The reaction,

Cr₂O3 + 2 Al → Al₂ O₃ + 2 Cr (ΔG⁰ = – 421 kJ)

is thermodynamically feasible as is apparent from the Gibbs energy value.

Why does it not take place at room temperature?

The reaction,

Cr₂O₃ + 2 Al → Al₂ O₃ + 2 Cr (ΔG⁰ = – 421 kJ)

Is thermodynamically feasible as is apparent from Gibbs energy value. The change in Gibbs free energy is related to the equilibrium constant, k as

ΔG = - RT ln K

A certain amount of energy activation is required even for such reactions which are thermodynamically feasible, therefore heating is required.

ΔG = ΔH + S

Increasing the temperature increases the value of TΔS, making the value of ΔG more and more negative. Therefore, the reaction becomes more and more feasible as the temperature is increased.