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From Klein's Organic chemistry [1, p. 780]:

The 1,2-adduct is believed to form more rapidly as a result of a proximity effect. Specifically, the carbocation and the bromide ion are initially very close to each other immediately after their formation in the first step of the mechanism. The bromide ion is simply closer in proximity to C2 than C4, so attack at C2 occurs more rapidly.

1,2-addition

Similarly, from Vollhardt's Organic Chemistry: Structure and Function [2, pp. 345–346]:

Ring size controls the speed of cyclic ether formation

A comparison of the relative rates of cyclic ether formation reveals a surprising fact: Three-membered rings form quickly, about as fast as five-membered rings. Six-membered ring systems, four-membered rings, and the larger oxacycloalkanes are generated more slowly.

Relative Rates of Cyclic Ether Formation

$$k_3 ≥ k_5 > k_6 > k_4 ≥ k_7 > k_8$$

$k_n =$ reaction rate, $n =$ ring size

[…]

A second enthalpic phenomenon, which has been called the proximity effect, operates, especially in 2-haloalkoxides.

The proximity effect was used to explain these two reactions. Is it applicable to every reaction?

What is the proximity effect?

References

  1. Klein, D. R. Organic Chemistry, 1st ed.; John Wiley: Hoboken, NJ, 2012. ISBN 978-0-471-75614-9.
  2. Vollhardt, K. P. C.; Schore, N. E. Organic Chemistry: Structure and Function, 7th ed.; W. H. Freeman: New York, 2014. ISBN 978-1-4641-2027-5.
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I will not attempt to define "proximity effects" in my answer. If you are looking for a definition, this is probably not the best answer. I think the term itself is quite vague and rather qualitative if you will. I've personally seen the concept of "proximity effects" being used more in chemical biology / enzymes chemistry context, than in organic chemistry.

In the first example, I am not even sure if there is indeed any proximity effect in operation. There could be; but it really depends on the solvent effects. How does one really separate the different factors in play? Doing gas phase simulations could give you an answer. The first example you have cited, can indeed be more easily be explained in terms of relative stabilities of the "carbocation" resonance forms: it is more favourable to have the positive charge at position 2 (allylic, secondary), than to have the positive charge at position 4 (allylic, primary). In terms of resonance or MO theory, this picture translates into position 2 being more electron deficient than position 4. A nucleophile would attack at the most electron deficient position (ceteris paribus), and this would be the fastest reaction path. Attack at position 2 leads to the kinetic product, as shown in the book you've cited. Whether there is any additional "proximity effects" - such as the nucleophile being "closer" to position 2 than position 4, is up for debate and cannot be concluded from this result. On the grounds of carbocation stabilisation alone, the observed product can be explained. Note however, that under the conditions, this reaction could be reversible and eventually generate the more stable thermodynamic product.

The second example, is perhaps a more appropriate application of 'proximity effects'. However, I'm not sure if it is appropriate to call this an 'enthalpic phenomenon'. I would have guessed a more probabilistic/entropic origin of this phenomenon. In this example, I would interpret proximity effect as simply the fact that if you are doing a 1-3 ring closure (to form a 3 membered ring), your two reaction centres are closer in space, and hence the reaction is probabilistically more likely than a 1-6 ring closure. I don't know how using the term 'proximity effect' helps give a better explanation, though. If you're interested, you should read up on rates of ring closures in a book of your choice, that discusses both enthalpic and entropic contributions to analyse the relative rates of ring closure.

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I think this "proximity effect" that you are referring to cannot be generalised. In fact, the "proximity effect" referred to in each example that you have provided is different from the other. My answer will just be supplementing the answer provided by Zachr as I believe he has explained the two examples with sufficient depth of analysis and clarity.

In the first case, I believe that the idea is that the product of the first step of the reaction, the protonation of the $\ce {C=C}$ bond is actually not just the carbocation intermediate in isolation, but a contact ion-pair, comprising the carbocation and the bromide counterion. This "ion pair" mechanism is discussed in greater detail with the appropriate reference to literature in this post by Orthocresol. As the positively-charged carbon atom and the bromide anion are in close proximity to each other, they can combine quickly to form the 1,2-adduct. This is certainly a kinetic phenomenon as this "ion pair" electrostatic combination just occurs more quickly.

In the second case, the "proximity effect" the author is referring to is that the nucleophilic oxygen atom is able to most easily access the electron-deficient that is closest to it. However, as Zachr has aptly pointed out, this can only be explained by analysing both the entropy of activation and the enthalpy of activation. To say that it is just an enthalpic effect, as written in the textbook source you have cited, would be myopic and more importantly, incorrect. In fact, it is the entropic factor that greatly favours the formation of the 3-membered ring. A detailed discussion of the these two factors can be found here.

While a generalisation that is useful cannot be provided, the "proximity effect" is certainly more of a kinetic phenomenon than a thermodynamic phenomenon, at least based on these two case studies.

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