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) \(\mathrm{B}_{3} \mathrm{~N}_{3} \mathrm{H}_{6}\)

Short answer

The empirical formulas for 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-by-step solution

Identify the greatest common divisor of the subscripts

The greatest common divisor (GCD) of 2 and 6 is 2.

Divide each subscript by the greatest common divisor

Divide 2 by 2 to get 1, and 6 by 2 to get 3. So, we will have \(\mathrm{Al}_{\frac{2}{2}}\mathrm{Br}_{\frac{6}{2}}\).

Write the empirical formula

The empirical formula for \(\mathrm{Al}_{2} \mathrm{Br}_{6}\) is \(\mathrm{AlBr}_{3}\). (b) \(\mathrm{C}_{8} \mathrm{H}_{10}\)

Identify the greatest common divisor of the subscripts

The greatest common divisor (GCD) of 8 and 10 is 2.

Divide each subscript by the greatest common divisor

Divide 8 by 2 to get 4, and 10 by 2 to get 5. So, we will have \(\mathrm{C}_{\frac{8}{2}}\mathrm{H}_{\frac{10}{2}}\).

Write the empirical formula

The empirical formula for \(\mathrm{C}_{8} \mathrm{H}_{10}\) is \(\mathrm{C}_{4}\mathrm{H}_{5}\). (c) \(\mathrm{C}_{4} \mathrm{H}_{8} \mathrm{O}_{2}\)

Identify the greatest common divisor of the subscripts

The greatest common divisor (GCD) of 4, 8, and 2 is 2.

Divide each subscript by the greatest common divisor

Divide 4 by 2 to get 2, 8 by 2 to get 4, and 2 by 2 to get 1. So, we will have \(\mathrm{C}_{\frac{4}{2}}\mathrm{H}_{\frac{8}{2}}\mathrm{O}_{\frac{2}{2}}\).

Write the empirical formula

The empirical formula for \(\mathrm{C}_{4} \mathrm{H}_{8} \mathrm{O}_{2}\) is \(\mathrm{C}_{2}\mathrm{H}_{4}\mathrm{O}\). (d) \(\mathrm{P}_{4} \mathrm{O}_{10}\)

Identify the greatest common divisor of the subscripts

The greatest common divisor (GCD) of 4 and 10 is 2.

Divide each subscript by the greatest common divisor

Divide 4 by 2 to get 2, and 10 by 2 to get 5. So, we will have \(\mathrm{P}_{\frac{4}{2}}\mathrm{O}_{\frac{10}{2}}\).

Write the empirical formula

The empirical formula for \(\mathrm{P}_{4} \mathrm{O}_{10}\) is \(\mathrm{P}_{2}\mathrm{O}_{5}\). (e) \(\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{Cl}_{2}\)

Identify the greatest common divisor of the subscripts

The greatest common divisor (GCD) of 6, 4, and 2 is 2.

Divide each subscript by the greatest common divisor

Divide 6 by 2 to get 3, 4 by 2 to get 2, and 2 by 2 to get 1. So, we will have \(\mathrm{C}_{\frac{6}{2}}\mathrm{H}_{\frac{4}{2}}\mathrm{Cl}_{\frac{2}{2}}\).

Write the empirical formula

The empirical formula for \(\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{Cl}_{2}\) is \(\mathrm{C}_{3}\mathrm{H}_{2}\mathrm{Cl}\). (f) \(\mathrm{B}_{3} \mathrm{N}_{3} \mathrm{H}_{6}\)

Identify the greatest common divisor of the subscripts

The greatest common divisor (GCD) of 3, 3, and 6 is 3.

Divide each subscript by the greatest common divisor

Divide 3 by 3 to get 1, 3 by 3 to get 1, and 6 by 3 to get 2. So, we will have \(\mathrm{B}_{\frac{3}{3}}\mathrm{N}_{\frac{3}{3}}\mathrm{H}_{\frac{6}{3}}\).

Write the empirical formula

The empirical formula for \(\mathrm{B}_{3} \mathrm{N}_{3} \mathrm{H}_{6}\) is \(\mathrm{BNH}_{2}\).

Key concepts

Molecular Formula

A molecular formula provides the exact count of each type of atom in a single molecule of a compound. It represents the actual number of atoms present in the compound, giving you the precise combination without simplification. For example, the molecular formula for glucose is \( \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6 \), which indicates six carbon atoms, twelve hydrogen atoms, and six oxygen atoms per molecule of glucose. This formula communicates the specific makeup of glucose without reduction.When comparing, an empirical formula is a simpler, reduced version that shows the smallest whole number ratio of atoms, like \( \mathrm{CH}_2\mathrm{O} \) for glucose. While molecular formulas give the actual composition, empirical formulas give a simplified snapshot, often used to understand proportions.

Chemical Subscripts

Chemical subscripts are the small numbers appearing to the right and slightly below each element symbol in a molecular formula. They indicate the number of atoms of that element in the molecule.For instance, in ethylene, which has a molecular formula \( \mathrm{C}_2 \mathrm{H}_4 \):

• The subscript '2' after carbon (C) suggests there are two carbon atoms.
• The subscript '4' after hydrogen (H) denotes there are four hydrogen atoms.

These subscripts play a critical role in determining the compound's formula mass and, in crafting the empirical formula, help interpret the ratio of the atoms within it. Without changing them, the chemical representation and understanding of a substance can significantly alter.

Greatest Common Divisor

The greatest common divisor (GCD) is crucial when simplifying molecular formulas to empirical formulas. It is the largest number that can evenly divide all the subscripts in a formula, simplifying them to their smallest relative values.Let's see how it works through an example:

• If you have a formula like \( \mathrm{C}_8 \mathrm{H}_{10} \), the GCD of 8 and 10 is 2.
• Divide each subscript by the GCD: 8 divided by 2 equals 4 and 10 divided by 2 equals 5.
• This gives you the empirical formula \( \mathrm{C}_4 \mathrm{H}_5 \).

Using the GCD allows chemists to effectively simplify formulas, providing a clear and concise way to represent chemical substances in their simplest form.