculation of unknown concentrations and is particularly effective for problems involving acid-base equilibria. When applying the ICE table, it's critical to ensure that the stoichiometry of the reaction is accurately represented, and all the changes are in accordance with the mole ratios dictated by the balanced chemical equation. Another important note while using the ICE table is that the change (C) is always the smallest value that can align with the stoichiometry; assuming it's x helps in solving for unknown concentrations when dealing with weak acids or bases in aqueous solutions.