Question

The heat of combustion of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l),\) is -1367 \(\mathrm{kJ} / \mathrm{mol}\). A bottle of stout (dark beer) contains up to \(6.0 \%\) ethanol by mass. Assuming the density of the beer to be \(1.0 \mathrm{~g} / \mathrm{mL},\) what is the caloric content due to the alcohol (ethanol) in a bottle of beer \((500 \mathrm{~mL})\) ?

Short answer

The caloric content due to the ethanol in a \(500 \mathrm{~mL}\) bottle of beer is approximately -889.1 kJ.

Step-by-step solution

Find the mass of ethanol in the bottle of beer

First, we need to find the mass of the entire bottle of beer. To do this, we can use the density and volume given in the problem. Density of beer = \(1.0 \frac{g}{mL}\) Volume of the beer bottle = \(500 mL\) Mass of the beer \(=\) Density of beer \(\times\) Volume of the beer bottle Mass of the beer \(=1.0 \frac{g}{mL} \times 500 mL = 500 g\) Since the beer contains \(6.0\%\) ethanol by mass, we can find the mass of ethanol in the bottle: Mass of ethanol \(= 6.0 \% \times 500 g = 0.06 \times 500 g = 30 g\)

Convert the mass of ethanol to moles

To convert the mass of ethanol to moles, we need the molar mass of ethanol. Molar mass of ethanol \(\mathrm{C}_2\mathrm{H}_5\mathrm{OH} = 2\times 12.01 + 6\times 1.01 + 1\times 16.00 = 46.07 \frac{g}{mol}\) Moles of ethanol \(= \frac{Mass\;of\;Ethanol}{Molar\;Mass\;of\;Ethanol} = \frac{30 g}{46.07 \frac{g}{mol}}\) Moles of ethanol \(= 0.651 mol\)

Calculate the caloric content due to ethanol using the heat of combustion

Now that we have the moles of ethanol, we can calculate the caloric content using the heat of combustion. Heat of combustion of ethanol \(= -1367 \frac{kJ}{mol}\) (The negative sign indicates energy is released during combustion) Caloric content due to ethanol \(= -1367 \frac{kJ}{mol} \times 0.651 mol = -889.1 kJ\) The caloric content due to the ethanol in a bottle of beer is approximately -889.1 kJ.

Key concepts

Understanding Ethanol

Ethanol, also known as ethyl alcohol, is a type of alcohol commonly found in beers, wines, and spirits. It is a simple alcohol with the chemical formula \(\mathrm{C}_2\mathrm{H}_5\mathrm{OH}\). Ethanol is not only used as a key ingredient in alcoholic beverages but also as an industrial solvent and as a renewable fuel additive to gasoline. It burns cleanly and is considered an important alternative fuel.

In the context of the exercise, ethanol acts as the primary fuel contributing to the caloric content of the beer. When ethanol undergoes combustion, it releases a significant amount of heat energy, which is used to calculate the caloric content. The heat of combustion provided in the problem statement is a vital figure used to determine how much energy is released per mole when ethanol is burned.

Molar Mass Explained

Molar mass is a crucial concept in chemistry that refers to the mass of one mole of a given substance, usually expressed in grams per mole \(\left(\frac{g}{mol}\right)\). In this context, calculating the molar mass of ethanol is necessary to convert grams of ethanol into moles. Understanding molar mass helps us bridge the gap between the macroscopic world we see and the molecular world where reactions occur.

To find the molar mass of ethanol \(\mathrm{C}_2\mathrm{H}_5\mathrm{OH}\):

• Carbon (C) has an atomic mass of 12.01 \(\frac{g}{mol}\).
• Hydrogen (H) has an atomic mass of 1.01 \(\frac{g}{mol}\).
• Oxygen (O) has an atomic mass of 16.00 \(\frac{g}{mol}\).

The calculation is as follows:

• 2 Carbon atoms: \(2 \times 12.01 = 24.02\)
• 6 Hydrogen atoms: \(6 \times 1.01 = 6.06\)
• 1 Oxygen atom: \(1 \times 16.00 = 16.00\)

Adding these gives us the molar mass of ethanol: \(24.02 + 6.06 + 16.00 = 46.08 \frac{g}{mol}\). This value is critical for converting the mass of ethanol into moles, an essential step for further calculations like determining the caloric content.

Density and Volume Relation

Density is a property of matter that describes how much mass is contained in a given volume, typically expressed as \(\frac{g}{mL}\) for liquids. This relationship allows us to calculate the mass of a substance when its density and volume are known. The formula is: mass = density \(\times\) volume.

This exercise uses the density of beer to find the total mass. Given that the density of the beer is \(1.0 \frac{g}{mL}\) and the volume of the beer bottle is 500 mL, you can calculate:

• Mass of beer = \(1.0 \frac{g}{mL} \times 500 mL = 500 g\).

Knowing that the beer is 6% ethanol by mass, we apply this percentage to the total mass to find the mass of ethanol:

• Mass of ethanol = 6% \(\times 500 g = 30 g\)

Understanding these calculations involving density and volume is key to determining the amount of ethanol in the beer, which is then used in subsequent steps to calculate energy content based on ethanol's heat of combustion.