A typical fat in the body is glyceryl trioleate, \(\mathrm{C}_{57} \mathrm{H}_{104} \mathrm{O}_{6}\). When it is metabolized in the body, it combines with oxygen to produce carbon dioxide, water, and \(3.022 \times 10^{4} \mathrm{~kJ}\) of heat per mole of fat. (a) Write a balanced thermochemical equation for the metabolism of fat. (b) How many kilojoules of energy must be evolved in the form of heat if you want to get rid of five pounds of this fat by combustion? (c) How many nutritional calories is this? (1 nutritional calorie = \(1 \times 10^{3}\) calories)

First we have to write the general equation for the metabolism of glyceryl trioleate. Glyceryl trioleate (C57H104O6) combines with oxygen (O2) to produce carbon dioxide (CO2), water (H2O), and heat. C57H104O6 + O2 -> CO2 + H2O + heat Next, balance the equation: C57H104O6 + 80O2 -> 57CO2 + 52H2O + 3.022 * 10^4 kJ (a) Here is the balanced thermochemical equation for the metabolism of glyceryl trioleate: C57H104O6 + 80O2 -> 57CO2 + 52H2O + 3.022 * 10^4 kJ/mol

We are given the weight of glyceryl trioleate in pounds, and we need to convert it into moles and calculate the heat evolved during combustion. First, we have to convert the weight to grams: 5 pounds * 454 g/pound ≈ 2270 g Now, we need to find the molar mass of glyceryl trioleate: Molar mass = (57 * C) + (104 * H) + (6 * O) Molar mass ≈ (57 * 12.01 g/mol) + (104 * 1.01 g/mol) + (6 * 16 g/mol) Molar mass ≈ 885.71 g/mol Next, we convert the weight of glyceryl trioleate to moles: Moles = weight/molar mass = 2270 g / 885.71 g/mol ≈ 2.56 moles Now, we can calculate the heat evolved: Heat evolved = moles * heat per mole = 2.56 moles * 3.022 * 10^4 kJ/mol ≈ 7.72 * 10^4 kJ (b) The heat evolved in the form of heat when burning 5 pounds of glyceryl trioleate is 7.72 * 10^4 kJ.

We have to convert the heat energy in kJ to nutritional calories. 1 nutritional calorie = 1 kcal = 10^3 cal 1 kJ = 0.239 kcal Heat in nutritional calories = heat in kJ * (0.239 kcal/kJ) = (7.72 * 10^4 kJ) * (0.239 kcal/kJ) ≈ 1.84 * 10^4 kcal (c) The heat evolved in the form of heat when burning 5 pounds of glyceryl trioleate is equivalent to 1.84 * 10^4 nutritional calories.