Problem 100 - Chemistry The Science in Context

Question

Food contains three main categories of compounds: carbohydrate, protein, and fat. Arctic explorers often eat a high-fat diet because fats have a high food value. The average food values for carbohydrate, protein, and fat are \(4 \mathrm{Cal} / \mathrm{g}, 4 \mathrm{Cal} / \mathrm{g},\) and \(9 \mathrm{Cal} / \mathrm{g},\) respectively. Consider a typical fat to have the chemical formula \(\mathrm{C}_{18} \mathrm{H}_{36} \mathrm{O}_{2},\) glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\) to be a representative carbohydrate, and the amino acid alanine \(\left(\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{NO}_{2}\right)\) a building block of proteins. Express the food values given above as \(\Delta H_{\text {comb }}\) in \(\mathrm{k} \mathrm{J} / \mathrm{mol}\).

Step-by-Step Solution

Answer

Answer: The energy content (ΔH_comb) values are 3012.77 kJ/mol for carbohydrates, 1490.16 kJ/mol for protein, and 10720.98 kJ/mol for fat.

Step 1: Conversion of food values from Cal/g to kJ/g

For conversion, we multiply each value given in Cal/g by 4.184 (1 Cal = 4.184 kJ). Carbohydrate: 4 Cal/g × 4.184 kJ/Cal = 16.736 kJ/g Protein: 4 Cal/g × 4.184 kJ/Cal = 16.736 kJ/g Fat: 9 Cal/g × 4.184 kJ/Cal = 37.656 kJ/g

Step 2: Calculation of molar mass for each representative molecule

For each representative molecule, we calculate their molar mass using the atomic masses of the elements: C = 12.01 g/mol, H = 1.008 g/mol, O = 16.00 g/mol, and N = 14.01 g/mol. Fat (C18H36O2) Molar mass = (18 × 12.01) + (36 × 1.008) + (2 × 16.00) = 284.48 g/mol Carbohydrate (C6H12O6) Molar mass = (6 × 12.01) + (12 × 1.008) + (6 × 16.00) = 180.18 g/mol Protein (C3H6NO2) Molar mass = (3 × 12.01) + (6 × 1.008) + (1 × 14.01) + (2 × 16.00) = 89.09 g/mol

Step 3: Calculation of ΔH_comb in kJ/mol for each compound

We will use the formula: ΔH_comb (kJ/mol) = Energy content (kJ/g) × Molar mass (g/mol) Fat: ΔH_comb = 37.656 kJ/g × 284.48 g/mol = 10720.98 kJ/mol Carbohydrate: ΔH_comb = 16.736 kJ/g × 180.18 g/mol = 3012.77 kJ/mol Protein: ΔH_comb = 16.736 kJ/g × 89.09 g/mol = 1490.16 kJ/mol Thus, the food values in ΔH_comb are as follows: Fat: 10720.98 kJ/mol Carbohydrate: 3012.77 kJ/mol Protein: 1490.16 kJ/mol

Key Concepts

Calorimetry

Calorimetry is a method used to measure the amount of heat involved in chemical reactions or changes of state. It helps us understand the energy content of different foods. When dealing with food, calories are a common measure, but in chemistry, we often convert these into the standard unit of Joules.

To convert food calories to kilojoules, we use the conversion factor where 1 Cal (or kilocalorie) equals 4.184 kJ. For example, carbohydrates and proteins have a calorific value of 4 Cal/g. By multiplying 4 Cal/g with 4.184 kJ/Cal, we get 16.736 kJ/g as their energy content in kilojoules. Fats, with 9 Cal/g, convert to 37.656 kJ/g.

This understanding is crucial as it allows us to quantify the energy released during combustion in a laboratory setting, making calorimetry a powerful tool for both chemistry and nutrition.

Molar Mass Calculation

Molar mass is the total mass of all atoms in a molecule, expressed in grams per mole (g/mol). Knowing the molar mass is important for various calculations in chemistry, including finding the enthalpy change during reactions.

To calculate molar mass, we sum the atomic masses of each element in a compound. For example, for a typical fat molecule like \(\text{C}_{18}\text{H}_{36}\text{O}_{2}\), we calculate:

• Carbon (C): \(18 \times 12.01\, \text{g/mol} = 216.18\, \text{g/mol}\)
• Hydrogen (H): \(36 \times 1.008\, \text{g/mol} = 36.29\, \text{g/mol}\)
• Oxygen (O): \(2 \times 16.00\, \text{g/mol} = 32.00\, \text{g/mol}\)

Adding these gives a molar mass of 284.48 g/mol.

Repeat similar calculations for carbohydrates \(\text{C}_{6}\text{H}_{12}\text{O}_{6}\) and proteins \(\text{C}_{3}\text{H}_{6}\text{NO}_{2}\) to find their molar masses. This forms a foundational step in calculating enthalpy of combustion.

Enthalpy of Combustion

The enthalpy of combustion \(\Delta H_{\text{comb}}\) is the heat change that occurs when one mole of a substance is burned completely in oxygen. It's expressed in kilojoules per mole (kJ/mol), explaining the energy a compound releases.

To find \(\Delta H_{\text{comb}}\), multiply the energy content per gram with the molar mass of the substance. For instance, the enthalpy of combustion for fat is calculated as:

• 37.656 kJ/g (energy content) \(\times\) 284.48 g/mol (molar mass) = 10720.98 kJ/mol

Repeat for carbohydrates and proteins to find their respective values.

This calculation helps in understanding why fats are so energy-rich, as they release more energy upon combustion compared to carbohydrates and proteins, making them a dense energy source, especially valuable in challenging environments.