Problem 93 - Chemistry The Central Science In Si Units

Question

The heat of combustion of fructose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6},\) is -2812 \(\mathrm{kJ} / \mathrm{mol}\). If a fresh golden delicious apple weighing \(120 \mathrm{~g}\) contains \(16.0 \mathrm{~g}\) of fructose, what caloric content does the fructose contribute to the apple?

Step-by-Step Solution

Answer

The fructose in the apple contributes approximately 60 dietary Calories.

Step 1: Calculate Moles of Fructose

First, find the molar mass of fructose by adding the atomic masses of C, H, and O: \(6 \times 12.01 + 12 \times 1.01 + 6 \times 16.00 = 180.18 \, \text{g/mol}\).Now, calculate the moles of fructose in 16.0 g: \(\frac{16.0 \, \text{g}}{180.18 \, \text{g/mol}} = 0.0888 \, \text{mol}\).

Step 2: Calculate Heat of Combustion for 16 g of Fructose

Using the heat of combustion value given, \(-2812 \, \mathrm{kJ/mol}\), multiply by the number of moles to find the total energy released:\(-2812 \, \mathrm{kJ/mol} \times 0.0888 \, \text{mol} = -249.7 \, \mathrm{kJ}\).

Step 3: Convert Energy from Kilojoules to Calories

Knowing that 1 calorie equals 4.184 joules, convert kilojoules to calories:\(-249.7 \, \text{kJ} = -249700 \, \text{J}\).Now, convert to calories: \(\frac{-249700 \, \text{J}}{4.184 \, \text{J/cal}} = -59688.95 \, \text{cal}\).

Step 4: Convert Calories to Dietary Calories

Understand that dietary Calories (uppercase C) are equivalent to kilocalories. Therefore, divide the calorie result by 1000 to convert:\(-59688.95 \, \text{cal} = -59.7 \, \text{kcal}\).In dietary terms, this means approximately 60 dietary Calories.

Key Concepts

Understanding Heat of Combustion

The heat of combustion is a critical concept in thermochemistry that refers to the amount of heat energy released when a substance is completely burned in the presence of oxygen. This process is typically exothermic, meaning it releases energy.
For example, the heat of combustion for fructose, a simple sugar, is given as -2812 kJ/mol. The negative sign indicates that energy is released during the reaction.
It's important to remember that this value is specific to the substance being burned. Each different material, whether it be a fuel like gasoline or a nutrient like glucose, has a unique heat of combustion.
Ultimately, understanding the heat of combustion lets us predict how much energy we can extract from burning certain substances. In the context of food, it shows us how much energy, or calories, the body might gain from a specific amount of food.

Molar Mass Calculation

Calculating the molar mass is an essential step in performing stoichiometric calculations. It involves adding up the atomic masses of all atoms in a molecule.
Taking fructose as an example, it is composed of six carbon atoms, twelve hydrogen atoms, and six oxygen atoms.
To find its molar mass, you simply multiply the number of each type of atom by the corresponding atomic mass and sum the total:

• Carbon: 6 atoms × 12.01 g/mol
• Hydrogen: 12 atoms × 1.01 g/mol
• Oxygen: 6 atoms × 16.00 g/mol

This adds up to a molar mass of 180.18 g/mol for fructose.
Understanding molar mass is a fundamental part of converting grams of a substance to moles, which is crucial in calculating reaction yields, concentrations, and energy release.

Energy Conversion

Energy conversion is a key idea when working with different units of energy measurement. Often, calculations in chemistry require switching between joules, kilojoules, calories, and dietary calories.
The three steps to converting energy are:

• First, determine the amount of energy in joules or kilojoules.
• Next, convert from kilojoules to calories. Since 1 calorie = 4.184 joules, multiply or divide accordingly.
• Finally, if needed, convert calories to dietary Calories. Note that 1 dietary Calorie (with uppercase C) is equivalent to 1 kilocalorie (kcal).

In the example involving fructose, the conversion from kilojoules to dietary Calories demonstrates how even small amounts of a substance can have measurable energy content.

Dietary Calories Explained

The concept of dietary Calories can often be confusing due to the way it is displayed on nutritional labels. Though often just written as "Calories," it actually refers to kilocalories (kcal).
When calculating the caloric content of food, the key is understanding the relationship between calories and dietary Calories.
1 kilocalorie equals 1000 regular calories, or the amount of energy needed to raise the temperature of 1 kg of water by 1 degree Celsius.
This conversion is particularly useful for assessing how food energy intake aligns with dietary needs. In the exercise example, fructose contributes to the total caloric intake of an apple, thus providing essential energy that the body can utilize. Overall, knowing how to convert and understand dietary Calories ensures you’re better equipped to assess the energy-providing capabilities of different foods.