Is each of the following substances likely to serve as an oxidant or a reductant: $(\mathbf{a}) \mathrm{Ce}^{3+}(a q),(\mathbf{b}) \mathrm{Ca}(s),(\mathbf{c}) \mathrm{ClO}_{3}^{-}(a q),\( (d) \)\mathrm{N}_{2} \mathrm{O}_{5}(g) ?$

In this case, Ce (Cerium) has an oxidation state of +3. It's important to notice that the maximum oxidation state for Cerium is +4. This means that Ce^{3+} is likely to gain an electron to reach its highest oxidation state and become Ce^{4+}. Therefore, Ce^{3+} will act as an oxidant.

In this case, Ca (Calcium) is a metal in its solid state with an oxidation state of 0. Metals typically lose electrons to form positive ions, i.e., they get oxidized. So, Ca is more likely to lose two electrons and become Ca^{2+} with an oxidation state of +2. Therefore, Ca(s) will act as a reductant.

In this case, we need to determine the oxidation state of Cl (Chlorine) in the ClO_{3}^{-} ion. The sum of oxidation states in this ion is equal to -1 (charge of the ion). Let x be the oxidation state of Cl. The oxidation state of three O (Oxygen) atoms is -2 each (total -6). As the sum of oxidation states is equal to -1, we get x - 6 = -1, leading to x = +5. Since the maximum oxidation state of Cl is +7, ClO_{3}^{-} is likely to gain two electrons (get reduced) and become Cl^{+7}. Therefore, ClO_{3}^{-} will act as an oxidant.

In this case, N_{2}O_{5} is a gaseous compound involving Nitrogen (N) and Oxygen (O). We need to determine the oxidation state of N. From the compound formula N_{2}O_{5}, it can be seen that there are two Nitrogen atoms and five Oxygen atoms. The sum of the oxidation states in this compound is equal to 0 (overall charge). Let x be the oxidation state of one N atom. The oxidation state of five O atoms is -2 each (total -10). Since there are two N atoms, the sum of oxidation states is 2x - 10 = 0, leading to x = +5. The maximum oxidation state for N is +5, which it already has in N_{2}O_{5}. Therefore, N is more likely to lose electrons (get oxidized) in this case. So, N_{2}O_{5}(g) will act as a reductant. In summary, Ce^{3+}(aq) and ClO_{3}^{-}(aq) act as oxidants, while Ca(s) and N_{2}O_{5}(g) act as reductants.