Question

What is the empirical formula of each of the following compounds? (a) \(\mathrm{P}_{4} \mathrm{O}_{10}\) (c) \(\mathrm{PbCl}_{4}\) (b) \(\mathrm{Cl}_{2} \mathrm{O}_{5}\) (d) \(\mathrm{HO}_{2} \mathrm{CC}_{4} \mathrm{H}_{8} \mathrm{CO}_{2} \mathrm{H}\)

Short answer

(a) \(\mathrm{P}_{2} \mathrm{O}_{5}\) (b) \(\mathrm{PbCl}\) (c) \(\mathrm{Cl}_{2} \mathrm{O}_{5}\) (d) \(\mathrm{HO}_{2} \mathrm{CC}_{4} \mathrm{H}_{8} \mathrm{CO}_{2} \mathrm{H}\)

Step-by-step solution

Identifying the subscripts in the molecular formula and their greatest common factors

For \(\mathrm{P}_{4} \mathrm{O}_{10}\), the subscripts are 4 and 10 and their greatest common factor is 2. For \(\mathrm{PbCl}_{4}\), the subscript is 4. For \(\mathrm{Cl}_{2} \mathrm{O}_{5}\), the subscripts are 2 and 5 and their greatest common factor is 1. For \(\mathrm{HO}_{2} \mathrm{CC}_{4} \mathrm{H}_{8} \mathrm{CO}_{2} \mathrm{H}\), the subscripts are 1, 2, 1, 4, 8, 2, and 1. The greatest common factor is 1.

Dividing the subscripts in the molecular formula by their greatest common factors

Dividing the subscripts in \(\mathrm{P}_{4} \mathrm{O}_{10}\) by their greatest common factor, 2, we get \(\mathrm{P}_{2} \mathrm{O}_{5}\). For \(\mathrm{PbCl}_{4}\), dividing the subscript 4 by itself we get \(\mathrm{PbCl}\). For \(\mathrm{Cl}_{2} \mathrm{O}_{5}\), dividing the subscripts by their greatest common factor, 1, we get \(\mathrm{Cl}_{2} \mathrm{O}_{5}\). For \(\mathrm{HO}_{2} \mathrm{CC}_{4} \mathrm{H}_{8} \mathrm{CO}_{2} \mathrm{H}\), dividing the subscripts by their greatest common factor, 1, we get \(\mathrm{HO}_{2} \mathrm{CC}_{4} \mathrm{H}_{8} \mathrm{CO}_{2} \mathrm{H}\).

Writing the empirical formulas

The empirical formula of \(\mathrm{P}_{4} \mathrm{O}_{10}\) is \(\mathrm{P}_{2} \mathrm{O}_{5}\). The empirical formula of \(\mathrm{PbCl}_{4}\) is \(\mathrm{PbCl}\). The empirical formula of \(\mathrm{Cl}_{2} \mathrm{O}_{5}\) is \(\mathrm{Cl}_{2} \mathrm{O}_{5}\). The empirical formula of \(\mathrm{HO}_{2} \mathrm{CC}_{4} \mathrm{H}_{8} \mathrm{CO}_{2} \mathrm{H}\) is \(\mathrm{HO}_{2} \mathrm{CC}_{4} \mathrm{H}_{8} \mathrm{CO}_{2} \mathrm{H}\).

Key concepts

Molecular Formula

The molecular formula of a compound provides the exact number of each type of atom in a molecule. For instance, in the compound \( \mathrm{P}_{4} \mathrm{O}_{10} \), it reveals that there are four phosphorus atoms and ten oxygen atoms in a single molecule. Molecular formulas are important because they provide details about the structure of a molecule, offering more comprehensive insight compared to its empirical formula.

Understanding the molecular formula helps determine the molecular weight, which is crucial for various chemical calculations and reactions. For example, when reacting different compounds, knowing the exact number of atoms gives precise stoichiometric predictions.

While the molecular formula tells us the exact makeup of a molecule, it is also used to derive the empirical formula. The empirical formula is essentially the simplest ratio of the atoms in a compound. This is particularly useful when the composition data initially retrieved from experimental analysis is ambiguous or when comparing many compounds with similar molecular formulas.

Greatest Common Factor

The greatest common factor (GCF) in chemistry is used to simplify the molecular formula to find its empirical formula. By identifying the highest number that divides all subscripts of the molecular formula, we aim to reduce these numbers to their simplest integral ratio.

For example, in the case of \( \mathrm{P}_{4} \mathrm{O}_{10} \), the subscripts are 4 and 10. The GCF here is 2. Dividing each subscript by 2 gives the empirical formula \( \mathrm{P}_{2} \mathrm{O}_{5} \). For compounds like \( \mathrm{Cl}_{2} \mathrm{O}_{5} \), the GCF is 1, indicating that its empirical formula is the same as the molecular formula.

The GCF helps in simplifying complex molecular structures into more manageable, comprehensible ratios without changing the fundamental characteristics of the original compound. Understanding this concept is crucial for accurately simplifying and studying large organic or biochemical molecules.

Subscripts in Chemistry

Subscripts in chemistry are small numbers written at the lower right-hand corner of elemental symbols in chemical formulas. They indicate the number of atoms of each element present in a molecule.

These subscripts are essential for understanding not only the composition of a molecule but also its quantitative aspects in reactions. For example, in a compound like \( \mathrm{PbCl}_{4} \), the subscript '4' next to chlorine (Cl) signifies four chlorine atoms are bonded to a single lead (Pb) atom. This information becomes vital when calculating the molar mass or when balancing chemical equations, ensuring the conservation of mass.

Subscripts can also depict structural insights into how atoms are arranged or bonded. In cases of molecules like \( \mathrm{HO}_{2} \mathrm{CC}_{4} \mathrm{H}_{8} \mathrm{CO}_{2} \mathrm{H} \), subscripts help in deducing the structural formula from the molecular formula, further enriching the understanding of a compound's dynamics.