Problem 84 - Chemistry: The Central Science

Question

The heat of combustion of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)\), is \(-1367 \mathrm{~kJ} / \mathrm{mol}\). A batch of Sauvignon Blanc wine contains \(10.6 \%\) ethanol by mass. Assuming the density of the wine to be \(1.0 \mathrm{~g} / \mathrm{mL}\), what caloric content does the alcohol (ethanol) in a 6-oz glass of wine (177 mL) have?

Step-by-Step Solution

Answer

The caloric content of the ethanol in a 6-oz glass of wine (177 mL) is approximately 132,951.32 calories.

Step 1: Convert volume of wine in the glass to mass

Given that the density of wine is 1.0 g/mL, we will use the formula for density to calculate the mass of wine in the glass. Density = Mass / Volume Therefore, Mass = Density * Volume Given the volume of wine in the glass is 177 mL, we can calculate the mass: Mass of wine = \(1.0 \frac{g}{mL} \times 177 mL = 177 g\)

Step 2: Calculate mass of ethanol in the glass of wine

We know the wine contains 10.6% ethanol by mass. So, we can find the mass of ethanol in the glass by using this percentage. Mass of ethanol = (Percentage of ethanol) * (Mass of wine) Mass of ethanol = \(\frac{10.6}{100} \times 177 g = 18.74 g\)

Step 3: Determine the number of moles of ethanol

To determine the number of moles of ethanol, use the molar mass of ethanol. The molar mass of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)\), is approximately 46 g/mol. Number of moles = \(\frac{\text{Mass of ethanol}}{\text{Molar mass of ethanol}}\) Number of moles = \(\frac{18.74\, g}{46\, g/mol} = 0.4074\, moles\)

Step 4: Calculate the energy liberated by the combustion of ethanol

Now we can calculate the total energy liberated by the combustion of ethanol using the given heat of combustion (-1367 kJ/mol). Energy liberated = Heat of combustion × Number of moles Energy liberated = \((-1367\, \frac{kJ}{mol}) \times 0.4074\, moles = -556.6974\, kJ\) Since the energy is released during combustion, the energy content will be positive. Therefore, the caloric content of ethanol in the glass of wine is 556.6974 kJ.

Step 5: Convert the energy to calories

To convert the energy in kilojoules to calories, use the conversion factor: \(1\, kJ = 239\, calories\). Caloric content = Energy liberated × Conversion factor Caloric content = \(556.6974\, kJ \times 239\, \frac{cal}{kJ} \approx 132951.3226\, calories\) The caloric content of the ethanol in a 6-oz glass of wine (177 mL) is approximately 132,951.32 calories.

Key Concepts

Heat of Combustion

In thermochemistry, the heat of combustion refers to the energy released as heat when a compound undergoes complete combustion with oxygen. This is an essential concept when examining the energy derived from fuels such as ethanol.The specific heat of combustion for ethanol is \(-1367\, \text{kJ/mol}\).

• This signifies that when one mole of ethanol combusts, 1367 kJ of energy is released.
• It's important to note the negative sign, indicating energy release or an exothermic reaction.

During combustion, chemical bonds are broken, and new ones are formed, releasing energy in the process.Understanding this allows us to calculate energy outputs from alcoholic beverages accurately.

Ethanol

Ethanol, often found in alcoholic beverages, is a simple alcohol with the chemical formula \(\text{C}_2\text{H}_5\text{OH}\).It plays a significant role in the caloric content of these drinks.

• Ethanol is volatile, flammable, and produces heat when burned or metabolized.
• In beverages, ethanol concentration is often expressed as a percentage by mass.

In the context of the exercise, ethanol constitutes 10.6\% of the wine by mass.Understanding the mass percentage helps us calculate the actual mass of ethanol in any given volume.

Caloric Content

The caloric content indicates the amount of energy a food or drink provides when consumed.In thermodynamics, this is often calculated in calories or kilojoules.

• In our exercise, the caloric content is obtained from the combustion energy of ethanol contained in the wine.
• We first compute the kilojoules released and then convert this energy into calories using the conversion factor: \(1 \text{ kJ} = 239\, \text{calories}\).

Through this calculation, we discovered that a 6-oz glass of wine contains approximately 132,951.32 calories from ethanol alone.By understanding caloric content, consumers can make more informed decisions about their intake.

Density Calculation

Density is a key concept in physics defined as mass per unit volume, usually expressed in \(\text{g/mL}\) for liquids.It is crucial for converting between the volume of a liquid and its mass.

• The density of wine in this exercise is 1.0 \(\text{g/mL}\),which simplifies calculations.
• By multiplying the volume of the wine (177 mL) by its density, we can determine its mass.This step is necessary for further calculations related to ethanol content.

Grasping the principles of density allows for accurate translations between volume and mass, integral for chemistry and many related fields.