Problem 9 - Algebra: Pure and Applied

Question

In Exercises 1 through 11 find the number of essentially different ways in which we can do what is described. Replace each of the hydrogen atom in a molecule of benzene with a fluorine, chlorine, or bromine atom. (A molecule of benzene consists of six carbon atoms in a regular hexagon, with one hydrogen atom bonded to each.)

Step-by-Step Solution

Answer

There are 81 essentially different substitution patterns for replacing benzene's hydrogen atoms with F, Cl, or Br.

Step 1: Understanding Benzene Structure

A benzene molecule consists of a hexagonal ring formed by six carbon atoms. Each carbon atom is bonded to one hydrogen atom.

Step 2: Identify the Substituents

The problem asks us to replace each hydrogen atom with either a fluorine (F), chlorine (Cl), or bromine (Br) atom. This gives us three choices for each hydrogen atom.

Step 3: Calculate Total Combinations Without Considering Symmetry

Since there are six hydrogen atoms, each with 3 substitution choices, there are initially \(3^6 = 729\) possible combinations of substituents.

Step 4: Consider Symmetrical Arrangements

Benzene's hexagonal structure has symmetry, meaning some combinations are identical in terms of substitution pattern. Use Burnside's Lemma or Polya's Enumeration Theorem to account for these symmetries.

Step 5: Symmetry Groups in Benzene

The symmetry group of benzene includes rotations and reflections. It contains 12 elements: 6 rotations (0, 60, 120, 180, 240, and 300 degrees) and 6 reflections through axes of symmetry.

Step 6: Calculate Using Polya's Enumeration Theorem

Using Polya's Enumeration Theorem, we calculate the number of distinct configurations: Let's consider the group actions. For each of the 12 symmetries, calculate the number of substitution patterns fixed: - Identity rotation: all 729 configurations remain unchanged.- 60-degree rotations: no configurations unchanged.- 120-degree rotations: 3 remain unchanged.- 180-degree rotations: 9 unchanged.- 240-degree rotations: 3 remain unchanged.- 300-degree rotations: no configurations unchanged.- Axial reflections (6 lines): 27 per line of symmetry.Summing these fixed configurations gives: \\[\frac{1}{12}(729 + 0 + 3 + 9 + 3 + 0 + 6 \times 27) = \frac{1}{12}(729 + 81 + 162) = \frac{972}{12} = 81\]

Step 7: Final Calculation

The number of essentially different ways to replace the hydrogen atoms in benzene with fluorine, chlorine, or bromine, considering benzene's symmetry, is 81.

Key Concepts

Group Theory in Chemistry

Group theory is a branch of mathematics that deals with symmetries and transformations. In chemistry, it helps us understand how molecules behave under various symmetrical operations. Group theory allows chemists to identify symmetrical characteristics in molecular structures, which can influence physical and chemical properties.

• **Symmetry Operations**: These include rotations around an axis, reflections through a plane, and inversions around a point.
• **Symmetry Elements**: Points, lines, or planes concerning which symmetry operations are performed.

Understanding symmetry helps chemists predict molecular behavior, such as how a molecule will react with other substances. When applied to benzene, group theory considers the rotational and reflective symmetry of its hexagonal structure.
This plays a crucial role in determining substitution patterns, as certain configurations might repeat or overlap due to symmetrical properties.

Benzene Symmetry

Benzene is a fundamental compound in organic chemistry, known for its distinct symmetric hexagonal shape. The symmetry of benzene is rich and includes several operations: - **Rotational Symmetry**: Benzene can be rotated around its center without changing its appearance. It has 6-fold rotational symmetry at 60-degree increments. - **Mirror Symmetry**: It also exhibits reflection symmetry along axes passing through opposite carbon atoms.
Because of these symmetrical properties, many substitution patterns of benzene appear similar to each other. When considering substitutions involving different atoms like fluorine, chlorine, or bromine, the symmetry can reduce the number of unique substitutive configurations. This reduction in configurations is elegantly handled using mathematical approaches like Polya’s Enumeration Theorem.

Substitution Patterns in Molecules

Substitution patterns refer to the different ways in which substituents can replace atoms in a molecule. In benzene, each hydrogen atom can be substituted with atoms like fluorine, chlorine, or bromine. Initially, this suggests a large number of combinations.
However, due to benzene’s symmetry, many of these combinations are not unique. Each substitution changes how the molecule functions, which affects properties like reactivity and stability.

• **Number of Substituents**: Determines how many atoms will be replaced in the molecule.
• **Type of Substituents**: Different atoms or groups of atoms can change the molecule significantly.

By applying Polya's Enumeration Theorem, one can efficiently calculate the number of distinct substitution patterns, accounting for symmetry, which greatly simplifies the complexity involved in these molecular modifications.