In what sense you want to distinguish stereoisomers? For example, both substances adsorb differently in a chromatographic column and can be separated and identified by Gas Chromatography. They have different boiling and fusion heats, different physical properties...
The diastereoisomers can be distinguished using 1H NMR. The H-H coupling constant for the Cis isomer will be smaller than that of the Trans one due to antiperiplanar orientation in the latter giving rise to larger coupling constant. Hope this helps.
I agree with Thresen Mathew that 1H NMR is a good option. My suggestion is to perform also an experiment of NOE like NOESY or NOESY 1D.
I agree with Eduardo Miguez , that beside the chemical stability of the cis- isomer is expected to be higher. the trans-isomer may add Br2 /CCl4 to give 1,4-dibromocyclooctane. heating may be helpful.
The idea of temperature variation is very good. The key point is the solvent to be used in the experiment. If it is used chloroform (boiling point about 67 Celsius), the experiment should be done in a safe temperature (about 10 degrees below the boiling point), in which case it may not be possible to observe some change. We can work at a lower temperature than ambient, close to zero or less. Another possibility is to use the solvent effect, obtaining spectra in different solvents with boiling point higher or lower freezing point
I don't think variable temperature NMR will help in this case. VT experiments are done to distinguish or identity usually the conformational isomers for e.g. rotamers in an amide, that will get frozen at low temp in one or other conformation, or get completely merged at high temp.
In this case, in addition to the conventional 1D H-NMR, and noe, an irradiation experiment also will help to find the correct chemical shift of every proton, and thereby calculate the exact coupling constants. Noe will help to understand the spacial arrangements of the functional groups.
If you have a separation issue of the mixture, column chromatography or hplc on a chiral column might help. The diasteromers will have different retention times even on a regular silicagel.
I suggest to you to do a H,C-HSQC experiment and look at the cross-peaks of the central CH2 groups of the five-membered rings, i. e. positions 3 and 7. These groups are easily identified as they give the lower intensity CH2 signal in the 1D 13C-NMR spectrum.
In the cis isomer the two protons at C-3/C-7 have different chemical shifts, in the trans-isomer they are isochronous (chemically equivalent) because the molecule possesses a C2-axis passing through C3 and C-7. (Hopefully the shift difference is large enough in the cis-isomer.)
Please tell us the outcome of the experiment.
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