Problem 12
Question
How many bonds does the hydrocarbon molecule \(\mathrm{C}_{n} \mathrm{H}_{2 n+2}\) have? Assume a carbon molecule has degree four.
Step-by-Step Solution
Verified Answer
The hydrocarbon molecule CnH2n+2 has \(6n+2\) bonds.
1Step 1: Determine the number of carbon and hydrogen atoms in the molecule
In the given hydrocarbon molecule, CnH2n+2, there are n carbon atoms and 2n+2 hydrogen atoms.
2Step 2: Calculate the total number of bonds formed by carbon atoms
Since each carbon atom has a degree of 4 and there are n carbon atoms in the molecule, the total number of bonds formed by carbon atoms will be 4n.
3Step 3: Calculate the total number of bonds formed by hydrogen atoms
Hydrogen atoms have a degree of 1, which means they form one bond. Since there are 2n+2 hydrogen atoms, the total number of bonds formed by hydrogen atoms is 2n+2.
4Step 4: Calculate the total number of bonds in the molecule
To calculate the total number of bonds in the molecule, add the total number of bonds formed by carbon and hydrogen atoms:
Total number of bonds = Bonds formed by carbon atoms + Bonds formed by hydrogen atoms
Total number of bonds = 4n + (2n+2)
5Step 5: Simplify the expression for the total number of bonds
Combine the terms in the expression to simplify it:
Total number of bonds = 6n + 2
The hydrocarbon molecule CnH2n+2 has 6n+2 bonds.
Key Concepts
Graph TheoryMolecule DegreeApplication in Chemistry
Graph Theory
Graph theory is a field of mathematics that studies the properties of graphs, which are structures used to model pairwise relations between objects. A graph is composed of vertices (also called nodes) and edges that connect pairs of vertices. In graph theory, a degree of a vertex refers to the number of edges connected to it.
This concept is fundamental in various fields, including computer science for network analysis, operations research for optimization, and, notably, chemistry for molecular structure representation. In chemistry, graph theory is employed to model molecules where atoms are represented as vertices and bonds as edges. The degree of an atom within a molecule corresponds to its valency, or the number of chemical bonds it can form.
Understanding graph theory helps in deciphering molecule structures, predicting chemical reactions, and synthesizing new compounds. The abstraction of these complex chemical structures into simpler graph models allows for a more straightforward analysis and manipulation of the information they carry.
This concept is fundamental in various fields, including computer science for network analysis, operations research for optimization, and, notably, chemistry for molecular structure representation. In chemistry, graph theory is employed to model molecules where atoms are represented as vertices and bonds as edges. The degree of an atom within a molecule corresponds to its valency, or the number of chemical bonds it can form.
Understanding graph theory helps in deciphering molecule structures, predicting chemical reactions, and synthesizing new compounds. The abstraction of these complex chemical structures into simpler graph models allows for a more straightforward analysis and manipulation of the information they carry.
Molecule Degree
The 'degree' of an atom in a molecule is a concept borrowed from graph theory, which signifies the number of chemical bonds an atom forms in a molecule. It's also known as the atom's valency. This is particularly important in organic chemistry, where the degree determines the structure and reactivity of molecules.
For example, in hydrocarbons, carbon typically has a degree of four, allowing it to form four bonds with other atoms. Hydrogen, on the other hand, has a degree of one, as it can form only one bond. This principle is the basis for determining molecular structures and is crucial for predicting the behavior of organic molecules.
By examining the molecule degree along with the concept of saturation (where saturated hydrocarbons have single bonds and unsaturated have double or triple bonds), chemists can deduce the types of bonds and the overall structure of the molecule.
For example, in hydrocarbons, carbon typically has a degree of four, allowing it to form four bonds with other atoms. Hydrogen, on the other hand, has a degree of one, as it can form only one bond. This principle is the basis for determining molecular structures and is crucial for predicting the behavior of organic molecules.
By examining the molecule degree along with the concept of saturation (where saturated hydrocarbons have single bonds and unsaturated have double or triple bonds), chemists can deduce the types of bonds and the overall structure of the molecule.
Application in Chemistry
The application of discrete mathematics, especially graph theory, in chemistry is pivotal for understanding molecular structures and predicting molecular properties. Chemical graph theory uses the language of graphs to describe the bonding patterns within a molecule.
With applications ranging from the simple enumeration of molecules' bonds, such as in the provided hydrocarbon example CnH2n+2, to complex reactions, graph theory supports chemists in visualizing the composition of molecules. It is particularly useful in organic chemistry for characterizing the structures of hydrocarbons, where the degree of a molecule provides insight into potential reactions and stability of the molecule.
Moreover, graph theoretical models are crucial for computational chemistry, where they are used to simulate and predict the behavior of molecules without the need for expensive laboratory experiments. This not only aids in the educational comprehension of chemical concepts but also fosters innovation and discovery in chemical research and development.
With applications ranging from the simple enumeration of molecules' bonds, such as in the provided hydrocarbon example CnH2n+2, to complex reactions, graph theory supports chemists in visualizing the composition of molecules. It is particularly useful in organic chemistry for characterizing the structures of hydrocarbons, where the degree of a molecule provides insight into potential reactions and stability of the molecule.
Moreover, graph theoretical models are crucial for computational chemistry, where they are used to simulate and predict the behavior of molecules without the need for expensive laboratory experiments. This not only aids in the educational comprehension of chemical concepts but also fosters innovation and discovery in chemical research and development.
Other exercises in this chapter
Problem 11
Let \(n\) denote the number of vertices of a tree and \(e\) the number of edges. Verify that \(e=n-1\) for each tree. IMAGE IS NOT AVAILABLE TO COPY
View solution Problem 12
Let \(n\) be a positive integer and key an arbitrary positive integer \(\leq n .\) Using binary search, write an algorithm to find key and the number of guesses
View solution Problem 13
Among seven identical coins lies a heavier counterfeit coin. Write an algorithm to identify the false coin using an equal-arm balance and minimum weighings.
View solution Problem 13
Let \(T\) be a tree with vertices \(v_{1}, \ldots, v_{n} .\) Show that \(\sum_{i=1}^{n} \operatorname{deg}\left(v_{i}\right)=2 n-2\)
View solution