Answer
$\Delta S^{\circ}_{rxn}=14.7\,J/K$
The moles of gas is constant. But, the moles of solid increased (increase in randomness or entropy), and therefore $\Delta S^{\circ}_{rxn}$ is positive.
Work Step by Step
$\Delta S^{\circ}_{rxn}=\Sigma n_{p}S^{\circ}(products)-\Sigma n_{r}S^{\circ}(reactants)$
$=[2\,mol\times S^{\circ}(Cr,s)+3\,mol\times S^{\circ}(CO_{2},g)]-[1\,mol\times S^{\circ}(Cr_{2}O_{3},s)+3\,mol\times S^{\circ}(CO,g)]$
$=[2\,mol(23.8\,Jmol^{-1}K^{-1})+3\,mol(213.8\,Jmol^{-1}K^{-1})]-[1\,mol(81.2\,Jmol^{-1}K^{-1})+3\,mol(197.7\,Jmol^{-1}K^{-1})]$
$=14.7\,J/K$
The number of moles of gas is constant. But, the number of moles of solid increased (increase in randomness or entropy), and therefore $\Delta S^{\circ}_{rxn}$ is positive.
