Imperial College London

Henry Rzepa

[Linear free energy
relationships](https://en.wikipedia.org/wiki/Free-energy_relationship)
(LFER) are associated with the dawn of physical organic chemistry in the
late 1930s and its objectives in understanding chemical reactivity as
measured by reaction rates and equilibria.

The [Hammett equation](https://en.wikipedia.org/wiki/Hammett_equation)
is the best known of the LFERs, albeit derived "intuitively". It is
normally applied to the kinetics of aromatic electrophilic substitution
reactions and is expressed as;

**log K~R~/K~0~ = σ~R~ρ** (for equilibria) and extended to **log
k~R~/k~0~ = σ~R~ρ** for rates.

The equilibrium constants are normally derived from the ionisation of
substituted benzoic acids, with K~0 ~being that for benzoic acid itself
and K~R ~that of a substituted benzoic acid, with σ~R~ being known as
the substituent constant and ρ the reaction constant. The concept
involved obtaining the substituent constants by measuring the ionisation
equilibria. The value of σ~R ~is then assumed to be transferable to the
rates of reaction, where the values can be used to obtain reaction
constants for a given reaction. The latter would then be assumed to give
insight into the electronic nature of the transition state for that
reaction.

The term log k~R~/k~0 ~(the ratio of rates of reaction) can be related
to ΔΔG = -RT ln k~R~/k~0 ~and this latter quantity can be readily
obtained from quantum calculations, where ΔΔG is the difference in
computed reaction activation free energies for two substituents (of
which one might be R=H). The most interesting such Hammett plots are the
ones where a discontinuity becomes apparent. The plot comprises two
separate linear relationships, but with different slopes. This is
normally taken to indicate a change of mechanism, on the assumption that
the two mechanisms will have different responses to substituents. 

A test of this is available via the calculated activations energies for
acid catalyzed cyclocondensation to give
furanochromanes\[cite\]10.1039/c8sc04302g\[/cite\] which is a two-step
reaction involving two transition states **TS1** and **TS2**, either of
which could be rate determining. A change from one to the other would
constitute a change in mechanism. In this example, **TS1** involves
creation of a carbocationic centre which can be stabilized by the
substituent on the Ar group; **TS2** involves the quenching of the
carbocation by a nucleophilic oxygen and hence might be expected to
respond differently to the substituents on Ar. As it happens, the
reaction coordinate for **TS2** is not entirely trivial, since it also
includes an accompanying proton transfer which might perturb the
mechanism.

[![](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2019/01/prins.svg){.aligncenter
.size-large .wp-image-20446 decoding="async"
width="400"}](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2019/01/prins.svg)

Fortunately for this reaction we have available full FAIR data (DOI:
[10.14469/hpc/3943](https://doi.org/10.14469/hpc/3943){target="data"
rel="noopener noreferrer"}), which includes not only the computed free
energies for both sets of transition states but also the entropy-free
enthalpies for comparison. This allows the table below to be generated.
For each substituent, the highest energy point is in bold, indicating
the rate limiting step. The span of substituents corresponds to a range
of rate constants of almost 10^10^, which in fact is rarely if ever
achievable experimentally.

<table style="width:100%;" data-border="1">
<colgroup>
<col style="width: 16%" />
<col style="width: 16%" />
<col style="width: 16%" />
<col style="width: 16%" />
<col style="width: 16%" />
<col style="width: 16%" />
</colgroup>
<tbody>
<tr class="header">
<th colspan="6">Highest free energy overall route for HCl catalysed
mechanism,
trans stereochemistry</th>
</tr>
<tr class="odd">
<th>Sub</th>
<th>ΔH‡/ΔG‡</th>
<th>Reactant</th>
<th>ΔH‡/ΔG‡, TS1</th>
<th>ΔH‡/ΔG‡, TS2</th>
<th>RDS</th>
</tr>
&#10;<tr class="odd">
<td>p-NH2</td>
<td>0.2/6.36</td>
<td>0.0/0.0</td>
<td>0.15/4.0</td>
<td>0.2/6.4</td>
<td>TS2/TS2</td>
</tr>
<tr class="even">
<td>p-OMe</td>
<td>2.7/8.48</td>
<td>0.0/0.0</td>
<td>2.7/8.45</td>
<td>2.1/8.48</td>
<td>TS1/TS2</td>
</tr>
<tr class="odd">
<td>p-Me</td>
<td>5.5/10.00</td>
<td>0.0/0.0</td>
<td>5.5/9.9</td>
<td>3.9/10.00</td>
<td>TS1/TS2</td>
</tr>
<tr class="even">
<td>p-Cl</td>
<td>7.7/12.28</td>
<td>0.0/0.0</td>
<td>7.7/12.28</td>
<td>5.9/11.84</td>
<td>TS1/TS1</td>
</tr>
<tr class="odd">
<td>p-H</td>
<td>7.6/13.01</td>
<td>0.0/0.0</td>
<td>7.6/13.01</td>
<td>5.5/11.51</td>
<td>TS1/TS1</td>
</tr>
<tr class="even">
<td>p-CN</td>
<td>10.6/18.02</td>
<td>0.0/0.0</td>
<td>10.6 /17.61</td>
<td>10.5/18.02</td>
<td>TS1/TS2</td>
</tr>
<tr class="odd">
<td>p-NO2</td>
<td>12.4/19.85</td>
<td>0.0/0.0</td>
<td>12.4/18.24</td>
<td>12.0/19.85</td>
<td>TS1/TS2</td>
</tr>
</tbody>
</table>

