Q.Arrange the compounds of each set in order of reactivity towards $S_N2$ displacement:

The key to this problem lies in understanding what $pK_b$ measures and how structure affects basicity in amines.

Recall that $pK_b = -\log K_b$, so a lower $pK_b$ means a stronger base (higher $K_b$), while a higher $pK_b$ means a weaker base. The basicity of an amine depends on how readily the lone pair on nitrogen can accept a proton. Anything that increases electron density on nitrogen makes it more basic; anything that withdraws or delocalizes those electrons makes it less basic.

Let me analyze each amine:

1. Diethylamine, $(C_2H_5)_2NH$: A secondary aliphatic amine. The two ethyl groups are electron-donating through the inductive effect ($+I$), pushing electron density onto nitrogen. This makes the lone pair more available for protonation. Aliphatic amines are generally strong bases.

2. Aniline, $C_6H_5NH_2$: An aromatic amine. The lone pair on nitrogen is delocalized into the benzene ring through resonance. This delocalization spreads the electron density across the aromatic system, making it much less available for bonding with a proton. Aromatic amines are significantly weaker bases than aliphatic ones.

3. Trimethylamine, $(CH_3)_3N$: A tertiary aliphatic amine. Three methyl groups donate electrons through $+I$ effect. However, in aqueous solution, steric hindrance around nitrogen and solvation effects (the bulky methyl groups interfere with hydrogen bonding to water) make tertiary amines slightly less basic than secondary amines, though still quite basic overall.

4. N,N-dimethylaniline, $C_6H_5N(CH_3)_2$: An aromatic amine with two methyl groups on nitrogen. The lone pair is still delocalized into the benzene ring (resonance effect dominates), but the methyl groups partially counteract this by donating electrons. It's more basic than aniline but still much weaker than aliphatic amines.

Now I can rank them by basicity (and therefore by $pK_b$):

Basicity order: $(C_2H_5)_2NH > (CH_3)_3N > C_6H_5N(CH_3)_2 > C_6H_5NH_2$ …

The Lucas reagent is a mixture of concentrated hydrochloric acid and anhydrous zinc chloride ($\ce{ZnCl2}$). It works by converting alcohols into alkyl chlorides through an $\mathrm{S_N1}$ mechanism, and the key to understanding reactivity lies in carbocation stability.

When an alcohol reacts with Lucas reagent, the $\ce{ZnCl2}$ coordinates with the oxygen atom, making it a better leaving group. The alcohol then loses water to form a carbocation, which is immediately attacked by chloride ion. Since carbocation formation is the rate-determining step, the ease of forming a stable carbocation dictates how quickly the reaction proceeds.

Carbocation stability follows the order: tertiary > secondary > primary. This is because alkyl groups are electron-donating through hyperconjugation and inductive effects, stabilizing the positive charge.

Now let's identify X, Y, and Z:

1. Compound X: $\ce{CH3CH2CH2OH}$ (1-propanol)

   This is a primary alcohol. It would form a primary carbocation, which is highly unstable. Primary alcohols do not react with Lucas reagent at room temperature because the carbocation intermediate is too unstable to form readily.

2. Compound Y: $\ce{(CH3)3COH}$ (2-methyl-2-propanol or tert-butanol) …

Nucleophilic substitution reactions involve the replacement of a leaving group (often a halogen) by a nucleophile. The reactivity of a substrate in such reactions depends on several factors, primarily the strength of the bond to the leaving group and the stability of any intermediate formed (like a carbocation in $\mathrm{S_N1}$ reactions). Let's analyze why aryl halides are less reactive than alkyl halides by examining each statement.

1. Evaluating Statement I: If aryl cation is formed, it is not stabilized by resonance.

   • Nucleophilic substitution reactions can proceed via an $\mathrm{S_N1}$ mechanism, which involves the formation of a carbocation intermediate. For an aryl halide, this would mean the halogen atom (X) leaves, forming an aryl carbocation (e.g., a phenyl carbocation).
   • In an aryl carbocation, the positive charge resides on an $\mathrm{sp}^2$-hybridized carbon atom that is part of the aromatic ring.
   • This carbocation is highly unstable for two main reasons:
     • The positive charge is on an $\mathrm{sp}^2$ carbon, which is more electronegative than an $\mathrm{sp}^3$ carbon. More electronegative atoms are less able to accommodate a positive charge.
     • The empty p-orbital containing the positive charge is orthogonal (at $90^\circ$) to the $\pi$-electron system of the benzene ring. This means there is no effective overlap, and thus no resonance stabilization of the positive charge by the aromatic ring.
   • Because the formation of such an unstable aryl carbocation is energetically very unfavorable, the $\mathrm{S_N1}$ pathway is highly disfavored for aryl halides.
   • Therefore, statement I is a correct reason for the lower reactivity.

