Problem 26 - Discrete Mathematics with Applications

Question

Find the number of bonds in each hydrocarbon molecule. (Assume each carbon atom is bonded to four atoms.) $$K_{7}$$

Step-by-Step Solution

Answer

In the K7 hydrocarbon molecule with 7 carbon atoms, there are 22 bonds. This is calculated using the formula: Number of bonds = (n * 4) - (n - 1), where n is the number of carbon atoms. For K7, Number of bonds = (7 * 4) - (7 - 1) = 28 - 6 = 22.

Step 1: 1. Determine the number of carbon atoms in K7

K7 represents a hydrocarbon molecule with 7 carbon atoms.

Step 2: 2. Understand the bonding pattern in hydrocarbon molecules

In a hydrocarbon molecule, each carbon atom forms four bonds in total. If we look at two adjacent carbon atoms, they form one bond with each other, while the rest of their bonds are either with hydrogen atoms (in saturated hydrocarbons) or with other carbon atoms (in unsaturated hydrocarbons).

Step 3: 3. Calculate the number of bonds for K7

We can use the following formula to calculate the number of bonds in a hydrocarbon molecule based on the number of carbon atoms: Number of bonds = (n * 4) - (n - 1) where n is the number of carbon atoms. For K7 (n = 7): Number of bonds = (7 * 4) - (7 - 1) = 28 - 6 = 22 Thus, there are 22 bonds in the K7 hydrocarbon molecule.

Key Concepts

Bonds in Molecules

In any molecule, bonds are the important forces holding atoms together. These connections generally arise from electrons being shared or transferred between atoms.

Specifically, in hydrocarbons, the bonds are all about sharing electrons. Carbon and hydrogen atoms in these molecules form covalent bonds by sharing valence electrons. This ensures each atom achieves a full outer electron shell, resembling the configuration of noble gases which are very stable.

• Hydrocarbons are primarily made up of carbon (C) and hydrogen (H) atoms.
• Covalent bonds are the principle type of bond in hydrocarbons.
• The type of bonding dictates the molecule's properties and reactions.

With covalent bonds, each atom comes to have electrons akin to the stable noble gases, which is why these bonds are prevalent in organic molecules like hydrocarbons.

Carbon Bonding

Carbon is known for its unique ability to form a variety of bonds, which contributes to the diverse forms of hydrocarbons. A single carbon atom can make four covalent bonds in a tetrahedral arrangement. This is because carbon's atomic number is 6, meaning it has four valence electrons able to pair with the electrons of other atoms.

The potential bonds include:

• Single bonds: where carbon shares one pair of electrons with another atom. Seen in saturated hydrocarbons like alkanes.
• Double bonds: sharing two pairs of electrons, found in alkenes.
• Triple bonds: sharing three pairs of electrons, which occur in alkynes.

Paths of carbon bonding define the types and features of hydrocarbons. The diversity in bonding patterns is a central reason why carbon can construct stable, complex structures essential for life's chemistry.

Hydrocarbon Structure

A hydrocarbon's structure is fundamentally dictated by how the carbon atoms connect with each other and with hydrogen atoms. This results in various shapes and forms of hydrocarbon chains or rings. There are two main types of hydrocarbons: saturated and unsaturated.

Saturated hydrocarbons, or alkanes, have all carbon-carbon bonds as single bonds. This results in a more stable and less reactive structure.

Unsaturated hydrocarbons, such as alkenes and alkynes, have one or more double or triple bonds, respectively. These bonds introduce kinks or bends in the molecule, greatly influencing its reactivity and the kinds of reactions it can perform.

• Different structures determine how hydrocarbons interact and react chemically.
• Linear chains, branched chains, and rings are common formations in hydrocarbon structures.
• The presence of double or triple bonds can lead to geometric or structural isomerism.

Understanding these variations is crucial for the study of organic chemistry and the numerous applications of hydrocarbons in industries, such as fuel and materials production.