Value of n According to Hückel’s Rule for the Given Aromatic Molecule

Correct Answer: n = 3

How to Find the Value of n According to Hückel’s Rule

To determine the value of n according to Hückel’s rule, we first need to count the total number of π electrons present in the cyclic conjugated system. After finding the number of π electrons, the value is substituted into the Hückel aromaticity equation:

Number of π electrons = 4n + 2

In the given molecule, the complete cyclic conjugated system contains seven double bonds. Since every ordinary carbon-carbon double bond contributes two π electrons to the conjugated system, the total number of π electrons is obtained by multiplying seven by two.

Step-by-Step Application of Hückel’s Rule

Step 1: Count the Number of Double Bonds

The first step is to examine the complete cyclic conjugated pathway in the given structure. The molecule contains a total of seven double bonds participating in the conjugated π-electron system.

For aromaticity calculations, each carbon-carbon double bond contributes two π electrons. Therefore, the total number of π electrons can be calculated as:

Total π electrons = Number of double bonds × 2

Substituting the number of double bonds:

Total π electrons = 7 × 2

Total π electrons = 14

Thus, the given molecule contains a total of 14 π electrons in its conjugated cyclic system.

Step 2: Apply the Hückel 4n + 2 Rule

According to Hückel’s rule for aromaticity, a cyclic, planar and completely conjugated molecule is aromatic when the number of π electrons can be represented by the expression:

4n + 2

where n is a non-negative integer such as 0, 1, 2, 3 and so on.

Since the given molecule contains 14 π electrons, we can write:

4n + 2 = 14

Step 3: Solve the Equation for n

Now, the value of n can be calculated by solving the Hückel equation:

4n + 2 = 14

Subtracting 2 from both sides:

4n = 12

Dividing both sides by 4:

n = 3

Therefore, the required value of n is 3.

Why the Given Molecule Satisfies the Hückel Electron Count

The question specifically asks us to assume that the molecule is aromatic. Therefore, the π-electron count must satisfy the 4n + 2 rule. The seven double bonds in the structure provide 14 π electrons, and 14 can be expressed in the Hückel form as:

14 = 4(3) + 2

This calculation confirms that the 14 π electrons correspond to n = 3. Therefore, under the assumption given in the question, the molecule satisfies the required Hückel electron count for aromaticity.

Understanding the Possible Values in Hückel’s Rule

The Hückel formula produces a characteristic series of allowed π-electron numbers. When n = 0, the system contains 2 π electrons. When n = 1, it contains 6 π electrons. When n = 2, it contains 10 π electrons, and when n = 3, it contains 14 π electrons.

For the molecule shown in the question, the relevant calculation is:

4(3) + 2 = 12 + 2 = 14 π electrons

Since the molecule has seven double bonds and therefore 14 π electrons, the correct integer value in the Hückel equation is 3.

Final Answer

The given cyclic conjugated molecule contains seven double bonds. Each double bond contributes two π electrons, giving a total of:

7 × 2 = 14 π electrons

Applying Hückel’s rule:

4n + 2 = 14

4n = 12

n = 3

Therefore, the value of “n” according to Hückel’s rule is 3.

Correct Answer: n = 3