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Suppose we have to compare the $\mathrm{S_N2}$ reactivity order for cyclopropyl and cyclopentyl chloride. How can we do that?

I think that cyclopentyl should be more reactive towards $\mathrm{S_N2}$ because of larger bond angle and therefore lesser steric hindrance to the atttacking nucleophile but I am not sure about this.

Also, can the trend be generalised i.e. does increasing ring size lead to faster $\mathrm{S_N2}$?

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TL;DR - Bond angle strain gets a lot worse in going from cyclpropyl chloride to the transition state for the 3-membered ring example, than for cyclopentyl chloride going to the corresponding transition state.

The rate for each reaction will be determined by the free energy barrier in each case. Cyclopropyl chloride is strained because the carbons "want" to be tetrahedral, with sp3-hybridized orbitals (109° bond angles), but are forced to have much more acute (60°) bond angles within the ring. However, at the transition state, the carbon undergoing nucleophilic attack "wants" to be trigonal planar, with sp2-hybridized orbitals (120°), but is still geometrically-constrained to 60° bond angles within the ring. This makes the free energy barrier for SN2 very large.

bond angles in cyclopropyl chloride

Cyclopentyl chloride has bond angles close to the ideal 109°. There will be some bond angle strain at the transition state, but the effect won't be nearly as bad as for the three-membered ring.

bond angles in cyclopentyl chloride

Now, can the effect be generalised? Mostly. As expected, cyclobutyl chloride reacts faster than cyclopropyl, but slower than 5 or 6-membered rings. Five vs six is a tricky one. Cyclopentyl chloride is actually slightly more reactive. If you look at cyclohexane, it is very stable in the chair conformation. All nice bond angles, no eclipsing interactions. At the transition state for SN2 with cyclohexyl chloride, bond angle strain is introduced and there is steric interaction between the incoming nucleophile and the adjacent axial hydrogens. Cyclopentyl chloride doesn't start off quite so perfect - there are some eclipsing interactions. Also, the transition state may actually even relieve some of these eclipsing interactions.

Beyond six-membered rings, things get very complicated quickly. You could apply the same approach, but the number of possible conformations quickly rises from 8-membered and above. It becomes difficult to take all the possible factors and conformations into account, so without high-level calculations, it is difficult to make predictions.

If you have access, the reactivity of 3-6 member cycloalkyl chlorides was included in a publication by Rablen et. al..[1] It includes calculated energy barriers for SN2 reactions of these cycloalkyl chlorides with cyanide ion as nucleophile.

Reference:

  1. Rablen, P. R.; Mclarney, B. D.; Karlow, B. J.; Schneider, J. E. How Alkyl Halide Structure Affects E2 and SN2 Reaction Barriers: E2 Reactions Are as Sensitive as SN2 Reactions. J. Org. Chem. 2014, 79 (3), 867–879 DOI: 10.1021/jo4026644.
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There is a literature study[1] on the rates of reaction of cycloalkyl bromides with sodium benzenethiolate [PhSNa] in the aprotic, polar solvent dimethylformamide [DMF] at 0 °C. Bromide is a good leaving group and the thiolate is an excellent nucleophile for an SN2 reaction. You are correct that the cyclopentyl halide is more reactive than the cyclopropyl halide. Indeed, their rates bracket the examples studied by over a million-fold! Why is the cyclopropyl halide the slowest of the group? Well, you said it — bond angle strain. Clearly, the idealized C-C-C bond angle of 120° in the transition state cannot be attained. The rate of the cyclobutyl halide displacement is sped up by the greater C-C-C bond angle by a factor of ~10,000 relative to the cyclopropyl system.

The cyclopentyl halide reacts about 10% more rapidly than the acyclic model, 3-bromopentane. This difference cannot be predicted a priori. While 3-bromopentane is not constrained by bond angle issues, the "floppy" methyl groups can inhibit the attack of the nucleophile. This problem is absent in the cyclopentyl bromide, which is a "tied back" version of 3-bromopentane. The internal C-C(Br)-C bond angle of the cyclopentyl bromide ring is ≈104°, which is still not close to the bond angle of 120° in an idealized SN2 transition state.[2] The cyclopentyl bromide with its "envelope flap" shape does not present any severe steric interactions for the nucleophile.
pseudo-first-order relative rates for the substitution of bromocycloalkanes with PHSNa in DMF at 0 °C

At first sight cyclohexyl bromide might be expected to have the fastest rate because the C-C(Br)-C bond angle is ≈112°. However, steric factors now come into play. The chair conformation of bromocyclohexane having the equatorial C1-Br bond has a pair of axial hydrogens at C3 and C5 that impede the attack of benzenethiolate. The observation[3] that cis-4-tert-butyl-1-bromocyclohexane (axial C-Br) reacts with benzene thiolate in aqueous ethanol 58 times faster than with its trans-counterpart lends credence to the reaction proceding through the chair conformation having the axial C-Br bond, thereby avoiding steric issues incurred with the equatorial C-Br bond.

Bromocycloheptane has a C-C-C bond angles of ≈115° that facilitate a favorable TS but this benefit is countered by the onset of transannular steric factors. Nonetheless, this displacement is ≈100 times faster than the cyclohexyl example. Steric transannular effects come into serious play in the cyclooctyl, cyclodecyl and cyclododecyl ring systems. In summary, bromocyclopentane reacts the fastest of the cycloalkyl bromides having the best balance between angle strain and steric factors. With bromocyclohexane as some what of an outlier, steric factors dominate in rings greater than 5-membered, while angle strain prevails in rings smaller than 5-membered.

Notes and References

  1. Masson, E.; Leroux, F. The Effect of Ring Size on Reactivity: The Diagnostic Value of ‘Rate Profiles’. Helv. Chim. Acta 2005, 88 (6), 1375–1386 DOI: 10.1002/hlca.200590110.
  2. Note: Recall that the degree of bond-breaking and bond-making need not be equal in the transition state [TS]. The TS certainly would be expected to be early given the nucleophilicity of the thiolate and the electrophilcity of the C-Br bond. Therefore, an angle less than 120° may well suffice for this displacement.
  3. Eliel, E. L.; Haber, R. G. Conformational Analysis. VII. Reaction of Alkylcyclohexyl Bromides with Thiophenolate. The Conformational Equilibrium Constant of Bromine. J. Am. Chem. Soc. 1959, 81 (5), 1249–1254 DOI: 10.1021/ja01514a058.
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From a thermodynamic point of view, cyclopropyl ring would look to be more reactive because of higher ring strain. But the answer comes from looking at the transition state, which determines the Gibbs energy of activation. In transition state, for any general $\mathrm{S_N2}$ reaction, the non reacting ones substituents would be forced to be in a plane in this case the ring and another hydrogen, but the carbon atoms in cyclopentane ring are already at an angle close to $120^\circ$ and they would require much less "force" in order to get to the transition state, but for cyclopropane ring, should go through a change in bond angle, which would much more "force", and hence $\Delta G^\ddagger$ for cyclopentane ring would be lower than for cyclopropane ring and thereby, $k$ for cyclopentane ring would also be higher.

I am not an expert in chemistry, so please correct me if I am wrong.

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  • $\begingroup$ Yep, cyclopropyl ringwill not be able to bear the partial positive charge generated on the ring due to the difference in the relative rates of bond breaking(for the leaving group) and bond formation(for the nucleophile) in accordance with Bayer's strain theory(which is quite valid uptil five membered rings). So it will be less reactive than cyclopentyl ring $\endgroup$ – YUSUF HASAN Oct 14 '18 at 11:08

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