Problem 94 - Chemistry The Central Science In Si Units

Question

The heat of combustion of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(t),\) 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 mL)?

Step-by-Step Solution

Answer

The caloric content due to the alcohol in 500 mL of beer is approximately 890 kJ.

Step 1: Determine mass of the beer

First, calculate the mass of the beer using its given density and volume. Since the density is \(1.0\, \text{g/mL}\) and the volume is \(500\, \text{mL}\), the mass of the beer is \(500\, \text{g}\).

Step 2: Calculate mass of ethanol in beer

Next, find the mass of ethanol in the beer. Since the beer contains \(6.0\%\) ethanol by mass, the mass of ethanol is \(0.06 \times 500\, \text{g} = 30 \, \text{g}.\)

Step 3: Determine moles of ethanol

We need to convert the mass of ethanol into moles. The molar mass of ethanol, \(\mathrm{C}_2\mathrm{H}_5\mathrm{OH}\), is approximately \(46.08\, \text{g/mol}\). Thus, the number of moles is \(\frac{30\, \text{g}}{46.08\, \text{g/mol}} \approx 0.651\, \text{mol}.\)

Step 4: Calculate caloric content from ethanol

Now, calculate the energy released by the ethanol. The heat of combustion of ethanol is \(-1367\, \text{kJ/mol}\). Therefore, the total energy released is \(0.651\, \text{mol} \times -1367\, \text{kJ/mol} \approx -890\, \text{kJ}.\) This negative sign indicates energy released.

Key Concepts

Understanding Ethanol

Ethanol is a type of alcohol commonly found in alcoholic beverages like beer, wine, and spirits. It is also known as ethyl alcohol or grain alcohol. Ethanol is a volatile, flammable, and colorless liquid with a slightly sweet flavor. It has the molecular formula \(\mathrm{C}_2\mathrm{H}_5\mathrm{OH}\).

• Used not only in beverages but also as a fuel source and industrial solvent.
• Ethanol in beverages is often identified by its volume percentage in products like beer, where 6% indicates that 6% of the beer's mass is ethanol.
• It plays a crucial role in the caloric content of alcoholic drinks due to its energy-dense properties.

In our problem, the ethanol in beer contributes majorly to its caloric content, and knowing its percentage by mass helps in calculating its energy release through combustion. Ethanol's combustion releases heat, a process we tap into when discussing caloric content.

Exploring Caloric Content

Caloric content refers to the amount of energy provided by a food or drink to the body. It is usually measured in kilojoules (kJ) or calories (cal) and helps us understand how much energy we would obtain from consuming the beverage.

• The term 'calorie' in food packaging and nutritional contexts often refers to kilocalories (kcal).
• When ethanol burns, it releases energy, which can be quantitatively measured using its heat of combustion, an important aspect when calculating the caloric content of alcoholic drinks.
• The concept helps us evaluate how much fuel (energy) ethanol provides compared to other nutrients or ingredients in food and beverages.

For our example, based on the heat of combustion, the ethanol in beer contributes significantly to its overall energy content, which is why alcoholic drinks with higher ethanol content also have higher caloric values.

Role of Density in Calculations

Density is a measure of how much mass exists in a given volume of a substance. It helps us convert between mass and volume, which is crucial when dealing with liquids like beer.

• Expressed as mass per unit volume, such as grams per milliliter (g/mL).
• For beer, given a density of \(1.0 \, \text{g/mL}\), we assume a straightforward conversion: 1 liter of beer weighs 1 kilogram.
• Knowing the density allows us to calculate the mass of a given volume accurately, essential for further computations like finding the mass of ethanol.

In the context of our problem, the density was used to first derive the total mass of the beer from its volume. From there, calculating the mass of ethanol in it was made possible, laying the ground for finding the energy content based on the heat of combustion. Understanding density and its role in these calculations ensures that we arrive at precise and meaningful answers.