Question

Write the empirical formulas of the following compounds: (a) \(\mathrm{C}_{2} \mathrm{~N}_{2},\) (b) \(\mathrm{C}_{6} \mathrm{H}_{6},\) (c) \(\mathrm{C}_{9} \mathrm{H}_{20},\) (d) \(\mathrm{P}_{4} \mathrm{O}_{10}\) (e) \(\mathrm{B}_{2} \mathrm{H}_{6^{-}}\)

Short answer

The empirical formulas are: (a) \( \mathrm{CN} \), (b) \( \mathrm{CH} \), (c) \( \mathrm{C}_9 \mathrm{H}_{20} \), (d) \( \mathrm{P}_2 \mathrm{O}_5 \), (e) \( \mathrm{BH}_3 \).

Step-by-step solution

Understand the Empirical Formula

The empirical formula is the simplest whole-number ratio of atoms of each element in a compound. It may or may not be the same as the molecular formula which denotes the actual number of atoms in a molecule.

Analyzing Compound (a) \( \mathrm{C}_2 \mathrm{N}_2 \)

For \( \mathrm{C}_2 \mathrm{N}_2 \), both carbon and nitrogen have subscripts of 2. To find the simplest ratio, divide by the greatest common divisor (GCD), which is 2. This results in the formula \( \mathrm{CN} \).

Analyzing Compound (b) \( \mathrm{C}_6 \mathrm{H}_6 \)

For \( \mathrm{C}_6 \mathrm{H}_6 \), both carbon and hydrogen have subscripts of 6. Divide by their GCD (which is 6) to get the formula \( \mathrm{CH} \).

Analyzing Compound (c) \( \mathrm{C}_9 \mathrm{H}_{20} \)

The subscripts are 9 (for carbon) and 20 (for hydrogen). The GCD is 1, thus the empirical formula remains the same as the molecular formula: \( \mathrm{C}_9 \mathrm{H}_{20} \).

Analyzing Compound (d) \( \mathrm{P}_4 \mathrm{O}_{10} \)

The subscripts for phosphorus and oxygen are 4 and 10, respectively. The GCD is 2. So, divide both by 2, resulting in the formula \( \mathrm{P}_2 \mathrm{O}_5 \).

Analyzing Compound (e) \( \mathrm{B}_2 \mathrm{H}_6^{-} \)

The subscripts for boron and hydrogen in \( \mathrm{B}_2 \mathrm{H}_6 \) are 2 and 6. The GCD of these numbers is 2. Divide both by 2 to get \( \mathrm{BH}_3 \) (the charge isn't affected by the empirical formula calculation).

Key concepts

Chemical Compounds

Chemical compounds are substances formed when two or more elements combine in fixed proportions. When atoms of different elements bond together, they form compounds, which possess unique properties different from their constituent elements. These compounds could be quite diverse, ranging from simple compositions like water (H₂O) to complex ones such as vitamin C (C₆H₈O₆). Understanding chemical compounds involves recognizing that:

• Elements combine in specific and fixed ratios.
• The properties of a compound are different from the properties of its elements.
• Compounds are classified based on the types of bonds holding the atoms together (ionic, covalent, metallic, etc.).

A chemical compound is represented by a chemical formula, which summarizes the types of atoms and the number of each type in a molecule. This allows chemists to communicate specific compositions across the globe.

Subscripts in Chemistry

Subscripts in chemistry are small numbers found in chemical formulas, used to indicate the number of atoms of each element present in a compound. These numbers are written directly after the chemical symbol for each element and play a crucial role in defining the molecular composition of a compound. For example, in water (H₂O), the subscript '2' indicates the presence of two hydrogen atoms bonded to one oxygen atom. Key points about subscripts include:

• They show the exact count of atoms in the simplest whole-number ratio.
• Subscripts must always be whole numbers, as atoms cannot be divided.
• Changing a subscript within a chemical formula changes the compound itself; for example, H₂O (water) versus H₂O₂ (hydrogen peroxide).

In empirical formulas, which represent the simplest ratio, the subscripts are often reduced by finding their greatest common divisor.

Greatest Common Divisor in Chemistry

The greatest common divisor (GCD) in chemistry is used to reduce the subscripts of the elements in a chemical formula to their simplest form. This concept simplifies understanding complex compounds by finding the largest number that divides all the subscripts without a remainder, hence obtaining their simplest ratio. Here's how GCD is utilized:

• Identify the subscripts in a chemical formula.
• Calculate the GCD of these numbers.
• Divide each subscript by the GCD to find the simplest formula.

For instance, in the formula C₆H₆, the GCD of both subscripts (6 for carbon and 6 for hydrogen) is 6. Dividing each by the GCD results in the empirical formula CH, which represents the most basic element ratio.

Molecular Formula vs Empirical Formula

Understanding the difference between molecular formula and empirical formula is crucial in chemistry. They both represent chemical compounds but in distinct ways:

• Molecular Formula: It shows the actual number of atoms of each element in a molecule. For instance, benzene has the molecular formula C₆H₆, indicating it contains six carbon and six hydrogen atoms.
• Empirical Formula: It presents the simplest whole-number ratio of the atoms in a compound. Using benzene again, its empirical formula is CH, demonstrating that the ratio of carbon to hydrogen is 1:1.

To transition from a molecular formula to an empirical formula, one must calculate the greatest common divisor for the subscripts and divide them. Knowing both forms provides a deeper insight into understanding and analyzing chemical substances.