Problem 93 - Chemistry: The Central Science in SI Units, Global Edition

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 total caloric content of the fructose in the apple is \((\frac{2812 \mathrm{kJ/mol}}{180.18 \mathrm{g/mol}}) \times 16 \mathrm{g} \times 0.239 \mathrm{kcal/ kJ}\) = 75.3 kcal.

Step 1: Calculate the number of moles of fructose in the apple

Given mass of fructose in the apple is 16 g. We can calculate the number of moles (n) using the formula: n = mass / molar mass The molar mass of fructose, \(\mathrm{C}_{6} \mathrm{H}_{12}\mathrm{O}_{6}\), is: \(6 \times 12.01 \: (for \: Carbon) + 12 \times 1.01 \: (for \: Hydrogen) + 6 \times 16.00 \: (for \: Oxygen) = 72.06 + 12.12 + 96 = 180.18 \mathrm{g/mol}\) Now, let's find the number of moles of fructose, n = \( \frac{16}{180.18}\)

Step 2: Convert the heat of combustion to energy content per gram of fructose

Heat of combustion of fructose is given as -2812 kJ/mol. To convert this to energy content per gram, we can divide the heat of combustion by the molar mass of fructose. Energy content per gram = \(\frac{2812 \mathrm{kJ/mol}}{180.18 \mathrm{g/mol}}\)

Step 3: Calculate the total energy content of the fructose present in the apple

Now that we know the energy content per gram of fructose and the number of moles of fructose present in the apple, we can calculate the total energy content by multiplying these values: Total energy content = Energy content per gram × mass of fructose = \( (\frac{2812 \mathrm{kJ/mol}}{180.18 \mathrm{g/mol}}) \times 16 \mathrm{g}\) Finally, as 1 kJ is equal to 0.239 kcal, we can convert our final answer in kJ to kcal: Total caloric content = Total energy content in kJ × 0.239 kcal / kJ

Key Concepts

Fructose

Fructose is a type of simple sugar, also known as a monosaccharide. It is naturally found in various fruits, vegetables, and honey. In its pure form, fructose is very sweet, even sweeter than glucose or sucrose (table sugar). This is why it's often used as a sweetener in foods and beverages. In the process of energy metabolism, fructose is absorbed from the intestine and converted into glucose or stored as fat in the liver. Eating foods rich in fructose can provide a quick source of energy because fructose rapidly enters metabolic pathways. Knowing how much fructose a food item contains is important when calculating its energy contribution.

Molar Mass

Understanding molar mass is crucial when dealing with chemical reactions and substances. The molar mass of a compound refers to the mass of one mole of that compound. In simpler terms, it is the mass of 6.022 x 10\(^ {23} \) molecules or atoms of a substance, which is Avogadro's number. For molecules like fructose, calculating the molar mass involves adding up the atomic masses of all the atoms in the molecule. As presented, the molar mass of fructose (C\(_6\)H\(_{12}\)O\(_6\)) is 180.18 g/mol. This value helps us determine the number of moles in a given sample by using the formula: \( n = \frac{\text{mass}}{\text{molar mass}} \). This relationship is foundational in converting between grams and moles, allowing us to understand the material's substance and reactivity.

Energy Content

Energy content, specifically in the context of food, refers to the amount of energy that can be derived from consuming a particular amount of food. It's usually measured in calories or kilojoules. The heat of combustion, such as that of fructose, gives us the energy produced when a substance is fully combusted in the presence of oxygen. This energy is crucial for all bodily functions, from simple tasks like breathing to active exercise. For fructose, converting its heat of combustion from a per mole basis to an energy content per gram offers a practical way to determine how much energy each gram of fructose in food provides. In this way, you can calculate how much energy the fructose in a fresh apple would contribute to its total caloric content. Using the formula: \( \text{Energy content per gram} = \frac{\text{Heat of combustion}}{\text{Molar mass}} \) helps in making these valuations. Such calculations assist in diet planning and understanding food energy, contributing to overall nutritional knowledge.