Problem 10 - General Chemistry: Principles and Modern Applications

Question

Lactic acid, \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{COOH},\) is found in sour milk. A solution containing \(1.00 \mathrm{g} \mathrm{NaCH}_{3} \mathrm{CH}_{2} \mathrm{COO}\) in 100.0 \(\mathrm{mL}\) of \(0.0500 \mathrm{M} \mathrm{HC}_{3} \mathrm{H}_{5} \mathrm{O}_{3}\) has a \(\mathrm{pH}=4.11 .\) What is \(K_{\mathrm{a}}\) of lactic acid?

Step-by-Step Solution

Answer

The Ka of lactic acid is \(1.413x10^{-5}\).

Step 1: Find the concentration of sodium lactate

First, you need to find the concentration of sodium lactate in the solution. The molar mass of \(\mathrm{NaCH}_{3} \mathrm{CH}_{2} \mathrm{COO}\) is about 112 g/mol. Given that there is 1g of sodium lactate in the solution, the molarity, M, of \(\mathrm{NaCH}_{3} \mathrm{CH}_{2}\mathrm{COO}\) is \(1g/(112 g/mol) =0.00893 mol\). Since it's in 100 mL of solution, the concentration is \(0.00893 mol/0.1 L = 0.0893 M\).

Step 2: Find the concentration of acid and hydronium ion

You are told that there is 0.050 M of lactic acid in the solution, so that is your \(\[HA\]\). The pH of the solution is given as 4.11, so to find the \([H_3O^+]\), use the formula \([H_3O^+] =10^{-pH} = 10^{-4.11}=7.94x10^{-5} M\).

Step 3: Substitute into Ka expression

Finally, substitute these values into the Ka expression: \(Ka = [H_3O^+][A^-]/[HA] = ( 7.94x10^{-5} M)(0.0893 M)/(0.0500 M) = 1.413x10^{-5}\).

Key Concepts

Lactic Acid

Lactic acid, chemically represented as \(\text{CH}_{3}\text{CH}_{2}\text{COOH}\), is an organic compound commonly found in sour milk and other fermented products. It's a simple alpha-hydroxy acid, which means it has both a carboxylic group \(-COOH\) and a hydroxyl group \(-OH\) on an adjacent carbon. This structure allows lactic acid to participate in various biochemical reactions, especially in metabolic pathways.A key property of lactic acid is its ability to dissociate in aqueous solutions, releasing a hydrogen ion \(H^+\) and forming the lactate ion \(CH_3CH_2COO^-\). This process is crucial in determining the acid's strength and behavior in solutions. The strength of lactic acid, like other acids, can be measured by its **acid dissociation constant** \(K_a\). The larger the \(K_a\), the stronger the acid and the more readily it donates a proton to water to produce \(H_3O^+\).

pH Calculation

Understanding pH calculation is essential for measuring the acidity of a solution, such as one containing lactic acid. The **pH** is a numeric scale used to specify the acidity or basicity of an aqueous solution, calculated by the formula:\[pH = -\log[H_3O^+]\]In the provided solution, a pH of 4.11 indicates a certain concentration of hydronium ions \([H_3O^+]\). We calculate this by rearranging the pH formula to find \([H_3O^+]\):\[[H_3O^+] = 10^{-4.11} \approx 7.94 \times 10^{-5} \text{ M}\]This calculation helps signify how acidic or basic a solution is, where lower pH values represent higher acidity and higher concentrations of \(H_3O^+\) ions. It's important for tasks such as determining the acid dissociation constant \(K_a\), which provides deeper insight into the acid's dissociation in its environment and is a crucial step in many chemical analysis processes.

Concentration of Solutions

In chemistry, the concentration of solutions is a fundamental concept representing the amount of solute dissolved in a given volume of solvent. For this exercise, calculating the concentration is essential to determining the acid dissociation constant \(K_a\).When you're given the mass of a compound such as sodium lactate, \(\text{NaCH}_3\text{CH}_2\text{COO}\), converting this mass to moles is the first step. Utilizing the molar mass, you can calculate the moles:

• Molar mass of sodium lactate = 112 g/mol
• Mass = 1 g
• Moles = \( \frac{1 \, \text{g}}{112 \, \text{g/mol}} = 0.00893 \, \text{mol} \)

Since the solution's volume is 100 mL (or 0.1 L), the molarity \(M\) of sodium lactate is then calculated as:\[M = \frac{0.00893 \, \text{mol}}{0.1 \, \text{L}} = 0.0893 \, \text{M}\]These concentration values are essential for plugging into the \(K_a\) expression \(K_a = \frac{[H_3O^+][A^-]}{[HA]}\), where \([A^-]\) and \([HA]\) reflect the concentrations of the lactate ion and lactic acid respectively. By understanding and calculating these concentrations, one can further analyze and understand the behaviors of acidic and basic solutions.