how to find order of sn1 in decreasing order in attached image?

As usual, I tried to check the stability of carbocation.As a result, bromomethyl cyclopropane will show Dancing resonance hence most stable.then come 3-bromocyclopropene forming aromatic but answer was

3-bromocyclopropene>bromomethyl cyclopropane>bromocyclopropane>1-bromocyclopropene.

To determine the order of reactivity for SN1 reactions, we need to assess the stability of the carbocations formed during the reaction. You've already identified two key candidates: bromomethyl cyclopropane and 3-bromocyclopropene, and you're on the right track by considering resonance and aromaticity. Let's break down your reasoning and clarify the order of stability step-by-step.

Understanding Carbocation Stability

In SN1 reactions, the rate-determining step involves the formation of a carbocation. The more stable the carbocation, the faster the SN1 reaction will proceed. Stability is influenced by several factors, including:

• **Hybridization**: Tertiary carbocations (3°) are generally more stable than secondary (2°), which are more stable than primary (1°).
• **Resonance**: Carbocations that can delocalize their positive charge through resonance will be more stable.
• **Aromaticity**: Aromatic systems provide exceptional stability due to their cyclic, planar structure and delocalized π electrons.

Evaluating Your Compounds

Now, let's evaluate the compounds you've mentioned:

Bromomethyl Cyclopropane

This compound can undergo what you've termed "dancing resonance," which suggests that the positive charge can be delocalized. This delocalization enhances its stability significantly, making it a strong contender for the most stable carbocation.

3-Bromocyclopropene

In this case, 3-bromocyclopropene can form an aromatic carbocation, which is highly stable. Aromatic systems are particularly favorable because they benefit from the stability of resonance and the energy associated with aromaticity. Therefore, this compound also represents a very stable carbocation.

Bromocyclopropane

This compound would typically yield a secondary carbocation. While it can exhibit some stabilization through hyperconjugation, it doesn't have the resonance stabilization that bromomethyl cyclopropane or 3-bromocyclopropene possesses. Thus, it is less stable compared to the first two.

1-Bromocyclopropene

Here, the carbocation formed is primary, making it the least stable of the group. It lacks sufficient resonance or hyperconjugation to stabilize the positive charge effectively.

Determining the Reactivity Order

Considering the stability of each carbocation, the order of reactivity for the SN1 mechanism can be summarized as follows:

• 3-bromocyclopropene (most stable, aromatic)
• bromomethyl cyclopropane (stable due to resonance)
• bromocyclopropane (less stable, secondary carbocation)
• 1-bromocyclopropene (least stable, primary carbocation)

This leads to the conclusion that your final order of reactivity is indeed correct: 3-bromocyclopropene > bromomethyl cyclopropane > bromocyclopropane > 1-bromocyclopropene. Each step reflects a nuanced understanding of how structural features influence carbocation stability in the context of SN1 reactions.