Problem 77 - Chemistry The Science in Context

Question

Gasoline consists primarily of a mixture of the hydrocarbons \(\mathrm{C}_{6} \mathrm{H}_{14}, \mathrm{C}_{7} \mathrm{H}_{16}, \mathrm{C}_{8} \mathrm{H}_{18},\) and \(\mathrm{C}_{9} \mathrm{H}_{20} .\) What is the empirical formula of each compound?

Step-by-Step Solution

Answer

Answer: The empirical formulas for the given hydrocarbons are: 1. \(\mathrm{C}_{6}\mathrm{H}_{14}\): \(\mathrm{C}_{3}\mathrm{H}_{7}\) 2. \(\mathrm{C}_{7}\mathrm{H}_{16}\): \(\mathrm{C}_{7}\mathrm{H}_{16}\) 3. \(\mathrm{C}_{8}\mathrm{H}_{18}\): \(\mathrm{C}_{4}\mathrm{H}_{9}\) 4. \(\mathrm{C}_{9}\mathrm{H}_{20}\): \(\mathrm{C}_{9}\mathrm{H}_{20}\)

Step 1: Finding the empirical formula of \(\mathrm{C}_{6}\mathrm{H}_{14}\)

Since the molecular formula is \(\mathrm{C}_{6}\mathrm{H}_{14}\), there are 6 moles of carbon and 14 moles of hydrogen. To find the empirical formula, we must find the GCD of 6 and 14, which is 2. So, divide both elements by the GCD: \(\frac{6}{2}:\frac{14}{2}\), which simplifies to \(\mathrm{C}_{3}\mathrm{H}_{7}\). Therefore, the empirical formula is \(\mathrm{C}_{3}\mathrm{H}_{7}\).

Step 2: Finding the empirical formula of \(\mathrm{C}_{7}\mathrm{H}_{16}\)

Since the molecular formula is \(\mathrm{C}_{7}\mathrm{H}_{16}\), there are 7 moles of carbon and 16 moles of hydrogen. To find the empirical formula, we must find the GCD of 7 and 16, which is 1. So, divide both elements by the GCD: \(\frac{7}{1}:\frac{16}{1}\), which does not simplify further. Therefore, the empirical formula is \(\mathrm{C}_{7}\mathrm{H}_{16}\).

Step 3: Finding the empirical formula of \(\mathrm{C}_{8}\mathrm{H}_{18}\)

Since the molecular formula is \(\mathrm{C}_{8}\mathrm{H}_{18}\), there are 8 moles of carbon and 18 moles of hydrogen. To find the empirical formula, we must find the GCD of 8 and 18, which is 2. So, divide both elements by the GCD: \(\frac{8}{2}:\frac{18}{2}\), which simplifies to \(\mathrm{C}_{4}\mathrm{H}_{9}\). Therefore, the empirical formula is \(\mathrm{C}_{4}\mathrm{H}_{9}\).

Step 4: Finding the empirical formula of \(\mathrm{C}_{9}\mathrm{H}_{20}\)

Since the molecular formula is \(\mathrm{C}_{9}\mathrm{H}_{20}\), there are 9 moles of carbon and 20 moles of hydrogen. To find the empirical formula, we must find the GCD of 9 and 20, which is 1. So, divide both elements by the GCD: \(\frac{9}{1}:\frac{20}{1}\), which does not simplify further. Therefore, the empirical formula is \(\mathrm{C}_{9}\mathrm{H}_{20}\). In conclusion, the empirical formulas for the given hydrocarbons are: 1. \(\mathrm{C}_{6}\mathrm{H}_{14}\): \(\mathrm{C}_{3}\mathrm{H}_{7}\) 2. \(\mathrm{C}_{7}\mathrm{H}_{16}\): \(\mathrm{C}_{7}\mathrm{H}_{16}\) 3. \(\mathrm{C}_{8}\mathrm{H}_{18}\): \(\mathrm{C}_{4}\mathrm{H}_{9}\) 4. \(\mathrm{C}_{9}\mathrm{H}_{20}\): \(\mathrm{C}_{9}\mathrm{H}_{20}\)

Key Concepts

Hydrocarbons

Hydrocarbons are chemical compounds made up entirely of hydrogen and carbon atoms. These are some of the most basic, yet crucial, elements in organic chemistry. The simplicity of hydrocarbons makes them highly versatile in forming various complex structures. Hydrocarbons are broadly categorized into:

• Alkanes: Saturated hydrocarbons with single bonds, such as methane ( CH_4 ).
• Alkenes: Unsaturated hydrocarbons with one or more double bonds, like ethylene ( C_2H_4 ).
• Alkynes: Unsaturated hydrocarbons with one or more triple bonds, such as acetylene ( C_2H_2 ).

In this exercise, the hydrocarbons in question are alkanes, as they contain only single bonds. The hydrocarbons C_6H_{14} , C_7H_{16} , C_8H_{18} , and C_9H_{20} consist of chains of carbon atoms with hydrogen atoms filling the remaining valency. Found primarily in fuels like gasoline, these compounds are essential in energy production and as raw materials in the chemical industry.

Molecular Formula

The molecular formula represents the actual number of atoms of each element present in a molecule of a compound. It provides insight into the molecular composition and allows chemists to understand the size and structure of the compound. In our exercise, these molecular formulas are C_6H_{14} , C_7H_{16} , C_8H_{18} , and C_9H_{20} . Each formula indicates:

• The number following the element symbol (like C or H) specifies how many of those atoms are in each molecule.
• The molecular formula can be used to derive the empirical formula, which is the simplest whole number ratio of atoms in a compound.

The molecular formula is essential for determining the molar mass of the compound and is key to understanding how the molecule might interact with other compounds. It gives a complete picture of the potential size and structure that can affect physical and chemical properties.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a mathematical concept frequently utilized in chemistry for simplifying ratios, as seen when determining empirical formulas from molecular formulas. The GCD of two numbers is the largest number that can divide both without leaving a remainder. To compute the GCD:

• Identify the factors of each number.
• Compare these factors to determine the largest one common to both numbers.
• Use the GCD to simplify ratios to their lowest terms.

In the case of finding empirical formulas, the GCD of the number of carbon and hydrogen atoms in the molecular formula aids in reducing these numbers to the simplest ratio. For example, in C_6H_{14} , the GCD of 6 and 14 is 2, allowing us to reduce the molecular formula to C_3H_7 . This concept is a practical tool in chemistry for pinpointing the simplest equation that still represents the compound's elemental composition.