Calculation of Ring Strain In Cycloalkanes

Ring Strain In Cycloalkanes (1) – Calculation of Ring Strain

This post is all about how ring strain is calculated. If you want more specific details about ring strain in cyclopropane and cyclobutane, I suggest moving on to the next post.

In the last post we learned about one consequence of the fact that carbon can form rings – that we can form stereoisomers (cis / trans).  This post attempts to explain another very interesting consequence of ring formation: ring strain.

It all starts with this: it turns out you can learn a lot about molecules by burning them. In a controlled and quantified way, of course. : – )

Table of Contents

1. Heat of Combustion of Alkanes
2. Heat of Combustion As A Function of Chain Length
3. The Heat Of Combustion For A Very Long Chain Is About 157 kcal/mol Per CH₂
4. Cycloalkanes Have The General Formula (CH₂)_n
5. Any Deviation From An Average Value of 157 kcal/mol Per CH2 Represents Ring Strain
6. What Factors Might Lead To Ring Strain?
7. Notes
8. (Advanced) References and Further Reading

1. Heat of Combustion Of Alkanes

The molar heat of combustion, as you might recall from gen chem, is the energy released upon burning one mole of a substance. It’s a value of enthalpy [Δ H] – usually measured in kJ/mol or kcal/mol. We’re going to use kcal/mol here (See post: Why Do Organic Chemists Use Kilocalories) but to convert to kJ/mol, multiply by 4.184 and you get the same thing.

Let’s start with something simple. Imagine we had a series of straight chain alkanes. Ethane, propane, butane, all the way up to dodecane (12).

As we increase the number of carbons, the increase energy released upon combustion will increase. I like to use the analogy of an infinite staircase: with every step we go higher, we’re increasing the distance between us and the floor, and therefore increasing the energy released if we were to fall from the staircase down to the ground : – ) . 

2. Heat of Combustion As A Function Of Chain Length

Let’s ask a simple question: how does the heat of combustion change as we add extra carbons?

Do you think we would we expect to see an “equal spacing” of energies as we successively add a carbon (by analogy to a staircase with equally spaced steps)? Or would the spacing gradually change?  And, importantly, could we use this information to make predictions?

Let’s burn various straight chain hydrocarbons and measure how much energy (in kcal/mol) is released by each molecule.

We can then calculate the average energy released per carbon – this gives us the “spacing” between each step. This will help us predict the energy released for the next step. 

[Here, use your imagination as we burn hydrocarbons of varying length and measure the heat of combustion for each case].

Let’s now look at the data. When we do this, we see the kcal/mol per carbon starts high: 213 for methane (not shown) ,  186 (for ethane) and then starts reaching a limit a little bit lower than 160 kcal/mol. This is like a staircase where the steps start off large, but gradually decrease in height until they all become a uniform height. 

3. The Heat Of Combustion For A Very Long Straight Chain Alkane is About 157 kcal/mol Per CH₂

What’s up with the curve, by the way? Very subtle question. Answer is long, so it’s in Note 1.

• Short version: in a straight chain alkane of formula  CH₃(CH₂)_nCH_{3 ,} the heat released from burning the ends  [CH₃] is higher than that obtained from burning the interior CH₂ groups.
• As the chain gets longer, and the #CH₂ >> #CH₃ the relative contribution of the 2 CH₃ groups to the overall ΔH will decrease. Therefore in the limit of infinite n the value of ΔH_combustion will approach that of CH₂ (about 157 kcal/mol).  
  

4. Cycloalkanes Have The General Formula (CH₂)_n

So what does this have to do with cycloalkanes? Cycloalkanes are composed only of CH₂ groups. There’s no “ends” to worry about. So if  we repeated the same combustion experiment – starting with cyclopropane, and increasing CH₂ by one each time, we might expect only to see heats of combustion that go up only by the ΔH of CH₂ (157 kcal/mol).  

In other words, we’d naively expect to see a staircase with uniform step height, all the way up.

Science often begins by having these naive notions about how nature should work, and then doing the experiment and finding that reality is contrary to (and thus much more interesting than) our expectations. This is a prime example!

5. Any Deviation From A Value Of 157 kcal/mol Per CH₂ Tells Us About Ring Strain

Here’s what happens when we burn ‘em. Look at that “average step height” (aka ΔH_combustion/CH₂).

It’s MUCH larger than we expect for C=3 (cyclopropane, 166.3 kcal/mol) and C=4 (cyclobutane, 163.9 kcal/mol), hits a minimum at C=6 (cyclohexane 157.4 kcal/mol) and then nudges up again until hitting another minimum at C=12 (cyclododecane).

(Although not included in the graph, the step height is constant from there on out, at about 157.4 kcal/mol.)

So if we take 157.4 kcal/mol as a baseline, cyclopropane releases 8.8 “extra” kcal/mol per carbon, for a total of 26.4 kcal/mol. Cyclobutane releases 6.4 “extra” kcal/mol per carbon, for a total of 25.6 kcal/mol.

This “extra” energy means these molecules are actually more unstable than we expected (“higher up the staircase” than we thought – so there’s farther to fall to the ground).  We call this “extra” instability, “strain”. 

Note – we’d get similar results if we compared heats of formation instead of heats of combustion, but combustion is so much more vivid : – )

6. What Factors Might Lead To Ring Strain?

Like any good experiment, this raises a whole lot of new questions.

