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Is rotation about that bond free? I learned that sigma bonds rotate and I want to know if there are any exceptions for a bond like that.

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    $\begingroup$ Yes it rotates just fine. Sigma bonds that can't rotate (e.g. substituted biphenyls) are pretty unique. $\endgroup$
    – orthocresol
    Jun 28 at 11:47
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    $\begingroup$ @orthocresol There would probably be a higher barrier to rotation due to the conugation between the double bond and the benzene ring. $\endgroup$
    – S R Maiti
    Jun 28 at 13:40
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    $\begingroup$ @ShoubhikRMaiti Yes, it's slower, exactly like in a conjugated diene; but it's still quite far away from the 'restricted rotation' category. $\endgroup$
    – orthocresol
    Jun 28 at 14:05
  • $\begingroup$ If the bond didn't rotate and there was an ortho substituent present, then there would be two possible ortho substituted cis isomers (phenyl in plane) or a single racemate (phenyl out of plane). $\endgroup$
    – user55119
    Jun 28 at 17:25
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The molecule shown by the OP is cis (Z) cinnamic aldehyde - not the ordinary trans (E) cinnamaldehyde. Trans cinnamaldehyde melts at -7.5C, so is liquid at room temperature, and while the melting point of the cis wasn't available, it has been described as a yellowish liquid. The trans isomer is available from fine chemical suppliers at about 50 USD/kg, but the cis isomer is considerably more expensive: $750/100 mg from a Canadian company.

If rotation were hindered in cinnamaldehyde, there might be enough intramolecular attraction to raise the melting point. Let me shift the focus a little bit - to cinnamic acid.

Ordinary (trans) cinnamic acid has a melting point of $133$$\pu{136 ^\circ C}$, so rotation of the sigma bond in the solid trans isomer is probably very hindered, but could perhaps rotate in the liquid or in solution. NMR spectra[1] don't suggest a hindered rotation.

The barrier to rotation in a simple molecule like ethane is about $\pu{2.8 kcal/mol}$[2], and between $2$$\pu{3 kcal/mol}$ for most $\ce{C-C}$ bonds[3], vs about $15$$\pu{20 kcal/mol}$ for amide ($\ce{N-C}$) rotation, and $\pu{37 kcal/mol}$ for allyl cation. Allyl cation has quite strong pi bonding over the three carbon system, pretty much the maximum possible for a "sigma" bond - although calling the bond a sigma bond is really a stretch.

pi bond interaction

The rotation, or more generally, the molecular mobility, in cis-cinnamic acid, seems to be significantly greater than in the trans form. The CRC Handbook gives melting points for three forms of cis cinnamic acid: $\pu{42^\circ C}$, $\pu{58^\circ C}$ and $\pu{68^\circ C}$. Apparently the purification method is the distinction[4], but this just suggests that $\pu{68^\circ C}$ is the best melting point. Still, in the solid, free rotation of the phenyl ring seems unlikely, but steric crowding of the phenyl and carboxyl make the system more unstable and more easily rotatable, whether by melting (at the lower temperature) or in solution.

Knowing exactly how long the bond is between the phenyl and vinyl groups would suggest how high the barrier to rotation is. Structural data for the cinnamic acids did not come up in a search, but a larger molecule (N-(3-nitrophenyl)cinnamamide) with a trans cinnamic group at one end had a $\ce{C^3-C^4}$(or $\ce{A^1}$) bond length (using their numbering system) of $\pu{1.466 Å}$, somewhat shorter than I would expect[5].

N-(3-nitrophenyl)cinnamamide

But it would seem that you can rotate the bond fairly easily in the liquid or in solution, but you can freeze it in the solid.

References:

[1]: PubChem. Cinnamic acid https://pubchem.ncbi.nlm.nih.gov/compound/Cinnamic-acid (accessed Jun 28, 2021).

[2]: Quijano-Quiñones, R. F.; Quesadas-Rojas, M.; Cuevas, G.; Mena-Rejón, G. J. The Rotational Barrier in Ethane: A Molecular Orbital Study. Molecules 2012, 17 (4), 4661–4671.

[3]: Conjugation and resonance in organic chemistry https://www.masterorganicchemistry.com/2017/01/24/conjugation-and-resonance/ (accessed Jun 28, 2021).

[4]: allocinnamic acid https://www.chemicalbook.com/ProductChemicalPropertiesCB71268435_EN.htm (accessed Jun 28, 2021).

[5]: Lee, J.-S.; Zeller, M.; Warkad, S. D.; Nimse, S. B. Synthesis, Characterization, and Crystal Structure of N-(3-Nitrophenyl)Cinnamamide. Crystals (Basel) 2019, 9 (11), 599.

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    $\begingroup$ The structure shown by the OP is not an acid at all but rather cinnamaldehyde. $\endgroup$
    – Curt F.
    Jun 28 at 20:51
  • $\begingroup$ @Curt: Thanks! What an oversight on my part! I will edit to correct. $\endgroup$ Jun 29 at 13:37

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