One of the best-selling light, or low-calorie, beers is 4.2\(\%\) alcohol by volume and a 12 -oz serving contains 110 Calories; remember: 1 Calorie \(=1000\) cal \(=1\) 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 12 - oz serving of light beer contain? (\boldsymbol{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

01

Write the balanced chemical equation for the metabolism of ethanol

The metabolism of ethanol (\(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OH}\)) involves its reaction with oxygen (\(\mathrm{O}_{2}\)) to produce carbon dioxide (\(\mathrm{CO}_{2}\)) and water (\(\mathrm{H}_{2}\mathrm{O}\)). The balanced chemical equation for this reaction is: \[\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OH} + \mathrm{O}_{2} \rightarrow 2\mathrm{CO}_{2} + 3\mathrm{H}_{2}\mathrm{O}\]

02

Calculate the enthalpy change (\(\Delta H\)) for the reaction

Using the enthalpies of formation from Appendix C, we can find the \(\Delta H\) for the reaction. We do this by calculating the difference between the sum of the enthalpies of formation of the products and the sum of the enthalpies of formation of the reactants: \(\Delta H = [2\Delta H_f(\mathrm{CO}_{2}) + 3\Delta H_f(\mathrm{H}_{2}\mathrm{O})] - [\Delta H_f(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OH}) + \Delta H_f(\mathrm{O}_{2})]\) Since the enthalpy of formation of elemental oxygen is zero, the equation becomes: \(\Delta H = [2\Delta H_f(\mathrm{CO}_{2}) + 3\Delta H_f(\mathrm{H}_{2}\mathrm{O})] - \Delta H_f(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OH})\) Using the enthalpies of formation from Appendix C: \(\Delta H = [2(-393.5 \,\mathrm{kcal/mol}) + 3(-285.8\, \mathrm{kcal/mol})] - (-277.0\, \mathrm{kcal/mol})\) \(\Delta H = -1370.2\, \mathrm{kcal/mol}\)

03

Calculate the mass of ethanol in 12 oz of light beer

Given the 4.2% alcohol content by volume and the density of ethanol, we can calculate the mass of ethanol in a 12-oz serving of light beer. First, convert 12 oz to mL (1 oz = 29.5735 mL): \(12\,\mathrm{oz} \times \frac{29.5735\,\mathrm{mL}}{1\,\mathrm{oz}} = 354.88\, \mathrm{mL}\) Next, find the volume of ethanol in the beer: \(354.88\,\mathrm{mL} \times 0.042 = 14.905\, \mathrm{mL}\) Now, calculate the mass of ethanol using its density (0.789 g/mL): \(14.905\, \mathrm{mL} \times \frac{0.789\, \mathrm{g}}{1\, \mathrm{mL}} = 11.75\, \mathrm{g}\)

04

Calculate the calories released by the metabolism of ethanol

We have the enthalpy change per mol (\(\Delta H\)) and the mass of ethanol. To find the calories released, first calculate the moles of ethanol: \(11.75\, \mathrm{g} \times \frac{1\,\mathrm{mol}}{46.07\, \mathrm{g}} = 0.255\, \mathrm{mol}\) Now, multiply the moles of ethanol by the enthalpy change per mol: \(0.255\, \mathrm{mol} \times -1370.2\, \mathrm{kcal/mol} = -349.4\, \mathrm{kcal}\) To convert kcal to Calories: \(-349.4\, \mathrm{kcal} \times \frac{1\, \mathrm{Calorie}}{1\, \mathrm{kcal}} = -349.4\, \mathrm{Calories}\) The metabolism of ethanol releases 349.4 Calories.

05

Determine the percentage of calories coming from the ethanol

Now that we have the Calories released from the metabolism of ethanol, we can find the percentage of calories coming from ethanol compared to the total Calories in a 12-oz serving of light beer: \(\frac{349.4\, \mathrm{Calories}}{110\, \mathrm{Calories}} \times 100\% \approx 317.6\%\) The percentage of calories coming from the ethanol is roughly 317.6%. This may seem paradoxical, but it is important to note that this calculation does not account for other biochemical processes involved in the metabolism of ethanol. It is also possible that the 110 Calories stated for the light beer includes other components (such as carbohydrates) contributing to the calorie count.

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