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Chapter 5: Problem 127

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?

Short Answer

Expert verified
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 by step solution

01

(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: \[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} + \dfrac{3}{2}\mathrm{O}_{2} → 2\mathrm{CO}_{2} + 3\mathrm{H}_{2} \mathrm{O}\]
02

(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
03

(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
04

(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
05

(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.

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Most popular questions from this chapter

Two positively charged spheres, each with a charge of \(2.0 \times\) $10^{-5} \mathrm{C}\(, a mass of \)1.0 \mathrm{~kg}\(, and separated by a distance of \)1.0 \mathrm{~cm}$, are held in place on a frictionless track. (a) What is the electrostatic potential energy of this system? (b) If the spheres are released, will they move toward or away from each other? (c) What speed will each sphere attain as the distance between the spheres approaches infinity? [Section 5.1]

It is interesting to compare the "fuel value" of a hydrocarbon in a hypothetical world where oxygen is not the combustion agent. The enthalpy of formation of \(\mathrm{CF}_{4}(g)\) is \(-679.9 \mathrm{~kJ} / \mathrm{mol}\). Which of the following two reactions is the more exothermic? $$ \begin{aligned} \mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) & \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) \\ \mathrm{CH}_{4}(g)+4 \mathrm{~F}_{2}(g) & \longrightarrow \mathrm{CF}_{4}(g)+4 \mathrm{HF}(g) \end{aligned} $$

From the enthalpies of reaction $$ \begin{aligned} 2 \mathrm{C}(s)+\mathrm{O}_{2}(g) & \longrightarrow 2 \mathrm{CO}(g) & \Delta H=-221.0 \mathrm{~kJ} \\ 2 \mathrm{C}(s)+\mathrm{O}_{2}(g)+4 \mathrm{H}_{2}(g) & \longrightarrow & 2 \mathrm{CH}_{3} \mathrm{OH}(g) & \Delta H=-402.4 \mathrm{~kJ} \end{aligned} $$ calculate \(\Delta H\) for the reaction $$ \mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(g) $$

Without referring to tables, predict which of the following has the higher enthalpy in each case: (a) \(1 \mathrm{~mol} \mathrm{I}_{2}(s)\) or $1 \mathrm{~mol} \mathrm{I}_{2}(g)$ at the same temperature, (b) \(2 \mathrm{~mol}\) of iodine atoms or \(1 \mathrm{~mol}\) of \(\mathrm{I}_{2},(\mathbf{c}) 1 \mathrm{~mol} \mathrm{I}_{2}(g)\) and $1 \mathrm{~mol} \mathrm{H}_{2}(g)$ at \(25^{\circ} \mathrm{C}\) or \(2 \mathrm{~mol} \mathrm{HI}(g)\) at $25^{\circ} \mathrm{C},(\mathbf{d}) 1 \mathrm{~mol} \mathrm{H}_{2}(g)\( at \)100^{\circ} \mathrm{C}$ or \(1 \mathrm{~mol} \mathrm{H}_{2}(g)\) at \(300^{\circ} \mathrm{C}\).

A 1.50 -g sample of quinone $\left(\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{O}_{2}\right)$ is burned in a bomb calorimeter whose total heat capacity is \(8.500 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}\). The temperature of the calorimeter increases from 25.00 to \(29.49^{\circ} \mathrm{C}\). (a) Write a balanced chemical equation for the bomb calorimeter reaction. (b) What is the heat of combustion per gram of quinone and per mole of quinone?
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