The reaction: $Cr_{2}O_{3} + 2 Al \rightarrow Al_{2}O_{3} + 2 Cr$ $\quad (\Delta_{r}G^{\Theta} = -421 \ kJ)$ is thermodynamically feasible as is apparent from the Gibbs energy value. Why does it not take place at room temperature?

(N/A) The change in Gibbs energy is related to the equilibrium constant,$K$,as $\Delta G = -RT \ln K$.
At room temperature,all reactants and products of the given reaction are in the solid state. As a result,the activation energy for the reaction is very high,and the kinetics are unfavorable.
Furthermore,according to the equation $\Delta G = \Delta H - T \Delta S$,increasing the temperature increases the value of $T \Delta S$,making the value of $\Delta G$ more negative.
At higher temperatures,the reactants gain sufficient energy to overcome the activation barrier,and the reaction proceeds.