This question involves identifying which compounds become optically inactive after catalytic hydrogenation. Optical inactivity occurs when a molecule is achiral (lacks chiral centers) or possesses a plane of symmetry, making it unable to rotate plane-polarized light. Hydrogenation typically adds H₂ across double or triple bonds, potentially altering chirality.

Key concepts:

A chiral center (usually a carbon with four different substituents) is necessary for optical activity.
Hydrogenation can reduce double bonds, possibly creating new chiral centers or removing existing ones.
For a molecule to be optically inactive after hydrogenation, the product must have either no chiral centers or a plane of symmetry (e.g., meso compounds).

Step-by-step analysis (assuming standard structures for common options like alkenes with chiral centers or symmetric alkynes):

Step 1: Examine each compound's structure. Look for chiral centers and double/triple bonds that will be hydrogenated.

Step 2: Predict the hydrogenation product. For example, hydrogenation of an alkene like $C_{2} H_{5} CH = CH - CH (CH)_{3}$ might yield an alkane with a new chiral center.

Step 3: Check for optical activity in the product. If the product has chiral centers but is meso (internal plane of symmetry) or racemic, it is optically inactive.

Step 4: Common cases include symmetric alkynes (e.g., $CH \equiv_{3} C - C \equiv C - CH)_{3}$ hydrogenating to an alkane with a plane of symmetry) or certain cyclic alkenes becoming meso compounds.

Final Answer: Without the exact images, generally, compounds that upon hydrogenation yield a symmetric molecule (e.g., with a plane of symmetry) or lose all chirality will be optically inactive. For instance, a compound with a triple bond at a symmetric position often gives an inactive product.

Related Topics

Chirality and Optical Activity: A molecule is chiral if it cannot be superimposed on its mirror image, usually due to a chiral center. Optical activity is the ability to rotate plane-polarized light, which requires chirality.

Hydrogenation Reaction: Catalytic hydrogenation adds hydrogen across unsaturated bonds (e.g., alkenes, alkynes), converting them to saturated bonds. This process can change the molecular symmetry and chirality.

Meso Compounds: These are achiral compounds with chiral centers but possess an internal plane of symmetry, making them optically inactive despite having stereocenters.

Formulae

No specific mathematical formulae, but key structural concepts:

Chiral center: Carbon with four different groups.
Plane of symmetry: A mirror plane that divides the molecule into identical halves.