What is the total number of nodes in the ψ \psi 3 and ψ \psi 4 MOs of 1,3-butadiene?

All right. Hello everyone. So this question is asking us to identify the number of nodes in each of the molecular orbitals of the indicated carbons. So here our given starting material is two E four E hep the 246 trien one ilium. So let's go ahead and talk a little bit about the pie system that we have in this carbo cion, I should say. So in this particular case, we have a seven P system in the structure of our given. And the reason for this is because of our six pi orbitals from the three pi bonds and the additional carbo cion which contributes an extra P orbital. Now, for the purposes of this discussion, even though this is not proper Iupac naming, I'm going to number these carbons from left to right, meaning that the carbo cion is listed here as position number seven. This means that the indicated arrows are on carbons four and seven, which is the aforementioned carbo CIA. So here if we go ahead and number from left to right, then the molecular orbitals that are indicated by the arrows are those of sci four N cy seven in other words, the molecular orbitals of carbons four and seven. So here recall that in most cases, we're only interested in the vertical nodes of a given system. And the reason why we tend to be interested in the vertical nodes is because these are the ones that we use to actually draw out the orbital diagram. And in those cases, the number of vertical nodes is equal to n subtracted by one where N is the number of si but this is a little bit of a trick question, right? Because this question is asking about the total number of notes. This means that we cannot ignore the horizontal node. Recall that the horizontal node is the one that separates all of the lobes as follows. Essentially it's when all of the bottom halves are shaded and all of the top halves are unshaded, that's the horizontal node and each molecular orbital has one of these. So for this question, we have to actually count the horizontal note. And so going back to our little formula which is and subtracted by one for the total number of vertical nodes. If we add our horizontal node, then that means that the total number of nodes is simply equal to the number of P I. So here for 54, we have four notes in total. And for size seven, we have seven notes total. And there you have it with that being said, thank you so very much for watching. And I hope you found this helpful.

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16. Conjugated Systems

Orbital Diagram:6-atoms- 1,3,5-hexatriene

Organic Chemistry

16. Conjugated Systems

Orbital Diagram:6-atoms- 1,3,5-hexatriene

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3:55 minutes

Problem 14

Textbook Question

What is the total number of nodes in the $ψ$ ₃ and $ψ$ ₄ MOs of 1,3-butadiene?

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Understand the concept of molecular orbitals (MOs) in conjugated systems: In 1,3-butadiene, the π molecular orbitals are formed by the overlap of four p orbitals from the four sp²-hybridized carbon atoms. These MOs are labeled as ψ₁, ψ₂, ψ₃, and ψ₄, where ψ₁ is the lowest energy orbital and ψ₄ is the highest energy orbital.

Recall the relationship between nodes and molecular orbitals: The number of nodes in a molecular orbital increases with its energy level. Specifically, the number of nodes in a π molecular orbital is equal to (n - 1), where n is the index of the molecular orbital (e.g., ψ₁ has 0 nodes, ψ₂ has 1 node, etc.).

Determine the number of nodes in the c₃ (ψ₃) molecular orbital: Using the formula (n - 1), where n = 3 for ψ₃, calculate the number of nodes in this orbital.

Determine the number of nodes in the c₄ (ψ₄) molecular orbital: Similarly, using the formula (n - 1), where n = 4 for ψ₄, calculate the number of nodes in this orbital.

Add the total number of nodes: Sum the number of nodes in ψ₃ and ψ₄ to find the total number of nodes in the c₃ and c₄ molecular orbitals of 1,3-butadiene.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Molecular Orbitals (MOs)

Molecular orbitals are formed by the combination of atomic orbitals when atoms bond together. In the case of 1,3-butadiene, the MOs are crucial for understanding the distribution of electrons in the molecule. Each MO can hold a maximum of two electrons and can be classified as bonding, antibonding, or non-bonding based on their energy levels and stability.

Nodes in Molecular Orbitals

Nodes are regions in a molecular orbital where the probability of finding an electron is zero. The number of nodes in an MO is related to its energy level; higher energy MOs have more nodes. For example, the number of nodes can be calculated using the formula: number of nodes = n - 1, where n is the principal quantum number of the MO.

1,3-Butadiene Structure

1,3-Butadiene is a conjugated diene with the formula C4H6, consisting of four carbon atoms connected by alternating double bonds. Its structure allows for the formation of π molecular orbitals that are delocalized over the entire molecule. Understanding its structure is essential for determining the characteristics of its molecular orbitals, including the total number of nodes in the c3 and c4 MOs.

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