Problem 125 - Chemistry: An Atoms-Focused Approach

Question

Some people who prefer natural foods make their own apple cider vinegar. They start with freshly squeezed apple juice that contains about \(6 \%\) natural sugars. These sugars, which all have nearly the same empirical formula, \(\mathrm{CH}_{2} \mathrm{O},\) are fermented with yeast in a chemical reaction that produces equal numbers of moles of ethanol \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\right)\) and carbon dioxide. The product of fermentation, called hard cider, undergoes an acid fermentation step in which ethanol and dissolved oxygen gas react together to form acetic acid (CH \(_{3} \mathrm{COOH}\) ) and water. This acetic acid is the principal solute in vinegar. a. Write a balanced chemical equation for the fermentation of natural sugars to ethanol and carbon dioxide. You may use in the equation the empirical formula given in the preceding paragraph. b. Write a balanced chemical equation for the acid fermentation of ethanol to acetic acid. c. What are the oxidation states of carbon in the reactants and products of the two fermentation reactions? d. If a sample of apple juice contains \(1.00 \times 10^{2} \mathrm{g}\) of natural sugar, what is the maximum quantity of acetic acid that could be produced by the two fermentation reactions?

Step-by-Step Solution

Answer

Question: Determine the balanced chemical equations for the fermentation of natural sugars to ethanol and carbon dioxide, the acid fermentation of ethanol to acetic acid, the oxidation states of carbon in the reactants and products of the two fermentation reactions, and the maximum quantity of acetic acid produced by the two fermentation reactions given a sample of apple juice containing \(1.00 \times 10^{2} \mathrm{g}\) of natural sugar. Answer: The balanced chemical equation for fermentation of natural sugars to ethanol and carbon dioxide is: $$\mathrm{C_{2}H_{4}O_{2}} \rightarrow \mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{OH} + \mathrm{CO}_{2}$$ The balanced chemical equation for acid fermentation of ethanol to acetic acid is: $$\mathrm{C_{2}H_{6}O} + \mathrm{O}_{2} \rightarrow \mathrm{CH}_{3}\mathrm{COOH} + \mathrm{2H}_{2}\mathrm{O}$$ The oxidation states of carbon in the reactants and products are: - In \(\mathrm{C_{2}H_{4}O_{2}}\), each C has an oxidation state of 0. - In \(\mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{OH}\), one C has an oxidation state of -1, and the other has an oxidation state of +1. - In \(\mathrm{CO}_{2}\), C has an oxidation state of +4. - In \(\mathrm{C_{2}H_{6}O}\), one C has an oxidation state of -1, and the other has an oxidation state of +1. - In \(\mathrm{CH}_{3}\mathrm{COOH}\), one C has an oxidation state of -3, and the other has an oxidation state of +3. The maximum quantity of acetic acid that could be produced from \(1.00 \times 10^{2} \mathrm{g}\) of natural sugar is 100 g.

Step 1: Write the balanced chemical equation for the fermentation of natural sugars to ethanol and carbon dioxide

The empirical formula of natural sugars is given as \(\mathrm{CH}_{2} \mathrm{O}\). They undergo fermentation using yeast to produce equal moles of ethanol and carbon dioxide. The unbalanced chemical equation is: $$\mathrm{CH}_{2}\mathrm{O} \rightarrow \mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{OH} + \mathrm{CO}_{2}$$ To balance the equation, we need to make sure that the atoms in the reactants equal the atoms in the products. To balance this equation, we will add a coefficient of 2 to the natural sugar: $$\mathrm{C_{2}H_{4}O_{2}} \rightarrow \mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{OH} + \mathrm{CO}_{2}$$

Step 2: Write the balanced chemical equation for the acid fermentation of ethanol to acetic acid