For the free energies, you can see that **TS2** is the rate limiting
step for the first two electron donating substituents, and the last two
electron withdrawing ones, whilst **TS1** represents the rate limiting
step for the middle substituents. This represents **two** changes of
rate limiting step over the entire range of substituents. A different
picture emerges if only the enthalpies are used. Now **TS1** is rate
limiting for essentially all the substituents. The difference of course
arises because of significant changes to the entropy of the transition
states. The Hammett equation, and its use of  σ~R ~constants to try to
infer the electronic response of a reaction mechanism, does not really
factor in entropic responses. Nor is it often if at all applied using a
really wide range of substituents. So any linearity or indeed
non-linearity in Hammett plots may correspond only very loosely to the
underlying mechanisms involved.

Starting in the 1940s and lasting perhaps 40-50 years, thousands of
different reaction mechanisms were subjected to the Hammett treatment
during the golden era of physical organic chemistry, but very few have
been followed up by exploring the computed free energies, as set out
above. One wonders how many of the original interpretations will fully
withstand such new scrutiny and in general how influential the role of
entropy is.

::: {.wp_orcid_field}
[https://orcid.org/0000-0002-8635-8390](https://orcid.org/0000-0002-8635-8390){target="_blank"
rel="author"}
:::

Linear free energy relationships (LFER) are associated with the dawn of physical organic chemistry in the late 1930s and its objectives in understanding chemical reactivity as measured by reaction rates and equilibria.

Free energy relationships and their linearity: a test example.

Five years back, [I
speculated](https://www.ch.imperial.ac.uk/rzepa/blog/?p=11065) about the
mechanism of the epoxidation of ethene by a peracid, concluding that
kinetic isotope effects provided interesting evidence that this
mechanism is highly asynchronous and involves a so-called "hidden
intermediate". Here I revisit this reaction in which a small change is
applied to the atoms involved.

Below are two representations of the mechanism. The synchronous
mechanism involves five "curly arrows", two of which are involved in
forming a bond between oxygen and carbon, and three of which transfer a
proton to the group X (X=O). The second variation asynchronously stops
at the half way stage to form a pseudo ion-pair (the "hidden
intermediate") and the proton transfer only occurs in the second stage.
If the ethene is substituted with deuterium, experimentally an inverse
kinetic isotope effect is observed, which provides strong evidence that
at the transition state, no proton transfer is occurring

[![](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imino.svg){.aligncenter
.size-medium .wp-image-20355 decoding="async"
width="400"}](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imino.svg)

Before I go on, I should say that you will not find the mechanism as
shown in either variation above in very many text books, which tend to
practice "curly arrow economy" by employing only four arrows. I will not
pursue this aspect here, except to note that as drawn above, the
synchronous mechanism resembles that of a **pericyclic reaction** in a
variation known as coarctate, as I noted in the original post
(DOI: [10.14469/hpc/4807](https://doi.org/10.14469/hpc/4807)).

Now I introduce a veritable variation into this reaction, known as Payne
epoxidation\[cite\]10.1021/jo01062a004\[/cite\],^†^ which replaces the
peracid with a reagent generated by adding hydrogen peroxide to a
nitrile to generate a transient species which can be represented by X=NH
above. How does this change things? The model below also uses propene
rather than ethene (M062X/Def2-TZVPPD/SCRF=dichloromethane).^‡^ This
transition state (ΔG~298~ 31.3 kcal/mol) shows two C-O bond formations,
and as before the proton is clearly not yet transferred to the nitrogen
(X=NH). Because of this asynchrony, the reaction could also be called a
coarctate *pseudo-pericyclic* reaction.

