The reaction,
Cr2O3 + 2Al → Al2O3 + 2Cr (ΔG° = -421 kJ)
is thermodynamically feasible as is apparent from the Gibbs energy value.
Why does it not take place at room temperature?
The change in Gibbs energy is related to the equilibrium constant, K as
ΔG = -RT ln K
At room temperature, all reactants and products of the given reaction are in the solid state. As a result, equilibrium does not exist between the reactants and the products. Hence, the reaction does not take place at room temperature. However, at a higher temperature, chromium melts and the reaction takes place.
We also know that according to the equation
ΔG = ΔH - TΔS
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.