There are three salts that contain complex ions of chromium and have the molecular formula \(\mathrm{CrCl}_{3} \cdot 6 \mathrm{H}_{2} \mathrm{O}\). Treating \(0.27 \mathrm{~g}\) of the first salt with a strong dehydrating agent resulted in a mass loss of \(0.036 \mathrm{~g}\). Treating \(270 \mathrm{mg}\) of the second salt with the same dehydrating agent resulted in a mass loss of \(18 \mathrm{mg}\). The third salt did not lose any mass when treated with the same dehydrating agent. Addition of excess aqueous silver nitrate to \(100.0\) -mL portions of \(0.100 M\) solutions of each salt resulted in the formation of different masses of silver chloride; one solution yielded \(1430 \mathrm{mg} \mathrm{AgCl}\); another, \(2870 \mathrm{mg} \mathrm{AgCl}\); the third, \(4300 \mathrm{mg} \mathrm{AgCl}\). Two of the salts are green and one is violet. Suggest probable structural formulas for these salts, defending your answer on the basis of the preceding observations. State which salt is most likely to be violet. Would a study of the magnetic properties of the salts be helpful in determining the structural formulas? Explain.

Step by step solution

01

Analyze the mass loss after dehydration

First, let's analyze the mass loss after treating each salt with a strong dehydrating agent. We observe that: - \(0.27\,\text{g}\) of the first salt loses \(0.036\,\text{g}\). - \(270\,\text{mg}\) of the second salt loses \(18\,\text{mg}\). - The third salt does not lose any mass. The loss in mass corresponds to the water molecules contained in their coordination sphere, which are lost upon treatment with a dehydrating agent. We infer that the more significant the mass loss, the more water molecules are present in the complex ion.

02

Relate the mass loss to water molecules in the complex

Now let's determine the amount of water molecules in each complex ion based on the mass loss after dehydration: For the first salt, the mass loss of \(\text{H}_2\text{O}\) is \(0.036\,\text{g}\). Since the molar mass of water is \(18\,\text{g/mol}\), we calculate the moles of water lost as follows: \[\frac{0.036\,\text{g}}{18\,\text{g/mol}} = 0.002\,\text{mol}\] For the second salt, the mass loss is \(18\,\text{mg}\) or \(0.018\,\text{g}\). The moles of water lost are: \[\frac{0.018\,\text{g}}{18\,\text{g/mol}} = 0.001\,\text{mol}\] The moles of chromium complex in each case is: \[\frac{0.27\,\text{g}}{266.45\,\text{g/mol}} = 0.001\,\text{mol}\] \[\frac{270\,\text{mg}}{266.45\,\text{g/mol}} = 0.001\,\text{mol}\] Comparing the moles of water to the moles of chromium complex, we find that in: - The first salt, there are \(2\,\text{H}_2\text{O}\) molecules. - The second salt, there is \(1\,\text{H}_2\text{O}\) molecule. - The third salt, there are no \(\text{H}_2\text{O}\) molecules in the complex ion.

03

Examine the formation of silver chloride

We are given that the addition of aqueous silver nitrate to \(100.0\,\text{mL}\) portions of \(0.100\,\text{M}\) solutions of the salts yielded different amounts of silver chloride. This implies that the number of chloride ions in the complex ions is different for each salt. Based on the mass of precipitated silver chloride, we can calculate the moles of chloride ions in each case. The molar mass of silver chloride is \(143.32\,\text{g/mol}\). The calculated moles of chloride ions are: 1. \(\frac{1.43\,\text{g}}{143.32\,\text{g/mol}} = 0.01\,\text{mol}\) 2. \(\frac{2.87\,\text{g}}{143.32\,\text{g/mol}} = 0.02\,\text{mol}\) 3. \(\frac{4.3\,\text{g}}{143.32\,\text{g/mol}} = 0.03\,\text{mol}\) Comparing the moles of chloride ions to the moles of chromium complex (\(0.01\,\text{mol}\)), we find that: 1. The first salt has \(1\,\text{Cl}^{-}\) ion. 2. The second salt has \(2\,\text{Cl}^{-}\) ions. 3. The third salt has \(3\,\text{Cl}^{-}\) ions.

04

Propose the structural formulas

Based on our analysis: 1. The first salt contains two \(\text{H}_2\text{O}\) molecules and one chloride ion in the complex ion. Its structural formula is: \(\,\text{Cr}(\text{H}_2\text{O})_4\text{Cl}_2\text{Cl}\cdot2\text{H}_2\text{O}\). 2. The second salt contains one \(\text{H}_2\text{O}\) molecule and two chloride ions in the complex ion. Its structural formula is: \(\,\text{Cr}(\text{H}_2\text{O})_2\text{Cl}_4\cdot4\text{H}_2\text{O}\). 3. The third salt contains no \(\text{H}_2\text{O}\) molecules and three chloride ions in the complex ion. Its structural formula is: \(\,\text{Cr}(\text{H}_2\text{O})_3\text{Cl}_3\cdot3\text{H}_2\text{O}\).

05

Determine the violet salt and the utility of studying magnetic properties

Among the three salts, only the third salt has \(\,\text{Cr}(\text{H}_2\text{O})_3\text{Cl}_3\) which has octahedral geometry. This geometry tends to exhibit colors due to d-d transitions of its electrons. Thus, the third salt is most likely to be violet. Studying the magnetic properties can be helpful in determining the structural formulas. The magnetic behavior is determined by the number of unpaired electrons in the d orbitals of chromium. A difference in the number of unpaired electrons can provide further evidence supporting the structural formulas proposed above.

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