Which of the following carbocations is most stable? (a) (CH3)3C—C+H2 (b) (CH3)3C+ (c) CH3CH2C+H2 (d) CH3C+HCH2

Exercises · 8.38

Q. Which of the following carbocations is most stable?

(a) (CH3)3C—C+H2

(b) (CH3)3C+

(d) CH3C+HCH2CH3.

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Concept understanding — Inductive Effect

Inductive Effect on Acidity – From Intuition to Precision

Imagine you are holding a rope tied to a heavy box. If you pull the rope, the box moves toward you. Now imagine the rope is made of rubber bands — the pull still reaches the box, but it gets weaker the farther away you are. That is exactly how the inductive effect works inside a molecule.

The Core Intuition

An acid donates a proton ( $H^+$ ). After it does, the remaining part (the conjugate base) carries a negative charge. The stability of that negative charge determines how willing the molecule is to give up the proton. More stable conjugate base → stronger acid.

Now, some atoms or groups are electron-withdrawing — they pull electron density toward themselves through the sigma bonds. If such a group is attached near the acidic proton, it pulls some electron density away from the negative charge on the conjugate base. That spreads out (delocalises) the negative charge, making the conjugate base more stable. The acid becomes stronger.

Conversely, electron-donating groups push electron density toward the negative charge, concentrating it and making the conjugate base less stable. The acid becomes weaker.

Note

The inductive effect operates through sigma bonds only. It does not involve pi bonds or resonance. It is a permanent, through-bond polarisation.

The Precise Statement

Inductive effect on acidity: The acidity of a compound increases with the presence of electron-withdrawing groups (EWGs) near the acidic site, and decreases with electron-donating groups (EDGs). The effect is strongest when the group is closest to the acidic proton, and diminishes rapidly with distance.

Mathematically, for a series of substituted carboxylic acids:

\text{R-COOH} \quad \text{where R = substituent}

The acid dissociation constant $K_a$ changes as:

If R is electron-withdrawing (e.g., $-\text{Cl}$ , $-\text{NO}_2$ , $-\text{CF}_3$ ): $K_a$ increases → stronger acid.
If R is electron-donating (e.g., $-\text{CH}_3$ , $-\text{C}_2\text{H}_5$ ): $K_a$ decreases → weaker acid.

Why Distance Matters

The inductive effect falls off with distance because sigma bonds are localised. Each bond attenuates the effect by roughly a factor of 2–3. For example, compare:

Compound	$pK_a$	Explanation
$\text{CH}_3\text{COOH}$	4.76	Reference (no EWG)
$\text{ClCH}_2\text{COOH}$	2.86	Cl withdraws through one bond
$\text{Cl}_2\text{CHCOOH}$	1.29	Two Cl atoms, stronger withdrawal
$\text{Cl}_3\text{CCOOH}$	0.65	Three Cl atoms, strongest withdrawal
$\text{CH}_3\text{CH}_2\text{COOH}$	4.87	Ethyl group is electron-donating (slightly weaker acid)

Notice: $\text{ClCH}_2\text{COOH}$ is about 100 times stronger than acetic acid ( $\Delta pK_a \approx 1.9$ ). But if the Cl is moved further away:

Compound	$pK_a$
$\text{ClCH}_2\text{CH}_2\text{COOH}$	4.08
$\text{ClCH}_2\text{CH}_2\text{CH}_2\text{COOH}$	4.52

The effect fades as the chlorine moves farther from the carboxyl group. …

Why this formula?

Inductive Effect: Why the Key Ideas Hold

The inductive effect is a fundamental concept in organic chemistry that explains how electron density shifts along a sigma ( $\sigma$ ) bond due to differences in electronegativity. Let's break down why the key principles work — not just what they are.

1. The Core Idea: Polarization of $\sigma$ Bonds

What happens?

When two atoms with different electronegativities form a $\sigma$ bond, the bonding electrons are not shared equally. The more electronegative atom pulls electron density toward itself.

Why does this happen?

Electronegativity is a measure of an atom's ability to attract shared electrons.
The $\sigma$ bond is a region of high electron density between the nuclei.
The more electronegative atom's nucleus exerts a stronger electrostatic pull on these electrons.
Result: The bond becomes polarized — one end becomes slightly negative ( $\delta^-$ ), the other slightly positive ( $\delta^+$ ).

Key formula (conceptual):

$\delta^- \leftarrow \text{Atom A} \rightarrow \delta^+$

where A is more electronegative than B.

2. Why the Effect Transmits Along a Chain

The puzzle:

If the inductive effect is about a single bond, how does it affect atoms several bonds away?

The reasoning:

The $\delta^+$ on the less electronegative atom creates a partial positive charge.
This partial charge polarizes the next $\sigma$ bond in the chain.
The effect is relayed through successive bonds, like a chain of dominoes.

Why does it weaken with distance?

Each bond acts as a dielectric medium — it partially screens the charge.
The electrostatic influence falls off with distance according to Coulomb's law:

F \propto \frac{q_1 q_2}{r^2}

In a molecular chain, the effective distance $r$ increases, so the induced dipole in each subsequent bond is smaller.

Key result: Inductive effect is significant only up to 3–4 bonds away.

3. The Quantitative Measure: Inductive Effect Constant ( $\sigma_I$ )

What is $\sigma_I$ ?

It's a Hammett-type constant that quantifies the electron-withdrawing or electron-donating power of a substituent through sigma bonds only.

