How can one determines whether there is a free rotation around a carbon-carbon triple bond?

That could be determined spectroscopically if it occurred, but it doesn't happen according to what I have been able to determine online.

I am not sure how even spectoscopically you could determine if rotation had occurred as ethyne is a member of the

D(infinity h ) point group. Similarly in a cyclic or linear molecule containing a C-C triple bond there is free rotation around the adjacent C-C single bonds so the group roatates as a whole.
The spectra of acetylene (ethyne) are complex. If you have access to an academic library there is a recent paper by Lyulin et al who have studied the line spectra of acetylene in great detail. The paper is:

"Line parameters of acetylene in the 1.9 and 1.7 ìm spectral regions", O.M. Lyulin et al, Journal of Quantitative Spectroscopy and Radiative Transfer
Volume 109, Issue 10, July 2008, Pages 1856-1874.

To determine whether there is a free rotation around a carbon-carbon triple bond, we need to consider the nature of the bond and the molecular geometry.

Carbon-carbon triple bonds consist of one sigma bond and two pi bonds. The overlap of the pi orbitals above and below the bonding axis restricts the rotation around the bond. This means that the triple bond is relatively rigid and does not undergo free rotation like a single bond or even a double bond.

However, there are two important exceptions to this general rule, known as the conditions for free rotation around a carbon-carbon triple bond:

1. Linear carbon-carbon triple bond: If the carbon atoms involved in the triple bond are linearly aligned, i.e., in a straight line, free rotation is possible. In this case, the pi bonds are parallel and can be rotated freely around the carbon-carbon axis.

2. Hindered rotation due to bulky substituents: If the carbon atoms connected by the triple bond have bulky substituents, such as large alkyl groups or functional groups, steric hindrance can prevent free rotation around the triple bond.

So, to determine whether there is free rotation around a carbon-carbon triple bond, we need to evaluate the molecular structure and geometry. If the carbon atoms are linearly aligned or if there are bulky substituents that hinder rotation, then free rotation may not be possible. However, if the carbon-carbon triple bond is unaffected by these conditions, free rotation can occur.