• First, what could be a source of this “strain”? Why might cyclopropane and cyclobutane be much more unstable than we naively expected?
• Second, why is cyclohexane more unstrained than cyclopentane? For example, the interior angles of a pentagon (108°) are much closer to the ideal bond angles for a tetrahedron (109°) than a hexagon is (120°). So what gives?
• Third, what’s up with that increase in strain between C=6 (cyclohexane) and C=14 ? Why does strain go up from cyclohexane and then go back down again?

In the next post we’ll deal with cyclopropane and cyclobutane, and then future posts will go through the next two questions.

Notes

Note 1: In excess O₂, all C is converted to CO₂ and all H is converted to H₂O.

Burning  methane (CH₄) gives us 1 equiv CO₂ and 2 equivs of H₂O.   Ethane gives us 2 equivs CO₂ and 3 equivs of H₂O. Propane gives us 3 equivs CO₂ and 4 equivs H₂O. On a per-carbon basis, as we increase the # of carbons, the CO₂/H₂O ratio will approach 1. The ΔH_combustion on a per-carbon basis is highest for methane because proportionately more H₂O is formed, so therefore proportionately more energy is released. back to top

(Advanced) References and Further Reading

1. Ueber Polyacetylenverbindungen
   Adolf Baeyer
   Ber. 1885, 18 (2), 2269-2281
   DOI: 10.1002/cber.18850180296
   The original paper on ring strain by the legendary chemist Adolf von Baeyer. Even though this paper is titled on a completely different topic, ring strain is discussed at the very end of the paper.
2. Evaluation of strain in hydrocarbons. The strain in adamantane and its origin
   Paul von R. Schleyer, James Earl Williams, and Blanchard K. R.
   Journal of the American Chemical Society 1970, 92 (8), 2377-2386
   DOI: 1021/ja00711a030
   An early paper by Prof. P. v. R. Schleyer before he moved to Germany in the 1970’s. Adamantane was a pet topic of his, as one of his most highly-cited papers is a 1-page communication in JACS on the simple synthesis of adamantane. Table VII in this paper has a large collection of strain energies of various hydrocarbons, including cyclopropane and cyclobutane (28.13 and 26.90 kcal/mol, respectively).
3. Critical evaluation of molecular mechanics
   Edward M. Engler, Joseph D. Andose, and Paul v. R. Schleyer
   Journal of the American Chemical Society 1973, 95 (24), 8005-8025
   DOI: 1021/ja00805a012
   Table II in this paper contains a large table of enthalpies of formation and strain energies, both experimentally determined and theoretically calculated.
4. Enthalpy and kinetics of isomerization of quadricyclane to norbornadiene. Strain energy of quadricyclane
   David S. Kabakoff, Jean C. G. Buenzli, Jean F. M. Oth, Willis B. Hammond, and Jerome A. Berson
   Journal of the American Chemical Society 1975, 97 (6), 1510-1512
   DOI: 1021/ja00839a039
   This paper goes through a detailed thermochemical study of the isomerization of quadricyclane, and determines the strain energy at 96 ± 1 kcal/mol.
5. A survey of strained organic molecules
   Joel F. Liebman and Arthur Greenberg
   Chemical Reviews 1976, 76 (3), 311-365
   DOI: 1021/cr60301a002
6. The Concept of Strain in Organic Chemistry
   Kenneth B. Wiberg
   Angew. Chem. Int. Ed. 1986, 25 (4), 312-322
   DOI: 10.1002/anie.198603121
   Ring strain can also be called ‘angle strain’, resulting from the distortion of bond angles, increasing the energy content of the molecule. This paper also discusses the propellanes, an interesting class of small strained molecules. While [1.1.1]propellane can be isolated, [2.2.1] has not been obtained as a pure substance yet. This is due to the strength of the central bond towards homolytic cleavage, which provides a path for decomposition. This energy is strongly influenced by the difference in the strain energy between the reactant and the resulting diradical. In [1.1.1]propellane, the difference is 65 kcal/mol, while in [2.2.1]propellane, it is 5 kcal/mol.
7. Theoretical analysis of hydrocarbon properties. 1. Bonds, structures, charge concentrations, and charge relaxations
   Kenneth B. Wiberg, Richard F. W. Bader, and Clement D. H. Lau
   Journal of the American Chemical Society 1987, 109 (4), 985-1001
   DOI: 1021/ja00238a004
   Changes in hybridization are associated with changes in electronegativity. The greater the s character of a particular carbon orbital, the greater its electronegativity. As a result, carbon atoms that are part of strained rings are more electronegative than normal towards hydrogen.
8. Reactivity of Strained Compounds:  Is Ground State Destabilization the Major Cause for Rate Enhancement?
   Ariel Sella, Harold Basch, and Shmaryahu Hoz
   Journal of the American Chemical Society 1996, 118 (2), 416-420
   DOI: 1021/ja951408c
   Ring strain can cause qualitative changes in the nature of the bonds (hybridization), and these changes can increase reactivity.
9. The Thermochemistry of Cubane 50 Years after Its Synthesis: A High-Level Theoretical Study of Cubane and Its Derivatives
   Filipe Agapito, Rui C. Santos, Rui M. Borges dos Santos, and José A. Martinho Simões
   The Journal of Physical Chemistry A 2015, 119 (12), 2998-3007
   DOI: 10.1021/jp511756v
   A reevaluation of the thermochemical properties of cubane using computational methods. The authors here reevaluate the strain energy of cubane to be 667 kJ/mol (159 kcal/mol), which is pretty close to what has been determined before (154 kcal/mol).