Explain the formation of $H_2$ molecule on the basis of valence bond theory.

> [!TLDR] > Two hydrogen atoms form $H_2$ by overlapping their half-filled $1s$ orbitals along the internuclear axis, creating a covalent bond through electron pairing and energy stabilization. ## The Core Idea: Why Atoms Bond Valence bond theory rests on a simple premise: atoms bond when their atomic orbitals overlap, allowing electrons to pair up with opposite spins. This pairing lowers the system's energy compared to isolated atoms. The hydrogen molecule is the simplest case—two identical atoms, each bringing one electron in a $1s$ orbital, meeting to share. The driving force is **energy minimization**. When two hydrogen atoms approach each other, their electron clouds begin to interact. At the right distance, the attraction between each electron and both nuclei outweighs the repulsion between the two nuclei and between the two electrons. The result is a stable molecule with lower potential energy than the separated atoms. ## Step-by-Step Formation of $H_2$ 1. **Initial state: Two isolated hydrogen atoms** Each hydrogen atom has the electronic configuration $1s^1$. The single electron occupies the spherical $1s$ orbital, and its spin can be either $+\frac{1}{2}$ or $-\frac{1}{2}$. For bonding to occur, the two electrons must have opposite spins (Pauli exclusion principle). 2. **Approach and orbital overlap** As the two atoms move closer, their $1s$ orbitals begin to overlap. This overlap occurs along the internuclear axis—the straight line connecting the two nuclei. Because both orbitals are spherically symmetric, the overlap region forms between the nuclei, creating a zone of high electron density. 3. **Electron pairing and bond formation** The two electrons, now with opposite spins, occupy the overlapping region. This shared electron pair constitutes the covalent bond. In valence bond language, we say the bond arises from the **overlap of two half-filled $1s$ orbitals**. The electrons are no longer localized on individual atoms; they belon

Explain the formation of H₂ molecule on the basis of valence bond theory.

Exercises · 4.33

H_2

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Concept understanding — Orbital Hybridization Theory

Orbital Hybridization Theory – From Intuition to Precision

The Problem That Started It All

Imagine you are looking at a methane molecule, $\text{CH}_4$ . Carbon has four valence electrons: two in the $2s$ orbital and two in the $2p$ orbitals. If carbon used its pure atomic orbitals to bond, you would expect two bonds from the $2s$ (identical, but one direction) and two from the $2p$ (at $90^\circ$ to each other). That would give you three different bond types and bond angles of $90^\circ$ and something else.

But experiment says methane is perfectly tetrahedral: all four bonds are identical in length, strength, and energy, and the bond angle is $109.5^\circ$ , not $90^\circ$ . Something is fundamentally wrong with the "pure orbital" picture.

This is the puzzle that hybridization theory solves.

The Core Intuition

Think of atomic orbitals as shapes that an electron can occupy. The $s$ orbital is a sphere. The $p$ orbitals are dumbbells along the $x$ , $y$ , and $z$ axes. When an atom forms bonds, it wants to mix these shapes together to create new, hybrid shapes that point in directions that maximise bond strength and minimise repulsion.

It is like mixing primary colours to get new colours. You don't have to use red, blue, and yellow separately — you can blend them to get green, orange, or purple. Similarly, an atom can blend its $s$ and $p$ orbitals to get new hybrid orbitals that are better suited for bonding.

The key insight: hybridization is a mathematical mixing of atomic orbitals on the same atom to produce an equal number of new, equivalent hybrid orbitals. The number of hybrid orbitals formed always equals the number of atomic orbitals mixed.

The Precise Statement

s

The theory rests on three pillars:

Conservation of orbitals: Mixing $n$ atomic orbitals gives exactly $n$ hybrid orbitals. No orbitals are created or destroyed.
Energy averaging: The hybrid orbitals have energies that are intermediate between the original $s$ and $p$ energies.
Directionality: Hybrid orbitals point in specific directions to minimise electron pair repulsion, which directly determines molecular shape.

The Three Common Hybridizations

Hybridization	Orbitals Mixed	Number of Hybrids	Geometry	Bond Angle	Example
$sp$	one $s$ + one $p$	2	Linear	$180^\circ$	$\text{BeCl}_2$
$sp^2$	one $s$ + two $p$	3	Trigonal planar	$120^\circ$	$\text{BF}_3$
$sp^3$	one $s$ + three $p$	4	Tetrahedral	$109.5^\circ$	$\text{CH}_4$

Note

The superscript in $sp^2$ or $sp^3$ tells you how many $p$ orbitals were mixed. $sp^3$ means one $s$ and three $p$ orbitals were blended. It does not mean there are three $s$ orbitals — there is only one $s$ orbital per shell.

