I´m not an expert but I think you can determine which conformation you have with NOESY. With the same experiment or maybe HNMR (if each conformer shows different signals) performed at different temperatures you might observe at some point both conformers with their signals very well defined, and at other temperatures broad signals that indicate you are close to the required energy for the conformational change. I hope this is helpful for you.
variable temperature NMR is helpful.It is similar to to the process of finding the rotamers for which we dissolve the rotamers in DMSO and then we take the NMR.
By NMR. There are three types of information you can get from NMR, proton-proton coupling constants, NOEs, and carbon chemical shifts. The axial-axial, axial-equatorial, and equatorial-equatorial proton-proton coupling constants are different and will give you clues on what you have, specially if you have substituents in the cyclohexane that will minimize the overlapping of proton signals from the cyclohexane. NOE will give you information on the spacial disposition of the protons and if you have substituents such a methyl, the NOE will tell you if it is equarial or axial. The carbon chemical shifts can also tell you that. There are important effects, specially the gamma gauche effect, that make distinct chemical shifts for the carbons in the ring and the substitutents like a methyl to be axial or equatorial. Check out the book K. Pihlaja and E. Kleinpeter, Carbon-13 NMR Chemical Shifts in Structure and Stereochemical Analysis, Wiley-VCH, New York, 1994. If you want to observe transitions, you will have to change the temperature to be able to "freeze" those conformations. Good luck!
Well, everyone else seems to have covered your question exceptionally well. The classic experiment is with C6D11H at temperatures gradually increasing from around -100 C. At such low temperatures, the NMR can distinguish between the axial and equatorial conformer of the single H and gives two sharp peaks each integrating for 1H which gradually coalesce into a single peak at around -60 C because the "ring flip" is happening faster than the NMR can distinguish between the axial and equatorial conformation of the H. Deuterium is invisible in 1-H NMR.
Yes. Use VT H NMR and, at low temperature, look for what is referred to as "W coupling": a long-range coupling between the equatorial protons on C1 and C3 (for example). In other words, a "4 bond coupling".
All the friends gave good answers and suggestions. I suggest that you study the stereochemistry book by Eliel, pages 686 to 708, chapter no 11. you can found the answer.
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NMR is the best tools to detect the conformation of cyclohexane. As per my knowledge 1H NMR does at very lowest temperature e.g. -70 to -90 degree centigrade.