Chemistry - Why is 1,2,3,4,5,8-hexahydronaphthalene more stable than 1,4,4a,5,8,8a-hexahydronaphthalene?
Solution 1:
Disclaimer:
This post might be hard to read for some people. Some may even consider this a rant. You have been warned.
This answer will question the validity of the used models, as well as the basis for such exercises. (It will also point to a fundamental flaw of this specific exercise.) If you are a student, you might learn concepts that will later turn out to be inaccurate or outdated. This answer should give you some general context, but it will probably not be able to help you in an exam.
Introduction
I have previously stated in the comments how much I hate these exercises, I even called them really, really big garbage. There are several reasons for that. I am going to try to convince you step by step why I dislike these exercises, this one in particular.
Stability is not a very meaningful concept
There is a reason why there is no easy definition for stability. It is too fuzzy a concept with too many different connotations that basically everybody could understand something different. This is an ultimate flaw, and for these exercises, some kind of understanding of 'stability' is implied, but probably never properly defined.
Let's start with something tangible, one possible (and accepted) definition of stable; one that is so trivial you might have not even thought about it. Stable means something exists. In more official terms, i.e. in the gold book, it is called structural stability (10.1351/goldbook.ST07109):
Within the Born–Oppenheimer (adiabatic) approximation, this is associated with an energy minimum on a potential energy surface. Structural stability implies that any change in the coordinates of the nuclei can only increase the total energy.
I'm not absolutely happy with the above definition, as it can be read that only global minima are stable. If you start from a local minima, some changes in the coordinates will lead to a decrease in total energy; that is how different conformations work and eventually also reactions. So the changes implied by this definition should be reasonably small.
Obviously, this definition does not at all help us with the exercise. Both compounds are stable according to this definition, but we cannot say which one is 'more stable'.
From this we could go ahead and rank molecules according to their total energy. As long as we would be looking at isomers, this could actually work, but another question is how meaningful this approach is. I won't and cannot go into any detail or discussion how to obtain the total energy; for the purpose of this post just assume every molecule does have an exact total energy.
In the end the total energy is a number associated with a molecule. It will not tell you anything about the chemistry that molecule does. Almost all tangible and practical information is not contained in this number. Sure it offers an exact and consistent way of sorting things, but –in my opinion– an utterly useless one.
It is also problematic in a semantic way: you are using one term to describe a phenomenon which already has a name.
This definition also will not help in answering the question, because estimating this value is hard. Quantum Chemistry has plenty of options to estimate or calculate the expectation value of these compounds. Some are simple, based on empirical data, e.g. semi-empirical methods, some are a bit more complex, e.g. Hartree-Fock, Density Functional Approximations, some are very complex, e.g. Configuration Interaction, Coupled Cluster. Almost all of them you cannot do on a piece of paper in a reasonable timeframe.
That leads us to the point where we need another definition. The gold book actually has another definition for us (10.1351/goldbook.S05900):
As applied to chemical species, the term expresses a thermodynamic property, which is quantitatively measured by relative molar standard Gibbs energies. A chemical species $\ce{A}$ is more stable than its isomer $\ce{B}$ if $\Delta_\mathrm{r}G^\circ > 0$ for the (real or hypothetical) reaction $$\ce{A -> B}$$ under standard conditions. If for the two reactions: \begin{align} \ce{P &-> X + Y} & (\Delta_\mathrm{r}G^\circ_1)\\ \ce{Q &-> X + Z} & (\Delta_\mathrm{r}G^\circ_2) \end{align} $\Delta_\mathrm{r}G^\circ_1 > \Delta_\mathrm{r}G^\circ_2$, $\ce{P}$ is more stable relative to the product $\ce{Y}$ than is $\ce{Q}$ relative to $\ce{Z}$. Both in qualitative and quantitative usage the term stable is therefore always used in reference to some explicitly stated or implicitly assumed standard. The term should not be used as a synonym for unreactive or 'less reactive' since this confuses thermodynamics and kinetics. A relatively more stable chemical species may be more reactive than some reference species towards a given reaction partner.
This brings us back to the original dilemma, the implied understanding of stability. While we certainly can estimate the hypothetical relative molar standard Gibbs energy of the reaction, it actually won't tell us much about the chemistry. At least this definition will give us some handling on the exercise, i.e. the problem becomes finding 'numbers'.
