How could you determine experimentally whether the compound \(\mathrm{Ga}_{2} \mathrm{Cl}_{4}\) contains two gallium(II) ions or one gallium(I) and one gallium(III) ion? (Hint: Consider the electron configurations of the three possible ions.)

We must first find the electron configuration of Ga and Cl in their ground states and then consider their electron configurations as ions. The electron configuration of Ga (Z=31) in the ground state is: \[1s^2 2s^2 2p^6 3s^2 3p^6 3d^{10} 4s^2 4p^1\] For Cl (Z=17), the ground state electron configuration is: \[1s^2 2s^2 2p^5 3s^2 3p^5\] Next, we will consider the possible ion configurations: Gallium(I) ion (Ga⁺): \[1s^2 2s^2 2p^6 3s^2 3p^6 3d^{10} 4s^2\] Gallium(II) ion (Ga²⁺): \[1s^2 2s^2 2p^6 3s^2 3p^6 3d^{10} 4s^1\] Gallium(III) ion (Ga³⁺): \[1s^2 2s^2 2p^6 3s^2 3p^6 3d^{10}\] Chloride ion (Cl⁻): \[1s^2 2s^2 2p^6 3s^2 3p^6\]

We must analyze the possible compositions of the compound: 1. Two gallium(II) ions: Ga²⁺ + Ga²⁺ + 4Cl⁻ 2. One gallium(I) and one gallium(III) ion: Ga⁺ + Ga³⁺ + 4Cl⁻

We can design an experiment using a redox reaction to differentiate between the gallium(I)/gallium(III) and two gallium(II) possibilities. First, we will dissolve the compound \(\mathrm{Ga}_{2}\mathrm{Cl}_{4}\) in water or another suitable solvent to obtain a clear solution. Next, we will perform a redox reaction using a standardized solution of a reducing agent, such as potassium permanganate (KMnO₄) or potassium dichromate (K₂Cr₂O₇). By monitoring the amount of reducing agent required to fully react with the gallium ions in the solution, we can determine the oxidation states of gallium ions in the compound. For example, if we use potassium permanganate (KMnO₄) as the reducing agent, two possible reactions could occur, depending on the composition of the Ga ions in the sample: 1. For two gallium(II) ions: \[2\,\mathrm{Ga}^{2+} + \mathrm{MnO}_4^- + 8\,\mathrm{H}^+ \rightarrow 2\,\mathrm{Ga}^{3+} + \mathrm{Mn}^{2+} + 4\,\mathrm{H}_2\mathrm{O}\] 2. For one gallium(I) and one gallium(III) ion: \[\mathrm{Ga}^+ + \mathrm{Ga}^{3+} + \mathrm{MnO}_4^- + 6\,\mathrm{H}^+ \rightarrow 2\,\mathrm{Ga}^{3+} + \mathrm{Mn}^{2+} + 3\,\mathrm{H}_2\mathrm{O}\] The amount of reducing agent required will be different for different oxidation states of the gallium ions in the sample. By comparing the consumption of the reducing agent with the expected theoretical amounts for each possibility, we can determine which composition is correct.

By comparing the experimental consumption of the reducing agent with the expected theoretical amounts for each possible composition of \(\mathrm{Ga}_{2}\mathrm{Cl}_{4}\), we can determine whether the compound contains two gallium(II) ions or one gallium(I) and one gallium(III) ion.