Quantitative measurement of ring strain

Question

While looking for data about ring strain in cycloalkanes, I came across two types of data:

1. Ring strain per methylene group (let's denote this quantity by X): The difference between heat of combustion of the cycloalkane per methylene group and the reference value of $\pu{658.6 kJ/mol}$.
2. Total ring strain (let's denote this quantity by Y): The difference between molar heat of combustion of the cycloalkane and the reference value of $\pu{658.6 kJ/mol}$ multiplied by number of carbon atoms in the cycloalkane.

From what I have read till now, it appears that the first quantity is the better representation of ring strain. (I say this on the basis that Y has the same value for cyclopentane as well as cycloheptane but still cycloheptane is considered to be less strained as it has lower value of X).

I want to know why X is a better measure for ring strain than Y. When we talk about a ring, we consider it as a whole; then why do we calculate ring strain per methylene group?

Answer 1

Let's look at following table from a Yale website:

$$ \begin{array}{c|ccc} & \ce{(CH2)3} & \ce{(CH2)4} & \ce{(CH2)5} & \ce{(CH2)6} & \ce{(CH2)7} \\ \hline \Delta H_\mathrm{Combusion} \ (\pu{kcal/mol}) & -499.8 & -656 & -793.5 & -944.6 & -1108.3 \\ \Delta H_\mathrm{Combusion} \text{ per } \ce{-CH2}- \ (\pu{kcal/mol}) & -166.6 & -164 & -158.7 & -157.4 & -158.3 \\ \text{Ring strain per } \ce{-CH2}- \ (\pu{kcal/mol}) & 9.2 & 6.6 & 1.3 & 0 & 0.9 \\ \text{Total ring strain} \ (\pu{kcal/mol}) & 27.6 & 26.3 & 6.5 & 0 & 6.3 \\ \hline \end{array} $$

Now, just concentrate on ring strain values for cyclopropane and cyclobutane only:

1. Ring strain per $\ce{-CH2}-$ (X-values) are $9.2$ and $\pu{6.6 kcal/mol}$, respectively.
2. Total ring strains (Y-values) are $27.6$ and $\pu{26.3 kcal/mol}$, respectively.

According to Y-values, cyclopropane and cyclobutane are equally unstable (relatively speaking), while according to Y-values, cyclopropane is significantly less stable than cyclobutane. Which valuse should be more reliable? Let's look at experimental findings:

Although cyclopropanes are far less reactive than alkenes, they react with chlorine and bromine under polar conditions to form corresponding addition products, 1,3-dihalopropanes, as the predominant product (Ref.1). To my understanding, cyclobutane and higher cycloalkanes do not give this ring opening reaction.

Cyclopropane also reacts with HCl and HBr to form corresponding addition products, 1-halopropanes (Ref.2; Reactions of bromocyclopropane with hydrobromic acid has given mixture of 1,1-, 1,2- and 1,3-dibromopropanes). Again, to my understanding, cyclobutane and higher cycloalkanes do not give this ring opening reaction.

So which one you think should be more reliable and why?

References:

1. Joseph B. Lambert, Erik C. Chelius, William J. Schulz, Jr., Nancy E. Carpenter, “Polar bromination and chlorination of cyclopropane,” J. Am. Chem. Soc. 1990, 112(8), 3156–3162. (https://doi.org/10.1021/ja00164a043).
2. Choi Chuck Lee, Bo-Sup Hahn, Kwok-Ming Wan, D. J. Woodcock, “Reactions of cyclopropane with hydrochloric acid and of bromocyclopropane with hydrobromic acid,” J. Org. Chem. 1969, 34(10), 3210–3211. (https://doi.org/10.1021/jo01262a097).