Sometimes, but this is very wrong, stability is used as masquerading a different concept altogether, a kinetic concept. Stability may then wrongly be defined as the lower likelihood to react, in other words a lower reactivity means more stability. Now this brings back the already discussed semantic problem, but this time it is actually worse. There is also no reason to not use the proper terminology. The gold book defines reactive (reactivity) as follows (10.1351/goldbook.R05180):
As applied to a chemical species, the term expresses a kinetic property. A species is said to be more reactive or to have a higher reactivity in some given context than some other (reference) species if it has a larger rate constant for a specified elementary reaction. The term has meaning only by reference to some explicitly stated or implicitly assumed set of conditions. It is not to be used for reactions or reaction patterns of compounds in general. The term is also more loosely used as a phenomenological description not restricted to elementary reactions. When applied in this sense the property under consideration may reflect not only rate, but also equilibrium, constants.
So this is pretty much linked to the activation barrier. This is surely nothing that is asked about by the exercise, especially since no reaction conditions are mentioned. However, it felt necessary to be included here, since quite a few people will actually mix up the terms or even use them synonymously.
You might come across terms like air-stable which means you can handle a substance in a normal atmosphere without having to worry that it would decompose or react. This is exactly the problem that I mentioned before.
Another point worth mentioning is that some people use 'stable' to indicate whether a species is long-lived; this is also a wrong usage according to the above definitions. It becomes especially problematic when it is used synonymously with unreactive or even inert; but these are discussions for a later time.
In conclusion: There are two correct ways to define stable/stability. (1) Stable either means something exists as a local minimum on the potential energy surface. (2) As a relative concept, A is more stable than B if it costs energy to transform A into B.
Hyperconjugation and the problem of counting interactions
The gold book again gives us a definition for hyperconjugation (10.1351/goldbook.H02924), but I will only quote a portion:
In the formalism that separates bonds into σ and π types, hyperconjugation is the interaction of σ-bonds (e.g. C–H, C–C, etc.) with a π network. This interaction is customarily illustrated by contributing structures, [...] sometimes said to be an example of 'heterovalent' or 'sacrificial hyperconjugation', so named because the contributing structures contains one two-electron bond less than the normal Lewis formula [...].
The important takeaway from this is, that this is only valid within a formalism that separates into σ and π bonds. This concept does not (in this form) exist in Molecular Orbital Theory.
As previously mentioned, there are various ways to describe electronic structure. Many of them require more than the back of an envelope. Those descriptions are basically modeling the electron density at given nuclear positions; these are just mathematical tools to reproduce observables.
As such, hyperconjugation interactions are also just mathematical models and you cannot measure their stabilising factor. All numbers you might find online or in textbooks, are pretty much partitioning schemes of experimental data.
As a very simple example: You can measure the hydrogenation enthalpy of ethene and propene. You can then assume that the π-bond will (in first approximation) behave equally. The resulting energy difference can then be interpreted as the stabilising effect of hyperconjugation. You can do that for thousands of molecules and find some average energy value for these interactions.
That certainly is a valid scientific approach; the problem comes then if you neglect all the conditions that lead to this value and take it as a given constant. It should be painfully obvious, that different σ-bonds will interact differently with other bonds. There is no reasonable justification to assume that a hyperconjugation interaction will always have the same stabilising effect. It should therefore come as no surprise that a carbon-hydrogen σ-bond located between two carbon-carbon π-bonds will have a totally different stabilising effect than a terminal carbon-hydrogen σ-bond.
Unfortunately, this is taught as some kind of holy grail of organic chemistry. Counting hyperconjugation contributors is at best a zeroth order approximation, an educated guess, a ballpark figure. Surely, you can use it to do a rough estimation, but you should always be aware about how imprecise this is.
It is also important that for a complete description of the electronic structure, you always need all resonance contributors, n.b. hyperconjugation is a more specialised form of resonance. Since these are hypothetical structures, they cannot be assigned an energy and they cannot be weighted that way. It really does need more involved methods to accurately describe this.
Focusing on this criterion alone also only ensures that you are neglecting any and all other possible influences. Dispersion will certainly have a different effect in one molecule than it will have in the other. There will be other mesomeric, or inductive effects; the conformational space will also be of concern.
So while $\pi \approx 3$ is a good guess to roughly estimate the size of a round window, you probably shouldn't use it to built your house.
