Problem 21 - Discrete Mathematics with Applications

Question

Find the number of bonds in each hydrocarbon molecule. (Assume each carbon atom is bonded to four atoms.) A butane molecule \(\mathrm{C}_{4} \mathrm{H}_{10}\)

Step-by-Step Solution

Answer

In a butane molecule \(\mathrm{C}_{4} \mathrm{H}_{10}\), there are 13 bonds in total. This is obtained by calculating the number of C-C bonds (3), internal C-H bonds (4), and terminal C-H bonds (6).

Step 1: 1. Understand the bonding in hydrocarbon molecules

In hydrocarbon molecules, carbon atoms form bonds with other carbon atoms and hydrogen atoms. Given each carbon atom is bonded to four atoms, it can form up to four single bonds. In this case, we have a butane molecule, \(\mathrm{C}_{4} \mathrm{H}_{10}\), containing 4 carbon atoms and 10 hydrogen atoms. We will break down the bonding in the molecule to find the total number of bonds.

Step 2: 2. Determine the bonds between carbon atoms

In the butane molecule, the 4 carbon atoms are bonded together in a chain. Since there are 4 carbon atoms, there will be 3 bonds. C - C - C - C Here is the bond count between carbon atoms: C to C bonds: 3

Step 3: 3. Determine the bonds between carbon and hydrogen atoms

Now, we need to account for the bonds between carbon atoms and hydrogen atoms. For the 2 internal carbon atoms, each of them is already bonded to 2 other carbon atoms, so they can form 2 more bonds. That means each internal carbon atom will form bonds with 2 hydrogen atoms. The 2 terminal (end) carbon atoms are only bonded to 1 other carbon atom, so they can form 3 more bonds. Therefore, each terminal carbon atom will form bonds with 3 hydrogen atoms. C - C - C - C | | | | H H H H | | H H Here is the bond count between carbon and hydrogen atoms: Internal C to H bonds: 2 internal carbons * 2 bonds each = 4 Terminal C to H bonds: 2 terminal carbons * 3 bonds each = 6

Step 4: 4. Calculate the total number of bonds

Finally, we need to add the number of C-C bonds and the number of C-H bonds to find the total number of bonds in the butane molecule. Total bonds = C-C bonds + internal C-H bonds + terminal C-H bonds Total bonds = 3 + 4 + 6 Total bonds = 13 So, there are 13 bonds in a butane molecule.

Key Concepts

Hydrocarbon Molecules

Hydrocarbons are organic compounds made solely from hydrogen and carbon atoms. They come in various forms, including alkanes, alkenes, and alkynes.
The simplest hydrocarbons are alkanes. Butane is an example of an alkane, represented by the chemical formula \( \mathrm{C}_{4} \mathrm{H}_{10} \). It's composed of 4 carbon atoms and 10 hydrogen atoms.
There are important characteristics associated with hydrocarbon molecules:

• They are non-polar, which means they do not mix well with water.
• They are generally found in fuels, like gasoline and natural gas.
• The longer the carbon chain, the higher the boiling and melting points.

Understanding these properties helps predict how hydrocarbons will behave chemically and physically.

Carbon-Carbon Bonds

Carbon-carbon (C-C) bonds are the backbone of organic chemistry. In a butane molecule, the carbon atoms are connected in a straight chain.
Each carbon atom can form four covalent bonds, filling their outer shell and achieving stability.
In the butane example, there are 4 carbon atoms, but because they are in a chain, only 3 C-C bonds are required:

• Each bond represents a shared pair of electrons between two carbon atoms.
• These bonds give hydrocarbons like butane their stability and shape.

The strength and flexibility of C-C bonds allow hydrocarbons to form varied and complex compounds.

Carbon-Hydrogen Bonds

Carbon-hydrogen (C-H) bonds are also crucial in hydrocarbons like butane.
In these bonds, carbon shares its electrons with hydrogen, enabling them both to achieve a stable electron configuration.
In the structure of butane:

• Internal carbon atoms in butane connect with 2 hydrogen atoms each, while terminal carbon atoms join with 3 hydrogen atoms each.
• These bonds are single bonds, meaning they involve one pair of shared electrons.

C-H bonds are relatively strong due to the sharing of electrons. They play a significant role in defining the chemical properties and reactions of hydrocarbons.