Step 1: Understanding the Question:
The question asks us to identify the specific metal catalyst responsible for facilitating the hydrogenation of an alkene (ethene to ethane).
Step 2: Key Formula or Approach:
Catalytic hydrogenation involves the addition of molecular hydrogen ($\text{H}_2$) across a carbon-carbon double or triple bond.
Because hydrogen gas is relatively unreactive at room temperature, a transition metal catalyst is required to adsorb and dissociate the $\text{H}_2$ molecules on its surface.
Step 3: Detailed Explanation:
The general reaction is:
$$\text{CH}_2\text{=CH}_2 + \text{H}_2 \xrightarrow{\text{Catalyst}} \text{CH}_3\text{-CH}_3$$
The most common transition metals used for this heterogeneous catalysis are Nickel (Ni), Palladium (Pd), and Platinum (Pt).
Finely divided Nickel (often known as Raney Nickel) provides a very large surface area for the reaction to occur efficiently at slightly elevated temperatures and pressures.
Let's briefly look at the other options: $\text{MnO}_2$ is used for decomposing hydrogen peroxide or oxidizing alcohols. Fe is used in the Haber process for ammonia synthesis.
Step 4: Final Answer:
Finely divided Nickel is the catalyst used, which corresponds to option (C).