Question

Lactic acid \(\left(\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH}\right)\) has one acidic hydrogen. A 0.10 \(\mathrm{M}\) solution of lactic acid has a pH of \(2.44 .\) Calculate \(K_{a} .\)

Short answer

The acid dissociation constant (\(K_a\)) for lactic acid is approximately \(1.34 \times 10^{-4}\).

Step-by-step solution

Finding the \(\mathrm{H}^+\) concentration

We are given that a 0.10 M solution of lactic acid has a pH of 2.44. We can use the pH formula to find the concentration of hydrogen ions in the solution: \[pH = -\log [\mathrm{H}^+]\] Plugging in the pH value: \[2.44 = -\log [\mathrm{H}^+]\] Now, let's solve for the \(\mathrm{H}^+\) concentration: \[[\mathrm{H}^+] = 10^{-2.44}\] This gives us: \[[\mathrm{H}^+] \approx 3.65 \times 10^{-3} \, \mathrm{M}\]

Setting up the equilibrium equation

Lactic acid (denoted as \(\mathrm{HLa}\)) dissociates into hydrogen ions and lactate ions, as shown by the equilibrium reaction below: \[\mathrm{HLa} \rightleftharpoons \mathrm{H}^+ + \mathrm{La}^-\] The equilibrium constant for this reaction (\(K_a\)) can be represented as follows: \[K_a = \frac{[\mathrm{H}^+][\mathrm{La}^-]}{[\mathrm{HLa}]}\]

Finding the concentrations of lactic acid and lactate ions at equilibrium

We know the initial concentration of lactic acid (\([\mathrm{HLa}]_{\text{initial}} = 0.10\) M) and the change in concentration of \(\mathrm{H}^+\) ions during the dissociation (\([\mathrm{H}^+]_{\text{change}} = 3.65 \times 10^{-3}\) M). Since lactic acid loses one proton for each hydrogen ion produced, the change in lactate ion concentration is equal to the change in hydrogen ion concentration. Thus, we can find the concentrations of lactic acid and lactate ions at equilibrium by using the initial concentration and the change in concentration: \[ \begin{array}{c|c|c|c} & \mathrm{Initial} & \mathrm{Change} & \mathrm{Equilibrium}\\ \hline \mathrm{HLa} & 0.1 & -3.65 \times 10^{-3} & 0.1 - 3.65 \times 10^{-3}\\ \mathrm{H}^+ & 0 & 3.65 \times 10^{-3} & 3.65 \times 10^{-3} \\ \mathrm{La}^- & 0 & 3.65 \times 10^{-3} & 3.65 \times 10^{-3} \end{array} \]

Calculating \(K_a\) using the equilibrium concentrations

Now we can plug the equilibrium concentrations into the equilibrium constant expression: \[K_a = \frac{[\mathrm{H}^+][\mathrm{La}^-]}{[\mathrm{HLa}]}\] Substitute the equilibrium concentrations: \[K_a = \frac{(3.65 \times 10^{-3})(3.65 \times 10^{-3})}{(0.1 - 3.65 \times 10^{-3})}\] Now, perform the calculations: \[K_a \approx 1.34 \times 10^{-4}\] So, the acid dissociation constant (\(K_a\)) for lactic acid is approximately \(1.34 \times 10^{-4}\).

Key concepts

pH Calculation

The pH scale is a measure of the acidity or basicity of an aqueous solution. It can be calculated using the formula:
\[pH = -\text{log}[H^+]\].
The hydrogen ion concentration (\([H^+]\)) is inversely related to pH: low pH values correspond to high concentrations of hydrogen ions, indicative of an acidic solution, while high pH values indicate lower concentrations of hydrogen ions and a basic solution. Calculating pH involves taking the negative logarithm of the hydrogen ion concentration in molarity. For instance, if a 0.10 M solution of an acid has a pH of 2.44, this reflects a hydrogen ion concentration of: \[[H^+] = 10^{-2.44} \approx 3.65 \times 10^{-3} M\].
Understanding this relationship can help students navigate chemistry problems that involve finding the strength of acids or bases in solution.

Chemical Equilibrium

In the context of acid and base reactions, chemical equilibrium is the state where the rate of the forward reaction (acid dissociating into ions) equals the rate of the reverse reaction (ions recombining into the acid). This balance does not mean the reactants and products are in equal concentration, but rather that their concentrations have stabilized at a constant ratio.
The equilibrium constant (\(K\)) for a reaction is the ratio of the concentrations of the products raised to the power of their coefficients to the concentrations of the reactants raised to the power of their coefficients as shown in a balanced chemical reaction. For the dissociation of an acid, which releases hydrogen ions, the equilibrium constant is denoted as \(K_a\). It’s crucial to grasp that when the value of \(K_a\) is high, the acid is strong as it dissociates more in water; conversely, a low \(K_a\) values indicate a weaker acid.

Dissociation of Acids

The dissociation of acids in water is a fundamental concept in acid-base chemistry. When an acid dissociates, it donates a hydrogen ion (\(H^+\)), yielding a conjugate base. The extent to which an acid dissociates in a solution is determined by its \(K_a\), the acid dissociation constant. An acid like lactic acid, \(CH_3CH(OH)COOH\), will dissociate in water to produce hydronium ions (\(H^+\)) and its conjugate base, lactate ions (\(La^-\)).
The equilibrium concentration of each species can be represented in an ICE (Initial, Change, Equilibrium) table, allowing for the calculation of the \(K_a\) value by finding the ratio of the product of the equilibrated concentrations of products over the concentration of the undissociated acid. The strength of the acid is related to how much it dissociates, which is an important concept for predicting the behavior of acids in various chemical reactions and in different environments.

Lactic Acid

Lactic acid, a carboxylic acid represented by the formula \(CH_3CH(OH)COOH\), is known for its role in various biological processes, such as lactic acid fermentation. In an aqueous solution, lactic acid can donate a hydrogen ion (\(H^+\)) from its carboxyl group, turning into its conjugate base, lactate (\(La^-\)). This simple structure of lactic acid having one acidic hydrogen allows it to act as a monocarboxylic acid.
The acid dissociation constant (\(K_a\)) for lactic acid can be experimentally determined from its pH in solution and gives insight into its acid strength. Lactic acid's \(K_a\) value is typically lower than that of strong acids, signifying that it does not completely dissociate in water, making it a weak acid. Understanding properties of weak acids like lactic acid is essential in fields ranging from biochemistry to food science.