The reaction

\(Cr_2O_3 + 2Al \longrightarrow Al_2O_3 + 2Cr\, \ \Delta G^o = -421 kJ\)

does not take place at room temperature because:

Certain amount of activation energy is required

The correct answer is option 3. Certain amount of activation energy is required.

Let us look into each option:

1. \(Cr_2O_3\) is unstable: This statement is incorrect. Chromium(III) oxide (\(Cr_2O_3\)) is a stable compound under normal conditions and does not decompose spontaneously at room temperature.

2. Aluminium is less reactive than chromium: This statement is incorrect. Aluminium is more reactive than chromium. In the thermite reaction, aluminium reduces chromium(III) oxide to form chromium metal, demonstrating its higher reactivity compared to chromium.

3. Certain amount of activation energy is required: This statement is correct. The thermite reaction involves the reduction of chromium(III) oxide (\(Cr_2O_3\)) by aluminium (\(Al\)) to produce aluminium oxide (\(Al_2O_3\)) and chromium metal (\(Cr\)). While the reaction is thermodynamically favorable (as indicated by the negative standard Gibbs free energy change (\(\Delta G^o\))), it requires an activation energy to initiate the reaction. This activation energy barrier prevents the reaction from occurring spontaneously at room temperature.

4. Entropy of the reaction decreases: This statement is unlikely to be a primary reason for the reaction not occurring at room temperature. The entropy change (\(\Delta S\)) for the reaction depends on the difference in entropy between the products and reactants. While the specific entropy change for this reaction would need to be calculated, it is less likely to be a determining factor in the initiation of the reaction compared to the activation energy requirement.

So, option 3 is the correct explanation for why the reaction does not take place at room temperature.