Problem 127 - Chemistry: The Central Science in SI Units, Global Edition

Question

One of the best-selling light, or low-calorie, beers is \(4.2 \%\) alcohol by volume and a 355 -mL serving contains 110 Calories; remember: 1 Calorie \(=1000 \mathrm{cal}=1 \mathrm{kcal} .\) To estimate the percentage of Calories that comes from the alcohol, consider the following questions. (a) Write a balanced chemical equation for the reaction of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH},\) with oxygen to make carbon dioxide and water. (b) Use enthalpies of formation in Appendix \(\mathrm{C}\) to determine \(\Delta H\) for this reaction. \((\mathbf{c})\) If \(4.2 \%\) of the total volume is ethanol and the density of ethanol is \(0.789 \mathrm{~g} / \mathrm{mL},\) what mass of ethanol does a \(355-\mathrm{mL}\) serving of light beer contain? (d) How many Calories are released by the metabolism of ethanol, the reaction from part (a)? (e) What percentage of the 110 Calories comes from the ethanol?

Step-by-Step Solution

Answer

The balanced chemical equation for the reaction of ethanol with oxygen is \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} + \dfrac{3}{2}\mathrm{O}_{2} → 2\mathrm{CO}_{2} + 3\mathrm{H}_{2} \mathrm{O}\). The change in enthalpy (ΔH) for the reaction is -1308.5 kJ/mol. In a 355-mL light beer with 4.2% of ethanol, the mass of ethanol is 11.81 g. The Calories released by the metabolism of ethanol are approximately -80.1 Calories. Therefore, about 72.8% of the total 110 Calories in the light beer comes from the metabolism of ethanol.

Step 1: (a) Balanced Chemical Equation

To balance the reaction of ethanol with oxygen, we first need to write down the reactants and products: Ethanol + Oxygen → Carbon Dioxide + Water Now in terms of chemical formulas, we have: \[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} + \mathrm{O}_{2} → \mathrm{CO}_{2} + \mathrm{H}_{2} \mathrm{O}\] Next, we balance the equation: \[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} + \dfrac{3}{2}\mathrm{O}_{2} → 2\mathrm{CO}_{2} + 3\mathrm{H}_{2} \mathrm{O}\]

Step 2: (b) Calculation of ΔH

Use the enthalpies of formation to calculate the change in enthalpy (ΔH) for the reaction: \[\Delta H = \sum H_{\text{products}} - \sum H_{\text{reactants}}\] From Appendix C, we have the following values: ΔHf (ethanol, liquid) = -277.69 kJ/mol ΔHf (oxygen, gas) = 0 kJ/mol ΔHf (carbon dioxide, gas) = -393.51 kJ/mol ΔHf (water, liquid) = -285.83 kJ/mol Now, we substitute the values and sum the enthalpies for reactants and products: ΔH = 2(-393.51 kJ/mol) + 3(-285.83 kJ/mol) - (-277.69 kJ/mol) ΔH = -1308.5 kJ/mol

Step 3: (c) Mass of ethanol in 355-mL light beer

Given that 4.2% of the total volume is ethanol and the density of ethanol is 0.789 g/mL, we can calculate the mass of ethanol in a 355-mL serving of light beer: Mass of ethanol = (4.2% × 355 mL) × (0.789 g/mL) Mass of ethanol = 11.81 g

Step 4: (d) Calories released by the metabolism of ethanol

We know the reaction releases -1308.5 kJ/mol of energy. To determine the Calories released, first convert the energy to Calories: Energy released = -1308.5 kJ/mol × (1000 J/1 kJ) × (1 Cal/4184 J) ≈ -313.0 Cal/mol Now, find the Calories released by 11.81 g of ethanol: Molar mass of ethanol = 46.07 g/mol Moles of ethanol = 11.81 g ÷ 46.07 g/mol ≈ 0.256 mol Calories released = 0.256 mol × (-313.0 Cal/mol) ≈ -80.1 Calories

Step 5: (e) Percentage of Calories from ethanol

Finally, determine the percentage of the 110 Calories in the light beer that comes from the metabolism of ethanol: Percentage of Calories from ethanol = (-80.1 Calories ÷ 110 Calories) × 100% Percentage of Calories from ethanol ≈ 72.8% So, approximately 72.8% of the Calories in the 355-mL light beer comes from the metabolism of ethanol.

Key Concepts

Chemical Equations

In chemistry, chemical equations are symbolic representations of chemical reactions. They show the reactants and products in a chemical process. Chemical equations provide important information such as the relative amounts and types of atoms involved.

When writing chemical equations, it's crucial to balance them. This means adjusting the coefficients in front of compounds to ensure the same number of each type of atom on both sides of the equation. This obeys the Law of Conservation of Mass, which states that matter cannot be created or destroyed.

• Reactants are the starting substances in a chemical reaction.
• Products are the substances that are formed as a result of the reaction.
• Coefficients are numbers placed before compounds to balance the equation.

For example, the balanced reaction of ethanol combustion is: \[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} + \dfrac{3}{2}\mathrm{O}_{2} \rightarrow 2\mathrm{CO}_{2} + 3\mathrm{H}_{2} \mathrm{O}\]This shows ethanol (a reactant) reacting with oxygen to form carbon dioxide and water (products).

Thermodynamics

Thermodynamics is the branch of chemistry that deals with energy changes. It involves concepts like enthalpy, entropy, and free energy. In chemical reactions, thermodynamics helps us understand how energy is transferred.

When ethanol combusts, it releases energy. This energy release is quantified as the change in enthalpy (\(\Delta H\)). Enthalpy is a measure of the heat content in a system at constant pressure.

• In exothermic reactions, such as ethanol combustion, heat is released.
• In endothermic reactions, heat is absorbed.

To calculate \(\Delta H\) for a reaction, we use the enthalpies of formation for each compound involved. We apply the formula:\[\Delta H = \sum H_{\text{products}} - \sum H_{\text{reactants}}\]For the ethanol reaction, we found \(\Delta H\) to be \(-1308.5 \text{kJ/mol}\), indicating a significant release of energy. This energy release largely accounts for the calorie content derived from ethanol in metabolism.

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It enables scientists to predict the quantities of substances consumed and produced.

In our example, the stoichiometry of ethanol combustion helps determine how much heat is released per mole of ethanol burned. We convert masses to moles using molar mass, and then use the balanced chemical equation to calculate energy changes.

• Moles are the standard unit for expressing amounts of a substance in chemistry.
• The molar mass of ethanol (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\)) is \(46.07 \text{g/mol}\).

Given the mass of ethanol (11.81 g), we can convert this to moles:\[\text{Moles of ethanol} = \frac{11.81\text{ g}}{46.07\text{ g/mol}} \approx 0.256 \text{ mol}\]This information helps us calculate the total caloric energy released during ethanol metabolism.

Alcohol Metabolism

Alcohol metabolism refers to how the body processes and breaks down ethanol. This process releases energy that can be measured in Calories. The body converts ethanol into acetaldehyde, then into acetic acid, and finally into carbon dioxide and water.

The key purpose of understanding alcohol metabolism in this context is to calculate what percentage of the energy (Calories) in a food or drink comes from alcohol. It’s important for nutrition labels and dietary guidelines.

• Metabolism of ethanol primarily occurs in the liver.
• The energy released by the complete metabolism of ethanol can be calculated using \(\Delta H\) from chemical reactions.

In the case of the 355 mL serving of light beer, \(\approx 72.8\%\) of its caloric content (110 Calories) comes from alcohol, based on the amount of ethanol and the heat release calculations. This highlights how significant a contributor alcohol can be to the calorie content of drinks.