::: {#attachment_20360 .wp-caption .aligncenter style="width: 410px"}
![](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine-863x1024.jpg){.size-large
.wp-image-20360 decoding="async"
aria-describedby="caption-attachment-20360"
onclick="jmolApplet([400,400],'load wp-content/uploads/2018/12/imine1.log;frame 29;spin 3;vectors on;vectors 4;vectors scale 8.0;color vectors green;vibration 6;','c1');"
width="400"
srcset="https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine-863x1024.jpg 863w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine-253x300.jpg 253w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine-768x911.jpg 768w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine.jpg 1057w"
sizes="(max-width: 863px) 100vw, 863px"}

Asynchronous concerted mechanism. Click for 3D
:::

However, the proton transfer is nonetheless part of a concerted
mechanism, as shown by the IRC profile.
[![](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine.gif){.aligncenter
.size-full .wp-image-20361 decoding="async"
width="450"}](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine.gif)

[![](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine_tot_ener.svg){.aligncenter
.size-large .wp-image-20362 decoding="async"
width="540"}](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine_tot_ener.svg)

The gradient norm most clearly shows the "hidden ion-pair intermediate"
at IRC = -1, and the proton transfer only occurs after this point is
passed.[![](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine_rms_gnorm.svg){.aligncenter
.size-large .wp-image-20363 decoding="async"
width="540"}](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine_rms_gnorm.svg)

This is even more spectacularly illustrated with a plot of dipole moment
along the IRC;
[![](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine_mol_prop.svg){.aligncenter
.size-large .wp-image-20364 decoding="async"
width="540"}](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine_mol_prop.svg)

In truth, no real differences are yet revealed between the Payne reagent
and the peracid. In fact, this is a real surprise, since the NH of the
Payne reagent should be very much more basic than the carbonyl oxygen of
the peracid. But more exploration of the potential energy surface
reveals another transition state!

::: {#attachment_20366 .wp-caption .aligncenter style="width: 410px"}
![](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine2-1024x966.jpg){.size-large
.wp-image-20366 decoding="async"
aria-describedby="caption-attachment-20366"
onclick="jmolApplet([400,400],'load wp-content/uploads/2018/12/imine2.log;frame 115;spin 3;vectors on;vectors 4;vectors scale 8.0;color vectors green;vibration 6;','c2');"
width="400"
srcset="https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine2-1024x966.jpg 1024w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine2-300x283.jpg 300w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine2-768x725.jpg 768w, https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine2.jpg 1292w"
sizes="(max-width: 1024px) 100vw, 1024px"}

Stepwise mechanism. Click for 3D
:::

[![](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine2.gif){.aligncenter
.size-full .wp-image-20371 loading="lazy" decoding="async" width="518"
height="292"}](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine2.gif)This
is seen forming the two C-O bonds AFTER the proton transfer from oxygen
to nitrogen. It is 4.2 kcal/mol **lower** than the first transition
state, which corresponds to the scheme below.

[![](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine2.svg){.aligncenter
.size-large .wp-image-20368 decoding="async"
width="450"}](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine2.svg)

The new ion-pair shown above is 7.1 kcal/mol higher than the previous
reactant, but is so much more basic than before that the overall
activation energy is indeed lowered. Two distinctly separate IRCs can be
constructed for this alternative, the first a pure proton transfer (not
shown) and the second a pure C-O bond forming process (below). This
second step is both concerted and almost purely synchronous.

[![](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine2t_ener.svg){.aligncenter
.size-large .wp-image-20373 decoding="async"
width="450"}](https://www.ch.ic.ac.uk/rzepa/blog/wp-content/uploads/2018/12/imine2t_ener.svg)

So now we see how a small change to the reactant molecules (X=O to X=NH)
can induce a reaction for which two quite different mechanisms can
operate, an asynchronous one albeit with a hidden intermediate and a
fully stepwise one in which a quite different, but this time real,
intermediate is involved. Nevertheless for both the peracid mechanism
and the peroxyimine variation shown here, the proton transfer is NOT
involved in the rate limiting step. So for this variation too, inverse
kinetic isotope effects would be expected.

------------------------------------------------------------------------

^‡^FAIR data for the calculations at DOI:
[10.14469/hpc/4909](http://doi.org/10.14469/hpc/4909){target="_blank"
rel="noopener"} ^†^Thanks Ed for pointing this out.

::: {.wp_orcid_field}
[https://orcid.org/0000-0002-8635-8390](https://orcid.org/0000-0002-8635-8390){target="_blank"
rel="author"}
:::


Five years back, I speculated about the mechanism of the epoxidation of ethene by a peracid, concluding that kinetic isotope effects provided interesting evidence that this mechanism is highly asynchronous and involves a so-called “hidden intermediate”. Here I revisit this reaction in which a small change is applied to the atoms involved. Below are two representations of the mechanism.

Postagens de Rogue Scholar

Free energy relationships and their linearity: a test example.

Epoxidation of ethene: a new substituent twist.