An iron complex formed from a solution containing hydrochloric acid and bipyridine is purified and analyzed. It contains $9.38 \% \mathrm{Fe}, 60.53 \%\( carbon, \)4.06 \%\( hydrogen, and \)14.12 \%$ nitrogen by mass. The remainder of the compound is chlorine. An aqueous solution of the complex has about the same electrical conductivity as an equimolar solution of \(\mathrm{K}_{2}\left[\mathrm{CuCl}_{4}\right] .\) Write the formula of the compound, using brackets to denote the iron and its coordination sphere.

Given the following mass percentages and atomic masses: - Iron (Fe): \(9.38\%\) and atomic mass = \(55.85\) - Carbon (C): \(60.53\%\) and atomic mass = \(12.01\) - Hydrogen (H): \(4.06\%\) and atomic mass = \(1.008\) - Nitrogen (N): \(14.12\%\) and atomic mass = \(14.01\) - Chlorine (the remainder): \(100 - (9.38 + 60.53 + 4.06 + 14.12) = 11.91\%\) and atomic mass = \(35.45\) First, let's assume that the total mass of the compound is 100g. In this case, the masses of each element would be equal to their mass percentages. Then, find the amount in moles of each element. Moles of Iron (Fe): \[\frac{9.38}{55.85} \approx 0.168\] Moles of Carbon (C): \[\frac{60.53}{12.01} \approx 5.04\] Moles of Hydrogen (H): \[\frac{4.06}{1.008} \approx 4.03\] Moles of Nitrogen (N): \[\frac{14.12}{14.01} \approx 1.01\] Moles of Chlorine (Cl): \[\frac{11.91}{35.45} \approx 0.336\]

Next, find the mole ratio of the elements by dividing each mole quantity by the smallest mole quantity. \( \frac{0.168}{0.168}\): Fe \(\rightarrow 1\) \( \frac{5.04}{0.168}\): C \(\rightarrow 30\) \( \frac{4.03}{0.168}\): H \(\rightarrow 24\) \( \frac{1.01}{0.168}\): N \(\rightarrow 6\) \( \frac{0.336}{0.168}\): Cl \(\rightarrow 2\) Thus, the empirical formula of the complex is: \(\mathrm{Fe(C_{30}H_{24}N_{6})Cl_2}\).

Given that the complex has approximately the same electrical conductivity as an equimolar solution of \(\mathrm{K}_{2}\left[\mathrm{CuCl}_{4}\right]\), it indicates that there are 2 ions in the solution. Thus, the iron complex must be charged, i.e., it must be a cation. The cation will have a positive charge which will balance the 2 negative charges from the two chlorine atoms. The bipyridine ligands in the compound must be present around the central Fe(II) ion to form the coordination complex. Bipyridine has the following molecular formula: \(\mathrm{C_{10}H_{8}N_{2}}\). To find the number of bipyridine ligands in the complex, divide the numbers of carbon, hydrogen, and nitrogen atoms found in the empirical formula by the corresponding numbers in the bipyridine molecular formula: \( \frac{30}{10}\): C \(\rightarrow 3\) \( \frac{24}{8}\): H \(\rightarrow 3\) \( \frac{6}{2}\): N \(\rightarrow 3\) This indicates that there are 3 bipyridine ligands in the complex.

The final formula of the complex considering brackets to denote the iron and its coordination sphere can be given by: \(\mathrm{[Fe(bipy)_{3}]Cl_2}\) where "\(\mathrm{bipy}\)" represents the bipyridine ligand. The coordination sphere of the iron complex is represented within the brackets.