Question

Write the empirical formulas of the following compounds: (a) \(\mathrm{Al}_{2} \mathrm{Br}_{6}\) (b) \(\mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{4}\) (c) \(\mathrm{N}_{2} \mathrm{O}_{5},\) (d) \(\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}\) (e) \(\mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}\)

Short answer

(a) \( \text{AlBr}_3 \); (b) \( \text{NaSO}_2 \); (c) \( \text{N}_2\text{O}_5 \); (d) \( \text{K}_2\text{Cr}_2\text{O}_7 \); (e) \( \text{HCO}_2 \).

Step-by-step solution

Understand the Empirical Formula Concept

The empirical formula of a compound is the simplest whole-number ratio of atoms of each element present in the compound. It may not represent the actual number of atoms in a molecule but rather the simplest ratio.

Simplify Al2 Br6

Determine the greatest common divisor (GCD) of the numbers of atoms in \( \text{Al}_2\text{Br}_6 \). The ratio of Al to Br is 2:6, which simplifies to 1:3. Thus, the empirical formula is \( \text{AlBr}_3 \).

Simplify Na2 S2 O4

The subscripts are \(2, 2, \text{ and } 4\). The GCD is 2. Divide each subscript by 2, resulting in \( \text{Na}_1\text{S}_1\text{O}_2 \). Thus, the empirical formula is \( \text{NaSO}_2 \).

Simplify N2 O5

The ratio of N to O in \( \text{N}_2\text{O}_5 \) is already in its simplest whole-number ratio. Thus, the empirical formula is \( \text{N}_2\text{O}_5 \).

Simplify K2 Cr2 O7

The ratio of K to Cr to O in \( \text{K}_2\text{Cr}_2\text{O}_7 \) cannot be simplified further as it is already in its simplest form. Thus, the empirical formula is \( \text{K}_2\text{Cr}_2\text{O}_7 \).

Simplify H2 C2 O4

For \( \text{H}_2\text{C}_2\text{O}_4 \), the GCD is 2. Divide each subscript by 2 to get the simplest ratio: \( \text{H}_1\text{C}_1\text{O}_2 \). Thus, the empirical formula is \( \text{HCO}_2 \).

Key concepts

Simplification

When dealing with chemical formulas, simplification plays a crucial role in finding the empirical formula. Simplification involves reducing the subscripts of the atoms in a compound to the smallest possible whole numbers while maintaining the same ratio. This step does not alter the elemental composition but simply expresses it in the simplest form.
To simplify a chemical formula, follow these steps:

• Identify the subscripts that indicate the number of each type of atom present in the compound.
• Determine the greatest common divisor (GCD) of the subscripts.
• Divide each subscript by the GCD to achieve the simplest form.

This reduction process makes it easier to compare compounds and analyze their basic elemental ratios.

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental concept in math and chemistry. It is the largest integer that divides a set of numbers without leaving a remainder. In the context of finding empirical formulas, the GCD is used to simplify the ratios of atoms to the smallest possible whole numbers.
For example, consider the compound \( \mathrm{Al}_2 \mathrm{Br}_6 \):

• The subscripts are 2 and 6.
• The GCD of 2 and 6 is 2.
• Dividing each subscript by the GCD, we simplify the formula to \( \mathrm{AlBr}_3 \).

Using the GCD in this way ensures that the empirical formula represents the simplest whole-number ratio of elements, which is essential for understanding the basic proportions in chemical compounds.

Chemical Compounds

Chemical compounds are substances formed from two or more elements combined in fixed proportions. They possess distinct properties different from the elemental components. Chemical compounds can be represented by molecular or empirical formulas.
Empirical formulas showcase the simplest whole-number ratio of elements in a compound. While this ratio does not necessarily reveal the total count of atoms, it serves to highlight proportions and relationships between different elements. This can be especially useful in stoichiometric calculations, where precise ratios are crucial for determining reaction outcomes.
This understanding of chemical compounds through empirical formulas helps chemists deduce and communicate the proportions of elements within a substance accurately and efficiently.

Whole-Number Ratio

The concept of a whole-number ratio is central to empirical formulas. A whole-number ratio expresses the simplest integer relationship between the numbers of atoms of each element in a compound.
For example, in \( \mathrm{Na}_2 \mathrm{~S}_2 \mathrm{O}_4 \), the ratio of sodium, sulfur, to oxygen is initially 2:2:4. By identifying the GCD, which is 2, we can simplify this ratio to 1:1:2 to get the empirical formula \( \mathrm{NaSO}_2 \).
Whole-number ratios ensure that the empirical formula is as concise and clear as possible, providing the foundation for understanding the elemental makeup effectively. This simplicity is what allows scientists to quickly interpret the elemental balance within different compounds.