Which half-cell reaction would have the greatest tendency to occur at the cathode?

In electrochemistry, cathode is the electrode where the reduction reaction occurs. In a redox reaction, the reduction involves decreasing the atom's oxidation number, losing electrons in the process. The opposite is true for oxidation.

I think you forgot to include the choices. Nevertheless, by concept, the greatest tendency for reduction reaction will be based on empirical values. The standard electrical potential, denoted as E0, is gathered from experiments. Every known reduction reaction there is, has a known value of E0. Therefore, the greatest tendency to occur in the cathode is the reaction with the highest standard reduction potential starting from the upper left to the lower right reaction in the picture.

For cyclohexane compounds drawn in a chair conformation:   -axial bonds are vertical  -equatorial bonds are parallel to two bonds within the ring.  thus, as i assume the chair conformation is drawn:   -the x group is in an axial position, as are the five other vertical bonds.  -three axial bonds are on the top face of the ring  -the other three axial bonds are on the bottom face  the position of the second substituent on each ring carbon is equatorial.  gauche groups must be on an adjacent carbon such as carbon 6 and carbon 2, and also gauche groups are rotated by 60 degrees relative to each other.  drawing the newman projections for c1–c2 and c1–c6 aids in visualizing the groups that are gauche to the y group on c1.  click this link and watch some academic videos that may you more understand the topic