Problem 49 - Chemistry: The Central Science

Question

Write the empirical formula corresponding to each of the following molecular formulas: (a) \(\mathrm{Al}_{2} \mathrm{Br}_{6}\), (b) \(\mathrm{C}_{8} \mathrm{H}_{10}\), (c) \(\mathrm{C}_{4} \mathrm{H}_{8} \mathrm{O}_{2}\), (d) \(\mathrm{P}_{4} \mathrm{O}_{10}\), (e) \(\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{Cl}_{2}\), (f) \(\mathrm{B}_{3} \mathrm{~N}_{3} \mathrm{H}_{6}\).

Step-by-Step Solution

Answer

The empirical formulas corresponding to each of the given molecular formulas are: (a) \(\mathrm{AlBr}_{3}\) (b) \(\mathrm{C}_{4} \mathrm{H}_{5}\) (c) \(\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O}\) (d) \(\mathrm{P}_{2} \mathrm{O}_{5}\) (e) \(\mathrm{C}_{3} \mathrm{H}_{2} \mathrm{Cl}\) (f) \(\mathrm{BNH}_{2}\)

Step 1: Find the GCD of subscripts

Calculate the greatest common divisor of the subscripts for each of the given molecular formulas. (a) \(\mathrm{Al}_{2} \mathrm{Br}_{6}\) GCD(2, 6) = 2 (b) \(\mathrm{C}_{8} \mathrm{H}_{10}\) GCD(8, 10) = 2 (c) \(\mathrm{C}_{4} \mathrm{H}_{8} \mathrm{O}_{2}\) GCD(4, 8, 2) = 2 (d) \(\mathrm{P}_{4} \mathrm{O}_{10}\) GCD(4, 10) = 2 (e) \(\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{Cl}_{2}\) GCD(6, 4, 2) = 2 (f) \(\mathrm{B}_{3} \mathrm{N}_{3} \mathrm{H}_{6}\) GCD(3, 3, 6) = 3

Step 2: Divide the subscripts by their GCD

Divide the subscripts in each molecular formula by their greatest common divisor. (a) \(\mathrm{Al}_{\frac{2}{2}} \mathrm{Br}_{\frac{6}{2}}\), which simplifies to \(\mathrm{Al}_{1} \mathrm{Br}_{3}\) (b) \(\mathrm{C}_{\frac{8}{2}} \mathrm{H}_{\frac{10}{2}}\), which simplifies to \(\mathrm{C}_{4} \mathrm{H}_{5}\) (c) \(\mathrm{C}_{\frac{4}{2}} \mathrm{H}_{\frac{8}{2}} \mathrm{O}_{\frac{2}{2}}\), which simplifies to \(\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O}_{1}\) (d) \(\mathrm{P}_{\frac{4}{2}} \mathrm{O}_{\frac{10}{2}}\), which simplifies to \(\mathrm{P}_{2} \mathrm{O}_{5}\) (e) \(\mathrm{C}_{\frac{6}{2}} \mathrm{H}_{\frac{4}{2}} \mathrm{Cl}_{\frac{2}{2}}\), which simplifies to \(\mathrm{C}_{3} \mathrm{H}_{2} \mathrm{Cl}_{1}\) (f) \(\mathrm{B}_{\frac{3}{3}} \mathrm{N}_{\frac{3}{3}} \mathrm{H}_{\frac{6}{3}}\), which simplifies to \(\mathrm{B}_{1} \mathrm{N}_{1} \mathrm{H}_{2}\)

Step 3: Write the empirical formulas

Write down the empirical formula for each molecular formula. (a) \(\mathrm{AlBr}_{3}\) (b) \(\mathrm{C}_{4} \mathrm{H}_{5}\) (c) \(\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O}\) (d) \(\mathrm{P}_{2} \mathrm{O}_{5}\) (e) \(\mathrm{C}_{3} \mathrm{H}_{2} \mathrm{Cl}\) (f) \(\mathrm{BNH}_{2}\)

Key Concepts

Molecular Formula

A molecular formula provides a significant amount of information about a compound, specifically its exact number of each type of atom. For example, glucose has a molecular formula of \( C_6H_{12}O_6 \), which indicates that one molecule of glucose contains 6 carbon atoms, 12 hydrogen atoms, and 6 oxygen atoms. This explicit representation is crucial in chemistry as it helps understand the composition and the structure of the compound.

When delving into molecular formulas, it is essential to differentiate them from empirical formulas. The molecular formula represents the actual numbers of atoms in a molecule while the empirical formula is the simplest whole-number ratio of those atoms. For example, the empirical formula for glucose is \( CH_2O \) because the ratio of carbon to hydrogen to oxygen is 1:2:1, which brings us to the importance of simplifying molecular formulas to their empirical counterparts.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD), sometimes known as the greatest common factor (GCF), is the largest positive integer that evenly divides each number in a set without leaving a remainder. In the context of chemistry, GCD is used to simplify chemical formulas by finding the simplest ratio of elements within a compound.

For instance, if you have a compound with the molecular formula \( C_8H_{10} \) and you wish to find the empirical formula, the GCD of the subscripts (in this case, 8 and 10) is 2. This GCD is utilized to divide each subscript, effectively simplifying the molecular formula to its lowest terms, yielding the empirical formula \( C_4H_5 \). Recognizing and calculating the GCD is a critical step in chemical formula simplification, which streamlines many aspects of chemical analysis and synthesis.

Chemical Formula Simplification

Chemical formula simplification is about reducing a molecular formula to its most basic form, the empirical formula, by using the GCD of the subscripts. This process is like simplifying fractions in mathematics. Taking the molecular formula \( C_6H_{12}O_6 \) again, we find that the GCD of 6, 12, and 6 is 6. Dividing each subscript by 6, the empirical formula is determined to be \( CH_2O \).

Keeping It Simple

Often, students encounter complex molecular formulas especially in organic compounds. Simplification removes complexity and helps them understand the key components of a compound. As seen in the exercise with \( C_8H_{10} \) and \( C_6H_{12}O_6 \) A simplified formula is invaluable in stoichiometry, where chemists calculate reactants and products in chemical reactions. Additionally, simplifying a chemical formula can uncover symmetries and patterns in chemical behavior, easing the prediction of reaction outcomes or properties of the substance. The journey from a molecular to an empirical formula is not simply a mathematical exercise, but a path to deeper insight into the nature of the substance being studied.