Why does it have this form?

The inductive effect is additive — each substituent contributes independently.
For a substituent X attached to a carbon chain:

\sigma_I = \log \left( \frac{K_a(\text{X-CH}_2\text{COOH})}{K_a(\text{CH}_3\text{COOH})} \right)

where $K_a$ is the acid dissociation constant.

Why use acid dissociation?

The carboxyl group ( $-\text{COOH}$ ) is a sensitive probe.
An electron-withdrawing group (EWG) stabilizes the conjugate base ( $\text{R-COO}^-$ ) by dispersing its negative charge.
This increases $K_a$ (stronger acid).
An electron-donating group (EDG) destabilizes the conjugate base, decreasing $K_a$ .

Key formula:

$\sigma_I > 0$ for EWGs (e.g., $-\text{Cl}$ , $-\text{NO}_2$ )

$\sigma_I < 0$ for EDGs (e.g., $-\text{CH}_3$ , $-\text{C(CH}_3)_3$ )

4. Why Inductive Effect is Not Resonance

Common confusion:

Students often mix inductive and resonance effects.

The critical difference:

Property	Inductive Effect	Resonance Effect
Electron movement	Through $\sigma$ bonds only	Through $\pi$ bonds or lone pairs
Distance dependence	Dies off after 3–4 bonds	Can transmit over long distances in conjugated systems
Permanent or temporary	Permanent polarization	Can be temporary (delocalization)

Why this matters for exam problems:

In alkyl halides, the inductive effect of $-\text{Cl}$ explains the $\delta^+$ on carbon.
In benzene derivatives, the combined inductive and resonance effects determine reactivity.

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Carbocation stability is primarily determined by the number of electron-donating alkyl groups and the extent of hyperconjugation. The tertiary carbocation (CH3)3C+ has the highest number of alpha-hydrogens (9) and the strongest inductive effect, making it the most stable.

Carbocations are species with a positively charged carbon atom, meaning they are electron-deficient. To achieve stability, they seek to delocalise or neutralise this positive charge. This stabilisation is primarily achieved by electron-donating groups (EDGs) attached to the positively charged carbon. Alkyl groups are common EDGs, and they stabilise carbocations through two main effects:

Inductive Effect (+I Effect): Alkyl groups are slightly electron-releasing. They push electron density towards the electron-deficient carbocationic carbon, thereby dispersing the positive charge and stabilising it. The more alkyl groups attached, the stronger the cumulative +I effect.
Hyperconjugation: This is a special type of resonance involving the delocalisation of electrons from C-H sigma bonds of alkyl groups adjacent to the carbocationic carbon (alpha-hydrogens) into the empty p-orbital of the carbocation. The more alpha-hydrogens available, the greater the number of hyperconjugative structures, and thus, the greater the stability. Hyperconjugation is generally considered a more significant stabilising factor than the inductive effect for carbocations.

Tip

The general order of carbocation stability is Tertiary > Secondary > Primary. This trend is a direct consequence of both the increasing number of electron-donating alkyl groups (stronger +I effect) and the increasing number of alpha-hydrogens (more hyperconjugative structures) as we move from primary to tertiary carbocations.

Let's analyze each given carbocation:

Analyze (a) (CH3)3C—C+H2
- This is a primary carbocation because the positively charged carbon ( $C^+$ ) is directly bonded to only one other carbon atom (the tertiary carbon of the (CH3)3C group).
- Inductive Effect (+I): The bulky (CH3)3C group is a strong electron-donating group, exerting a significant +I effect towards the $C^+H_2$ group.
- Hyperconjugation: The carbon atom directly attached to the $C^+H_2$ group is the tertiary carbon of the (CH3)3C moiety. This alpha-carbon is bonded to three methyl groups and the $C^+H_2$ group, meaning it has 0 alpha-hydrogens. Therefore, no hyperconjugative stabilisation is possible for this carbocation.
Analyze (b) (CH3)3C+
- This is a tertiary carbocation because the positively charged carbon ( $C^+$ ) is directly bonded to three other carbon atoms (three methyl groups).
- Inductive Effect (+I): There are three methyl groups directly attached to the $C^+$ . Each methyl group is electron-donating, resulting in a very strong cumulative +I effect that effectively disperses the positive charge.
- Hyperconjugation: Each of the three methyl groups attached to the $C^+$ is an alpha-carbon. Each methyl group has 3 hydrogens. Total alpha-hydrogens = $3 \times 3 = \textbf{9}$ . This allows for 9 hyperconjugative structures, providing substantial stabilisation.
Analyze (c) CH3CH2C+H2
- This is a primary carbocation because the positively charged carbon ( $C^+$ ) is directly bonded to only one other carbon atom (the CH2 of the ethyl group).
- Inductive Effect (+I): The ethyl group (CH3CH2—) is an electron-donating group, exerting a moderate +I effect.
- Hyperconjugation: The carbon atom directly attached to the $C^+H_2$ group (the CH2 of the ethyl group) is an alpha-carbon. This alpha-carbon has 2 hydrogens. Therefore, 2 hyperconjugative structures are possible.
Analyze (d) CH3C+HCH2CH3
- This is a secondary carbocation because the positively charged carbon ( $C^+$ ) is directly bonded to two other carbon atoms (one methyl carbon and one ethyl carbon). …

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Q.Which of the following carbocations is most stable?

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