How It Works: The Methane Example

Carbon in its ground state has the configuration $1s^2 2s^2 2p_x^1 2p_y^1$ . Only two unpaired electrons — it should form only two bonds. But we know carbon forms four bonds.

Step 1: Promotion. One electron from the $2s$ orbital is promoted (excited) to the empty $2p_z$ orbital. This costs a small amount of energy, but it is more than compensated by the energy released when four strong bonds form instead of two.

Step 2: Hybridization. The one $2s$ orbital and three $2p$ orbitals mix to form four equivalent $sp^3$ hybrid orbitals. Each hybrid has 25% $s$ character and 75% $p$ character.

Step 3: Bonding. Each $sp^3$ hybrid overlaps with the $1s$ orbital of a hydrogen atom, forming four identical $\sigma$ bonds. The hybrids point to the corners of a tetrahedron, giving the $109.5^\circ$ angle. …

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Two hydrogen atoms form $H_2$ by overlapping their half-filled $1s$ orbitals along the internuclear axis, creating a covalent bond through electron pairing and energy stabilization.

The Core Idea: Why Atoms Bond

Valence bond theory rests on a simple premise: atoms bond when their atomic orbitals overlap, allowing electrons to pair up with opposite spins. This pairing lowers the system's energy compared to isolated atoms. The hydrogen molecule is the simplest case—two identical atoms, each bringing one electron in a $1s$ orbital, meeting to share.

The driving force is energy minimization. When two hydrogen atoms approach each other, their electron clouds begin to interact. At the right distance, the attraction between each electron and both nuclei outweighs the repulsion between the two nuclei and between the two electrons. The result is a stable molecule with lower potential energy than the separated atoms.

Step-by-Step Formation of $H_2$

Initial state: Two isolated hydrogen atoms

Each hydrogen atom has the electronic configuration $1s^1$ . The single electron occupies the spherical $1s$ orbital, and its spin can be either $+\frac{1}{2}$ or $-\frac{1}{2}$ . For bonding to occur, the two electrons must have opposite spins (Pauli exclusion principle).
Approach and orbital overlap

As the two atoms move closer, their $1s$ orbitals begin to overlap. This overlap occurs along the internuclear axis—the straight line connecting the two nuclei. Because both orbitals are spherically symmetric, the overlap region forms between the nuclei, creating a zone of high electron density.
Electron pairing and bond formation

The two electrons, now with opposite spins, occupy the overlapping region. This shared electron pair constitutes the covalent bond. In valence bond language, we say the bond arises from the overlap of two half-filled $1s$ orbitals. The electrons are no longer localized on individual atoms; they belong to the molecule.
Energy stabilization

The overlap increases electron density between the nuclei, which attracts both positively charged nuclei toward the shared electrons. This electrostatic attraction stabilizes the system. At the equilibrium bond length ( $r_e = 74 \text{ pm}$ for $H_2$ ), the attractive forces balance the repulsive forces (nucleus–nucleus and electron–electron), and the molecule reaches its lowest energy state.
Nature of the bond: $\sigma$ bond

The bond formed by head-on overlap along the internuclear axis is called a sigma ( $\sigma$ ) bond. It has cylindrical symmetry around the bond axis—if you rotate the molecule around the line joining the two nuclei, the electron density distribution looks the same. This is the strongest type of covalent bond because maximum overlap occurs along the axis.

\text{Bond energy of } H_2 = 436 \text{ kJ/mol}

Tip

The $H_2$ molecule is the only homonuclear diatomic molecule where valence bond theory gives a nearly perfect description without invoking resonance or hybridization, because both atoms contribute identical orbitals.

Key Features of the $H_2$ Bond

Property	Value / Description
Bond type	Single $\sigma$ bond
Bond length	$74 \text{ pm}$

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Q.Explain the formation of H2H_2H2​ molecule on the basis of valence bond theory.

The Core Idea: Why Atoms Bond

Step-by-Step Formation of H2H_2H2​

Key Features of the H2H_2H2​ Bond

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Q.Explain the formation of $H_2$ molecule on the basis of valence bond theory.

Step-by-Step Formation of $H_2$

Key Features of the $H_2$ Bond

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