The acid fermentation process involves the reaction between ethanol and dissolved oxygen gas which forms acetic acid and water. The unbalanced chemical equation is: $$\mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{OH} + \mathrm{O}_{2} \rightarrow \mathrm{CH}_{3}\mathrm{COOH} + \mathrm{H}_{2}\mathrm{O}$$ To balance the equation, we need to make sure that the atoms in the reactants equal the atoms in the products. To balance this equation, we will add a coefficient of 2 to ethanol and water: $$\mathrm{C_{2}H_{6}O} + \mathrm{O}_{2} \rightarrow \mathrm{CH}_{3}\mathrm{COOH} + \mathrm{2H}_{2}\mathrm{O}$$

Step 3: Find the oxidation states of carbon in the reactants and products of the two fermentation reactions

To find the oxidation states, we will follow the general rules for assigning oxidation numbers. 1. The oxidation state of oxygen is usually -2 2. The oxidation state of hydrogen is usually +1 For the fermentation reaction, \(\mathrm{C_{2}H_{4}O_{2}}\) to \(\mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{OH}\) and \(\mathrm{CO}_{2}\), we find the oxidation states as follows: - In \(\mathrm{C_{2}H_{4}O_{2}}\), each C is oxidized by 1 oxygen and reduced by 2 hydrogens; so x = 0. - In \(\mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{OH}\), C is -1 and C is +1, as they are reduced or oxidized by 3 hydrogens and 1 oxygen. - In \(\mathrm{CO}_{2}\), C is +4 because 2 oxygens reduced it. For the acid fermentation reaction, \(\mathrm{C_{2}H_{6}O}\) to \(\mathrm{CH}_{3}\mathrm{COOH}\), we find the oxidation states as follows: - In \(\mathrm{C_{2}H_{6}O}\), C is -1 and C is +1, like in the first equation. - In \(\mathrm{CH}_{3}\mathrm{COOH}\), C is -3, as it is reduced by 3 hydrogens, and C is +3, as it is oxidized by 3 oxygens.

Step 4: Calculate the maximum quantity of acetic acid that could be produced

Given a sample of apple juice containing \(1.00 \times 10^{2} \mathrm{g}\) of natural sugar, the maximum quantity of acetic acid produced can be calculated using stoichiometry. From the two balanced chemical equations, we have: $$\mathrm{C_{2}H_{4}O_{2}} \rightarrow \mathrm{CH}_3\mathrm{CH}_2\mathrm{OH} + \mathrm{CO}_2$$ $$\mathrm{C_{2}H_{6}O} + \mathrm{O}_{2} \rightarrow \mathrm{CH}_{3}\mathrm{COOH} + 2 \mathrm{H}_{2}\mathrm{O}$$ From the first equation, 1 mole of \(\mathrm{C_{2}H_{4}O_{2}}\) produces 1 mole of \(\mathrm{CH}_3\mathrm{CH}_2\mathrm{OH}\). From the second equation, 1 mole of \(\mathrm{C_{2}H_{6}O}\) produces 1 mole of \(\mathrm{CH}_3\mathrm{COOH}\). a) Find moles of \(\mathrm{C_{2}H_{4}O_{2}}\): The molar mass of \(\mathrm{C_{2}H_{4}O_{2}} = 12 + 12 + 4 + 32 = 60 \mathrm{g/mol}\) $$\text{moles of}\ \mathrm{C_{2}H_{4}O_{2}} = \frac{1.00 \times 10^{2}\ \mathrm{g}}{60\ \mathrm{g/mol}} = 1.67\ \text{moles}$$ b) Calculate moles of \(\mathrm{CH}_{3}\mathrm{COOH}\): In the last step, we found that 1 mole of \(\mathrm{C_{2}H_{4}O_{2}}\) will produce 1 mole of \(\mathrm{CH}_{3}\mathrm{COOH}\), so it will produce 1.67 moles of acetic acid. c) Convert moles of \(\mathrm{CH}_{3}\mathrm{COOH}\) to grams: The molar mass of \(\mathrm{CH}_{3}\mathrm{COOH} = 12 + 4 + 12 + 16 + 16 + 1 = 60 \mathrm{g/mol}\) $$\text{mass of}\ \mathrm{CH}_{3}\mathrm{COOH} = 1.67\ \mathrm{moles} \times 60\ \mathrm{g/mol} = 100\ \mathrm{g}$$ Thus, with a sample of apple juice containing \(1.00 \times 10^{2} \mathrm{g}\) of natural sugar, the maximum quantity of acetic acid that could be produced is 100 g.

