Determine the empirical formulas for the following compounds: (a) caffeine, \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{4} \mathrm{O}_{2}\) (b) fructose, \(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\) (c) hydrogen peroxide, \(\mathrm{H}_{2} \mathrm{O}_{2}\) (d) glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) (e) ascorbic acid (vitamin \(\mathrm{C}), \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{6}\)

The molecular formula of a chemical compound describes the exact number and type of atoms in a single molecule of that compound. It represents the actual number of atoms of each element that make up the molecule.

For instance, the molecular formula of caffeine is \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{4} \mathrm{O}_{2}\) meaning a single molecule of caffeine contains 8 carbon atoms, 10 hydrogen atoms, 4 nitrogen atoms, and 2 oxygen atoms. However, the molecular formula does not always reflect the simplest ratio of atoms; that's where the empirical formula becomes useful.

Stoichiometry is a section of chemistry that refers to the calculation of the quantities of reactants and products in chemical reactions. It is based on the conservation of mass, and the principle that elements are neither created nor destroyed in chemical reactions. Thus, stoichiometry can also be used to determine the quantitative relationship between reactants and products in terms of moles or masses.

In the case of determining empirical formulas, stoichiometry is essential for calculating the mole ratio of elements in a compound. This informs us about the simplest whole-number ratio of atoms in a compound, which can lead to the determination of the empirical formula.

Chemical composition refers to the identity and quantity of the elements that make up a chemical compound. It includes both qualitative (which elements are present) and quantitative (how much of each element is present) aspects. By understanding the chemical composition, scientists can predict how a compound will react, determine its properties, and synthesize it for various applications.

Knowing the chemical composition is the starting point for determining empirical formulas, as was demonstrated with caffeine and the other compounds in the provided exercises. For instance, the chemical composition of glucose, given as \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\), allows for the determination of its empirical formula after assessing the mole ratio of its elements.

The greatest common divisor (GCD), sometimes called the greatest common factor, is the largest positive integer that divides each of the numbers in a set without leaving a remainder. In chemistry, finding the GCD of the number of atoms of each element in a compound's molecular formula can simplify it to an empirical formula.

For example, in hydrogen peroxide's molecular formula \(\mathrm{H}_{2} \mathrm{O}_{2}\), the GCD is 2. By dividing each element's subscript by 2, we can find the empirical formula, which is \(\mathrm{HO}\). It's important to apply the GCD correctly to ensure the empirical formula reflects the simplest ratio of atoms in the molecule.

The simplest whole-number ratio of atoms in a compound is shown by its empirical formula. This formula gives the smallest possible integer values for the ratio of atoms, translating complex molecular information into more manageable and comparable units.

For instance, the empirical formula for glucose is derived by taking the molecular formula \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) and simplifying it based on the whole-number ratio of its elements, which gives us \(\mathrm{CH}_{2}\mathrm{O}\). The process of determining this involves dividing the subscripts by the greatest common divisor, to ensure that this ratio is as simple as it can be.