Question

Acetic acid \(\left(\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\right)\) can be prepared by the action of the acetobacter organism on dilute solutions of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\). The equation for the reaction is $$ \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}(a q)+\mathrm{H}_{2} \mathrm{O} $$ How many milliliters of a \(12.5 \%\) (by volume) solution of ethanol are required to produce \(175 \mathrm{~mL}\) of \(0.664 \mathrm{M}\) acetic acid? (Density of pure ethanol \(=0.789 \mathrm{~g} / \mathrm{mL}\).)

Short answer

Answer: Approximately 54.32 mL of the 12.5% ethanol solution is required to produce 175 mL of 0.664 M acetic acid.

Step-by-step solution

Determine the moles of acetic acid required

To find the moles of acetic acid produced, we can use the formula: Moles of acetic acid = Molarity × Volume Since the volume of acetic acid is given in milliliters, we need to convert it to liters before using the formula: \(175 \mathrm{~mL} = 0.175 \mathrm{~L}\) Now insert the values: Moles of acetic acid = \(0.664 \mathrm{~M} \times 0.175 \mathrm{~L} = 0.1162\,\text{moles}\)

Calculate the moles of ethanol required

Using the stoichiometry of the balanced chemical equation, we can determine the number of moles of ethanol required. The balanced equation indicates that one mole of ethanol reacts with one mole of acetic acid, so the stoichiometry is 1:1. Thus, the same number of moles of ethanol is required. Moles of ethanol = \(0.1162\,\text{moles}\)

Calculate the mass of ethanol required

Convert moles of ethanol into grams using the molar mass of ethanol. Molar mass of ethanol: \(C_{2}H_{5}OH = 2(12.01) + 6(1.01) + 16.00 + 1.01 = 46.07\,\text{g/mol}\) Mass of ethanol = Moles × Molar mass Mass of ethanol = \(0.1162\,\text{moles} \times 46.07\,\text{g/mol} = 5.35\,\text{g}\)

Calculate the volume of the pure ethanol required

Using the given density of ethanol, we can find the volume of pure ethanol required to produce the acetic acid. Volume of pure ethanol = \(\frac{\text{mass}}{\text{density}}\) Volume of pure ethanol = \(\frac{5.35\,\text{g}}{0.789\,\text{g/mL}} = 6.79\,\text{mL}\)

Determine the volume of the 12.5% solution of ethanol required

Now, we know the volume of pure ethanol (6.79 mL) that is required to produce the acetic acid. We need to find out how much of the 12.5% (by volume) ethanol solution is needed. To do this, divide the volume of the pure ethanol by the percentage of ethanol in the solution: Volume of 12.5% ethanol solution = \(\frac{\text{Volume of pure ethanol}}{\text{Percentage}}\) Note that the percentage is given in decimal form: \(12.5\% = \frac{12.5}{100} = 0.125\) Volume of 12.5% ethanol solution = \(\frac{6.79\,\text{mL}}{0.125} = 54.32\,\text{mL}\) So, approximately 54.32 mL of the 12.5% ethanol solution is required to produce 175 mL of 0.664 M acetic acid.

Key concepts

Stoichiometry

Stoichiometry is the mathematical relationship between the quantities of reactants and products in a chemical reaction. It involves using the balanced chemical equation to determine the amount of each substance involved in the reaction.

For the acetic acid production from ethanol, stoichiometry allows us to understand that for every mole of ethanol reacting, one mole of acetic acid is produced. This 1:1 ratio is crucial as it guides us on how much starting material we need to create a desired amount of product. It is especially helpful when scaling up reactions for industrial purposes or ensuring that chemical reactions are cost-effective and environmentally friendly.

Students can find this concept clearer by visualizing the reactants and products as puzzle pieces, where one piece of ethanol fits perfectly with one piece of oxygen to make one piece of acetic acid and one piece of water. The precise calculations stemming from stoichiometry ensure that all pieces fit together without any leftovers.

Molarity Calculation

Molarity is a measure of the concentration of a solute in a solution. It indicates how many moles of a substance are present in one liter of solution. The calculation of molarity is a fundamental skill in chemistry, allowing us to express concentrations for use in various chemical calculations, including stoichiometry and preparation of solutions.

The molarity formula is written as: \[ Molarity (M) = \frac{moles \text{ of solute}}{liters \text{ of solution}} \]
When solving problems that involve finding how much reactant is needed to produce a specific concentration of a product, you convert volumes to liters and then use the molarity of the desired product to find the moles required. Students should ensure correct units are used throughout the calculation to avoid mistakes. Understanding molarity also empowers students to predict the outcomes of mixing different solutions, which is vital in both laboratory and industrial processes.

Chemical Reaction of Acetic Acid

The chemical reaction of acetic acid production from ethanol is an example of an oxidation reaction where ethanol is oxidized by the acetobacter organism to yield acetic acid. The balanced equation for this reaction is: \[ C_2H_5OH_{(aq)} + O_2_{(g)} \longrightarrow HC_2H_3O_2_{(aq)} + H_2O \]
Understanding this chemical reaction involves recognizing that molecular oxygen acts as an oxidizing agent, converting ethanol to acetic acid and water. This type of reaction is widely used in the food industry, specifically in the production of vinegar, where ethanol in alcoholic beverages is converted to acetic acid under controlled conditions.

Additionally, understanding the chemical nature of acetic acid, its properties, and how it forms through such reactions, serves as a basis for learning about acidity, the pH scale, and the different functional groups present in organic compounds.