Key Concepts

Apple Cider Vinegar

Apple cider vinegar is a popular natural remedy made from fermented apple juice. It is known for its sharp, tangy flavor and pungent aroma. Made through a two-step fermentation process, apple cider vinegar has a rich history of use for both culinary and medicinal purposes. This transformation begins with the simple sugars found in apples. During the initial fermentation, these sugars are converted into ethanol, a type of alcohol. In the second stage, this ethanol is further fermented into acetic acid, the key component that gives vinegar its distinct sour taste.

Apple cider vinegar is praised for its potential health benefits, which are largely attributed to the acetic acid. It is thought to help with digestion, blood sugar control, and weight management. Moreover, apple cider vinegar is often used in cooking, from salad dressings to marinades, adding a bright and zesty note to various dishes.

Fermentation Process

Fermentation is a natural metabolic process that converts sugar to acids, gases, or alcohol. In the case of apple cider vinegar, fermentation is crucial for transforming apple juice into vinegar. This process occurs in two stages: alcohol fermentation and acid fermentation.

In the first stage, yeast is introduced to the apple juice. Yeast consumes the natural sugars and converts them into ethanol and carbon dioxide, yielding a product known as hard cider. This happens under anaerobic conditions, meaning without oxygen.

The second stage is aerobic, requiring oxygen. During this phase, acetobacter bacteria convert the ethanol into acetic acid. This two-step fermentation process is essential for the creation of apple cider vinegar, as each stage contributes specific changes both chemically and physically to the product.

Fermentation not only preserves the apple juice and gives it its characteristic flavor, but it also enhances nutrients and can produce beneficial compounds that are important for health.

Oxidation States

Oxidation states play a significant role in understanding chemical reactions, particularly those involving organic compounds like apple cider vinegar. In simple terms, an oxidation state is a measure of the degree of oxidation (loss of electrons) of an atom in a chemical compound.

During the fermentation processes, the initial compound \(\mathrm{CH}_{2}\mathrm{O}\), undergoes changes in oxidation states as it is converted into different products. In the first stage of fermentation, the sugars are transformed into ethanol \( \mathrm{C}_{2}\mathrm{H}_{6}\mathrm{O} \) and carbon dioxide \( \mathrm{CO}_{2} \). Here, carbon atoms undergo changes from their initial oxidation states as they either gain or lose electrons.

Similarly, in the second stage of fermentation, ethanol is oxidized to acetic acid (\(\mathrm{CH}_{3}\mathrm{COOH}\)). Understanding these oxidation states helps in predicting the reactivity and the changes occurring at each step of the process. It highlights the balance between different elements that allows for smooth completion of the chemical transformations.

Stoichiometry

Stoichiometry is like a recipe in a chemical sense; it's the study of quantitative relationships between reactants and products in a chemical reaction. In the context of apple cider vinegar production, understanding stoichiometry is essential to determine the exact amounts of ethanol and acetic acid produced from the natural sugars in apple juice.

The chemical equations used to describe the fermentation processes illustrate these stoichiometric relationships. For example, converting natural sugars to ethanol and carbon dioxide requires balancing each molecule's atoms—carbon, hydrogen, and oxygen. This ensures that no atoms are lost or gained in the process.

In practice, if you have a certain mass of apple juice composed of a specific percentage of sugars, stoichiometry allows you to calculate how much acetic acid can be produced from it. By understanding and applying stoichiometric principles, we can predict yields and even adjust conditions to optimize the production of desired compounds, like acetic acid in vinegar.