2. Evaluating Statement II: C–X has partial double bond character due to resonance.

   • Aryl halides exhibit resonance due to the presence of a lone pair of electrons on the halogen atom (X) and the $\pi$-electron system of the benzene ring.
   • The lone pair on the halogen can delocalize into the benzene ring, as shown by the resonance structures below:

    For example, with chlorine:

    (The full set of resonance structures would show the negative charge delocalized to the ortho and para positions of the ring, and a positive charge on the halogen, indicating a partial double bond between C and X.)
*   This resonance introduces a partial double bond character between the carbon atom of the benzene ring and the halogen atom.
*   A double bond is stronger and shorter than a single bond. This partial double bond character makes the C-X bond in aryl halides stronger and more difficult to break compared to the purely single C-X bond in alkyl halides.
*   Breaking the C-X bond is a crucial step in both $\mathrm{S_N1}$ (to form a carbocation) and $\mathrm{S_N2}$ (for nucleophilic attack and displacement) mechanisms. A stronger bond means higher activation energy for bond cleavage, thus reducing reactivity.
*   Therefore, statement II is a correct reason for the lower reactivity.

3. Evaluating Statement III: $\mathrm{sp}^3$-hybridized carbon is attached to the halogen.

* In aryl halides, the carbon atom directly bonded to the halogen is part of an aromatic ring (benzene ring). …

In S_N1 reactions, the rate-determining step is the formation of a carbocation intermediate. The more stable the carbocation, the faster the reaction. So, we need to compare the stability of the carbocations formed from X, Y, and Z. Look for factors like resonance (allylic or benzylic positions), hyperconjugation, and inductive effects.

1. Identify the structures from the reactions

   The problem shows three reactions (though not drawn here, we infer from typical patterns):

   • X reacts with AgNO₃ (a classic test for halide reactivity) to give a precipitate quickly.
   • Y reacts even faster.
   • Z reacts slowly or not at all. This suggests X, Y, Z are alkyl halides (or similar) with different carbocation stabilities.

2. Analyze carbocation stability for each

   • Y: Likely a tertiary halide or one that forms a resonance-stabilized carbocation (e.g., allylic or benzylic). Tertiary carbocations are more stable than secondary, which are more stable than primary.
   • X: Probably a secondary halide or one with moderate stabilization.
   • Z: Likely a primary or methyl halide, or one where the carbocation is destabilized (e.g., by electron-withdrawing groups). Primary carbocations are very unstable, so S_N1 is slow.

3. Order by decreasing carbocation stability …

The key concept here is carbocation stability. An $\mathrm{S_N1}$ reaction proceeds via a two-step mechanism: first, the leaving group departs, forming a carbocation intermediate; then, the nucleophile attacks this carbocation. The rate-determining step is the first step — formation of the carbocation. Therefore, anything that stabilizes the carbocation speeds up the reaction. Alkyl groups stabilize carbocations through hyperconjugation and inductive effects, so the more substituted the carbocation, the more stable it is. This gives the familiar stability order: tertiary > secondary > primary > methyl.

1. Identify the rate-determining step. In $\mathrm{S_N1}$, the slow step is the ionization of the alkyl halide to form a carbocation:

The rate depends only on the concentration of the alkyl halide (first-order kinetics), and crucially, on how easily the carbocation forms.

2. Relate carbocation stability to reaction rate.

   A more stable carbocation forms faster because the transition state leading to it is lower in energy (Hammond’s postulate: the transition state resembles the carbocation). So the order of reactivity for $\mathrm{S_N1}$ is exactly the order of carbocation stability.

3. Recall the stability order of carbocations.

   • Tertiary carbocation: three alkyl groups donate electron density via hyperconjugation and inductive effects → most stable.
   • Secondary carbocation: two alkyl groups → moderately stable.
   • Primary carbocation: only one alkyl group → very unstable.
   • Methyl carbocation: no alkyl groups → extremely unstable. Hence: tertiary > secondary > primary > methyl. …

The boiling point of an alkyl halide depends on two competing factors: molecular mass and molecular shape. Heavier molecules have stronger London dispersion forces, while branched molecules have smaller surface areas and weaker intermolecular contact.

When comparing alkyl halides, the halogen atom dominates the molecular mass because it is much heavier than the carbon skeleton. Bromine ($M_r = 80$) is significantly heavier than chlorine ($M_r = 35.5$), so bromides boil higher than chlorides of similar structure. Among isomers with the same halogen, branching reduces the boiling point because compact, spherical molecules have less surface contact than extended chains.

Let me identify each compound:

• I: $\mathrm{CH_3CH_2CH_2CH_2Cl}$ — 1-chlorobutane (straight chain, Cl)
• II: $\mathrm{CH_3CH_2CH_2CH_2Br}$ — 1-bromobutane (straight chain, Br)
• III: $\mathrm{CH_3CH_2CH(Br)CH_3}$ — 2-bromobutane (secondary, Br)
• IV: $\mathrm{(CH_3)_3CBr}$ — 2-bromo-2-methylpropane (tertiary, Br)

Now I'll rank them step by step:

1. Halogen effect dominates first: All three bromides (II, III, IV) will boil higher than the chloride (I), because bromine's greater mass and polarizability create stronger dispersion forces. So I is lowest.

2. Among the bromides, branching decides the order: All three have the same molecular formula for the bromobutanes (II and III are $\mathrm{C_4H_9Br}$; IV is also $\mathrm{C_4H_9Br}$).

   • II is a straight chain (1-bromobutane): maximum surface area, strongest intermolecular forces. …