A sample of sucrose, \(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\), is contaminated by sodium chloride. When the contaminated sample is burned in a bomb calorimeter, sodium chloride does not burn. What is the percentage of sucrose in the sample if a temperature increase of \(1.67^{\circ} \mathrm{C}\) is observed when \(3.000 \mathrm{~g}\) of the sample is burned in the calorimeter? Sucrose gives off \(5.64 \times 10^{3} \mathrm{~kJ} / \mathrm{mol}\) when burned. The heat capacity of the calorimeter is \(22.51 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}\).

Calorimetry is the science of measuring the change in heat associated with a chemical reaction. The bomb calorimeter is a sophisticated tool used in these measurements. It works by having a reaction take place in a sealed, oxygen-rich environment called a bomb. The bomb is submerged in water, and the entire apparatus is well-insulated. When a substance, such as sucrose, burns within the calorimeter, the heat released raises the temperature of the water surrounding the bomb.

The key advantage of a bomb calorimeter is that it operates at a constant volume, which allows for precise measurements of heat. The temperature change of the water is directly proportional to the heat of combustion of the substance. To determine the heat exchange, we use the specific heat capacity of the calorimeter, factoring in the mass of the water and the temperature change. However, keep in mind that any non-combustible contaminants present, such as sodium chloride (common table salt) in our exercise, do not influence the calorimeter’s reading since they do not react.

The heat of combustion refers to the amount of heat energy released when a substance is burned in the presence of oxygen. It is an essential concept in thermodynamics and stoichiometry, reflecting the energy content of a chemical compound. When dealing with problems involving calorimetry, such as determining the purity of a sample, knowing the heat of combustion enables us to calculate the amount of heat generated by burning a particular amount of the substance.

In chemistry, the heat of combustion is typically expressed in units of kJ/mol, signifying the energy released by burning one mole of a substance. In our example with sucrose, the heat of combustion is given as \(5.64 \times 10^{3} \mathrm{~kJ} / \mathrm{mol}\), making it the cornerstone for calculating how much sucrose was present in our sample by relating it to the heat measured by the bomb calorimeter.

Molar mass is a fundamental concept in chemistry, defined as the mass in grams of one mole of a substance. It's pivotal when converting between grams and moles, a common step in stoichiometric calculations. For any compound, the molar mass can be calculated by summing the atomic masses of all the atoms in the molecule, which are typically found on the periodic table.

In the given problem, the molar mass of sucrose (\(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\)) is calculated to be \(342.30 \mathrm{~g/mol}\). This value is crucial to convert the amount of heat released during the combustion (measured in kJ) to the mass of sucrose burned, represented in grams. Understanding how to find and apply molar mass correctly is essential for accurate quantitative analysis in chemistry.

Stoichiometry is the section of chemistry that deals with the quantitative relationship between reactants and products in a chemical reaction. It allows chemists to predict the amounts of substances consumed and produced in a given reaction. In calorimetry, stoichiometry helps relate the mass of a substance to the amount of heat released or absorbed.

In the context of the exercise, stoichiometry comes into play when we relate the heat capacity of the calorimeter and the temperature change to the amount of sucrose that reacted. Upon calculating the moles of sucrose burned using its combustion heat, stoichiometry principles are employed to convert moles to grams, further enabling the calculation of the sample's purity. Mastery of stoichiometry is essential for interpreting calorimetry experiments and understanding the mole concept, molar mass, and the Law of Conservation of Mass in quantitative terms.