[orgo814]

How can you tell whether a di-substituted cyclohexane is interconvertible with a ring flip in regards to the stereochemistry?

[discodermolide]

Because on ring flip the equatorial substituents become axial and the axial become equatorial. Make some models to see this.
Or I can draw you a picture?

[orgo814]

But why is  cis 1,2-dimethylcyclohexane interconvertible but cis 1,3-dimethylcyclohexane not?

[discodermolide]

In both molecules the substituents are both axial that is on the same side of the molecule one axial one equatorial. On ring flip they become equatorial and axial.
In the other case both the substituents are axial, on ring flip they become equatorial. 1,3-diaxial steric interactions raise the energy barrier to ring flip but it will happen if you put enough energy into it.

Make some models and you will see.

[mamid]

@discodermolide: the middle structure, the boat, is the 1,4 isomer.

[fledarmus]

Build models!

Look carefully at the cis 1,2-dimethylcyclohexane. Are the two methyl groups axial or equatorial?

What about cis 1,3-dimethylcyclohexane?

[discodermolide]

Dammit, sorry people, my mistake

[orgo814]

Ok- I get that all cyclohexanes can technically interconvert by a ring flip but why can't cis-1,2 dimethylcyclohexane be separated because it rapidly interconverts? It technically is chiral, has a mirror imagine (enantiomeric), but you can't seem to separate it because it rapidly interconverts... Is this the only molecule in basic organic chemistry that is capable of this? Keep in mind I'm only in organic chemistry 1

[discodermolide]

If you cool them down to liquid nitrogen temperatures you can see the two conformers by ¹H-NMR. Do the experiment properly and you can measure a rate constant for the flip. The NMR experiment shows two sets of signals for the axial and the equatorial protons.
Chiral tertiary amines and tertiary phosphines also do this "flip".

[orgo814]

Ok- is it only just cis-1,2-dimethylcyclohexane that is rapidly interconvertible or is it just cis-1,2-disubstituted cyclohexanes in general (i.e. cis-1,2-dibromocyclohexane)?

[discodermolide]

This is a general thing for cyclohexanes except when the barrier to ring flip is too high for example if one of the substituents is tertiary butyl the ring will be locked in one of the chair conformations due to steric constraints caused by the bulky t-butyl group stopping the ring flipping.

Make models and you will see.