The problem is the question
Now that we have gotten all of the general problems out of the way, there was one problem pointed out previously in the comments by user55119:
There is a cis/trans issue with the second structure. You might explore the heats of hydrogenation/combustion of hexenes that bear the substitution of the double bond. Welcome to ChemSE!
As if we hadn't already enough problems, the question is already ambiguous by specifying the molecules. In my personal opinion, this shows that the instructor did not spend enough time actually posing the exercise, really thinking about it. It shows that this exercise is targeted exactly at this specific counting exercise. To be honest, I would not have been surprised if the approximation had yielded a completely different result.
To be on the safe side, I performed a couple of calculations. First I ran a conformational analysis with CREST on the xtb level of theory, then I optimised the lowest structures with RI-BP86/def2-SVP in Orca. These are the relative electronic energies. (I did not bother with thermal corrections, since they are not accurate enough for this level, and qualitatively will probably not change the general order.
(1) 1-2-3-4-5-8-hexahydronaphtalene 0.0 kJ/mol
(2) cis-1-4-4a-5-8-8a-hexahydronaphtalene 48.8 kJ/mol
(3) trans-1-4-4a-5-8-8a-hexahydronaphtalene 26.8 kJ/mol
You can see from these numbers, especially between 2 and 3 where the number of hyperconjugation interactions is the same, that there are quite large differences. So using hydrogenation or combustion values is probably a better approach to this kind of question.
Additionally, those numbers don't tell you much about the molecules themselves or the chemistry they most likely will perform. In general I'd expect 1,4-cyclohexadiene derivatives to readily undergo oxidation to an aromatic species, but that is besides the point.
My personal opinion is that these counting models oversimplify chemistry beyond the point of usefulness. They do not further any understanding of molecular and electronic structure. It turns useful and educational models into formulaic use cases, which are applied without further understanding.
Solution 2:
@ Martin has addressed the shortcomings of assessing “stability” by counting hyperconjugating C-H bonds. If by “more stable” one means which one of the two isomers, 1,2,3,4,5,8-hexahydronaphthalene 1 or cis-1,4,4a,5,8,8a-hexahydronaphthalene 2, has the lower heat of combustion, formation or hydrogenation, then a decision can be made. Unfortunately, there does not appear to be any thermochemical data [NIST] on these compounds to answer this question.
If one assumes, not necessarily justifiably, that each double bond in these isomers is not affected by the presence of the other double bond, then one can model a solution based on known values for the heats of hydrogenation (ΔHho) of 1,2-cis-dialkyl and tetraalkyl-substituted double bonds. The more alkyl substituted a double bond, the less heat is produced upon hydrogenation, a trend that is readily verified with data from the NIST site. The heats of formation and hydrogenation of compounds are from the same source.
The ΔHho for an unstrained 1,2-cis-double bond is -28.3 kcal/mol as is seen in the hydrogenation of cis-2-butene to n-butane and cyclohexene to cyclohexane. The heat of hydrogenation of 2,3-dimethylbut-2-ene (tetramethylethylene) to 2,3-dimethylbutane is -25.7 kcal/mol. Using these two values, the heat of formation (ΔHfo) of 1 and 2 can be calculated given the ΔHfo of cis-decalin 3. Accordingly, diene 1 (ΔHfo = +13.5 kcal/mol) is more “stable” than diene 2 (ΔHfo = +16.5 kcal/mol)(blue numbers) by ~3 kcal/mol.
Notice that the monohydrogenation of cyclohexa-1,4-diene liberates only -25.2 kcal/mol upon formation of cyclohexene. This implies that the two double bonds, although non-conjugated, are not isolated in their reactivity. If they were isolated and equivalent in reactivity, the ΔHfo of cyclohexa-1,4-diene would be +27.1 kcal/mol. [Parenthetically, the conjugated cyclohexa-1,3-diene has a ΔHfo 1 kcal/mol greater than cyclohexa-1,4-diene.] Diene 1 has a similar arrangement of double bonds, albeit one of them is tetrasubstituted. Assuming that the -3.1 kcal/mol correction may be applied in the case of diene 1, the ΔHho would be lowered to -50.9 kcal/mol (red numbers). Whatever the true correction may actually be, the heat of formation of diene 1 will be lower than that for diene 2.