The reaction,
$\mathrm{Cr}_{2} \mathrm{O}_{3}+2 \mathrm{Al} \longrightarrow \mathrm{Al}_{2} \mathrm{O}_{3}+2 \mathrm{Cr} \quad\left(\Delta G_{\mathrm{o}}=-421 \mathrm{~kJ}\right)$
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
$\Delta G=-\mathrm{R} T \ln K$
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
$\Delta G=\Delta H-T \Delta S$,
Increasing the temperature increases the value of _11-12-08_Utpal_12_Chemistry_6_4_html_1dedd1a5.gif){kind=small}
making the value of _11-12-08_Utpal_12_Chemistry_6_4_html_m55291de.gif){kind=small}
more and more negative. Therefore, the reaction becomes more and more feasible as the temperature is increased.