Question

Give the empirical formula that corresponds to each molecular formula. (a) \(\mathrm{C}_{2} \mathrm{H}_{6}\) (b) \(\mathrm{N}_{2} \mathrm{O}_{4}\) (c) \(\mathrm{C}_{4} \mathrm{H}_{6} \mathrm{O}_{2}\) (d) \(\mathrm{NH}_{3}\)

Short answer

The empirical formulas are (a) \(\mathrm{CH}_{3}\), (b) \(\mathrm{NO}_{2}\), (c) \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{O}\), and (d) \(\mathrm{NH}_{3}\).

Step-by-step solution

Understanding the Concept of an Empirical Formula

An empirical formula represents the simplest whole-number ratio of the atoms of each element in a compound. To determine an empirical formula from a molecular formula, you divide the subscripts of each element in the molecular formula by the greatest common divisor for all elements present.

Identify the Greatest Common Divisor for (a) \(\mathrm{C}_{2} \mathrm{H}_{6}\)

For the molecular formula \(\mathrm{C}_{2} \mathrm{H}_{6}\), the greatest common divisor between the subscripts for carbon (2) and hydrogen (6) is 2. Divide the subscript of each element by 2 to obtain the empirical formula.

Calculate the Empirical Formula for (a)

Dividing the subscripts in \(\mathrm{C}_{2} \mathrm{H}_{6}\) by 2 gives us \(\mathrm{C}_{1} \mathrm{H}_{3}\). The subscript 1 is typically not written, so the empirical formula is \(\mathrm{CH}_{3}\).

Identify the Greatest Common Divisor for (b) \(\mathrm{N}_{2} \mathrm{O}_{4}\)

For the molecular formula \(\mathrm{N}_{2} \mathrm{O}_{4}\), the greatest common divisor between the subscripts for nitrogen (2) and oxygen (4) is 2. Divide the subscript of each element by 2 to obtain the empirical formula.

Calculate the Empirical Formula for (b)

Dividing the subscripts in \(\mathrm{N}_{2} \mathrm{O}_{4}\) by 2 gives us \(\mathrm{N}_{1} \mathrm{O}_{2}\), or more simply \(\mathrm{NO}_{2}\).

Identify the Greatest Common Divisor for (c) \(\mathrm{C}_{4} \mathrm{H}_{6} \mathrm{O}_{2}\)

For the molecular formula \(\mathrm{C}_{4} \mathrm{H}_{6} \mathrm{O}_{2}\), no single greatest common divisor exists for all subscripts (4, 6, and 2). However, we can still simplify by using the lowest common divisor, which is 2.

Calculate the Empirical Formula for (c)

Dividing each subscript by 2, we get \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{O}_{1}\) or simply \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{O}\).

Determining the Empirical Formula for (d) \(\mathrm{NH}_{3}\)

The molecular formula \(\mathrm{NH}_{3}\) already represents the simplest whole-number ratio of nitrogen to hydrogen (1:3), therefore the empirical formula is the same as the molecular formula: \(\mathrm{NH}_{3}\).

Key concepts

Molecular Formula

The molecular formula provides the exact number of atoms of each element in a molecule, which reflects the actual chemical composition of a substance. It represents the total number of atoms in a single molecule of a compound, usually written as subscripts beside each elemental symbol. For example, in the molecular formula of water, \( H_2O \), there are 2 atoms of hydrogen and 1 atom of oxygen.

To fully grasp what molecular formulas signify in chemistry, it's helpful to look at a specific example. Take glucose, its molecular formula is \( C_6H_{12}O_6 \), which indicates that each molecule comprises 6 carbon atoms, 12 hydrogen atoms, and 6 oxygen atoms. Knowing this allows chemists to predict properties and reactiveness of the substance.

Chemical Composition

Understanding the chemical composition of compounds is central to the field of chemistry. It's all about knowing what elements are present and in what quantities. The chemical composition specifies the kinds of atoms and the counts within a compound, thus defining the compound's characteristic properties.

When we say a compound like ethanol has the chemical composition \( C_2H_5OH \), we are stating that each molecule of ethanol consists of 2 carbon atoms, 6 hydrogen atoms, and 1 oxygen atom. Recognizing the chemical composition is significant for predicting how substances will interact in chemical reactions.

Stoichiometry

Stoichiometry is a quantitative branch of chemistry that involves the calculation of the relative quantities of reactants and products involved in chemical reactions. It's based on the conservation of mass and the concept of moles, which is a measure of the number of entities (usually atoms or molecules) in a sample.

The stoichiometric coefficient in a balanced chemical equation tells us in what ratio the molecules react or form. For example, in the complete combustion of methane, the stoichiometric ratio of \( CH_4:O_2:CO_2:H_2O \) is 1:2:1:2, highlighting that one molecule of methane reacts with two molecules of oxygen to produce one molecule of carbon dioxide and two molecules of water.

Greatest Common Divisor

The greatest common divisor (GCD) is a mathematical concept that refers to the largest positive integer that divides a set of numbers without leaving a remainder. In the context of chemistry and the conversion of molecular formulas to empirical formulas, we use the GCD to simplify the subscripts to the smallest whole numbers that maintain the ratio of elements.

For instance, when we determine the empirical formula for \( C_2H_6 \), we find that 2 is the GCD for the subscripts 2 and 6. Dividing each by this number yields the simplified ratio \( CH_3 \) for its empirical formula. GCD is essential for transforming complex molecular formulas into their most reduced form, the empirical formula.