Question

A manganese complex formed from a solution containing potassium bromide and oxalate ion is purified and analyzed. It contains \(10.0 \% \mathrm{Mn}, 28.6 \%\) potassium, \(8.8 \%\) carbon, and \(29.2 \%\) bromine by mass. The remainder of the compound is oxygen. An aqueous solution of the complex has about the same electrical conductivity as an equimolar solution of \(\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{\mathrm{S}}\right]\). Write the formula of the compound, using brackets to denote the manganese and its coordination sphere.

Short answer

The formula of the manganese complex is \(\text{K}_4[\text{Mn}(\text{C}_2\text{O}_4)_2\text{Br}_2]\), which is determined using the given percentages of elements and their molar masses, and assuming the same number of ionic species in its formula as \(\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{\mathrm{S}}\right]\).

Step-by-step solution

Calculate the number of moles of each element

Given the percentages of each element by mass, we will assume 100 grams of the compound. From this assumption, we can calculate the number of moles for each element using their respective molar masses: - Moles of Mn = \(\frac{10.0\ \text{g}}{54.94\ \text{g/mol}}\) = \(0.182\ \text{mol}\) - Moles of K = \(\frac{28.6\ \text{g}}{39.10\ \text{g/mol}}\) = \(0.731\ \text{mol}\) - Moles of C = \(\frac{8.8\ \text{g}}{12.01\ \text{g/mol}}\) = \(0.733\ \text{mol}\) - Moles of Br = \(\frac{29.2\ \text{g}}{79.90\ \text{g/mol}}\) = \(0.365\ \text{mol}\) For the remaining oxygen, we can find the mass and then calculate the moles: - Mass of O = \(100.0\ \text{g} - (10.0\ \text{g} + 28.6\ \text{g} + 8.8\ \text{g} + 29.2\ \text{g})\) = \(23.4\ \text{g}\) - Moles of O = \(\frac{23.4\ \text{g}}{16.00\ \text{g/mol}}\) = \(1.46\ \text{mol}\)

Normalize the mole ratios

To find the empirical formula, we need to normalize the mole ratios by dividing each by the smallest number of moles: - Mn: \(\frac{0.182}{0.182} = 1\) - K: \(\frac{0.731}{0.182} \approx 4\) - C: \(\frac{0.733}{0.182} \approx 4\) - Br: \(\frac{0.365}{0.182} = 2\) - O: \(\frac{1.46}{0.182} = 8\)

Write the empirical formula

Based on the normalized mole ratios, the empirical formula is: \( \text{K}_4\text{Mn}(\text{C}_2\text{O}_4)_2\text{Br}_2\)

Write the formula of the complex

According to the exercise, we need to write the formula of the compound using brackets to denote the manganese and its coordination sphere. Since the coordination sphere includes the oxalate ion, we have: \( \text{K}_4[\text{Mn}(\text{C}_2\text{O}_4)_2\text{Br}_2]\) This is the formula of the manganese complex.

Key concepts

Empirical Formula Calculation

Understanding how to calculate the empirical formula is a foundational concept in chemistry, especially when analyzing the composition of a compound. The empirical formula is the simplest integer ratio of elements in a compound. It doesn't reflect the exact number of atoms, but gives a proportional representation of each type of atom within the compound.

To calculate the empirical formula, one should begin by determining the amount of each element in grams. Using the assumption that we're dealing with a 100-gram sample simplifies the math, allowing us to treat percentage composition as direct gram measurements. The next step is to convert the mass of each element to moles using their atomic or molar masses. This yields a mole quantity for each element involved.

After finding the moles of each element, the key to determining the empirical formula is to divide each element's mole quantity by the smallest mole quantity found among the elements. This division helps normalize the ratios, making them easier to compare. The resulting values are rounded to the nearest whole number, as empirical formulas must consist of whole numbers of atoms. In some instances, multiplication of these whole numbers may be required to eliminate decimal places, allowing for an accurate empirical formula.

For example, in the exercise, once the moles of each element are known, dividing each by the smallest moles, in this case, manganese with 0.182 moles, yields the normalized mole ratio. This ratio is then used to express the empirical formula in the simplest integer terms.

Mole Ratio Normalization

Mole ratio normalization is a critical step in the empirical formula calculation process. After the moles of each constituent element are determined, the normalization process involves creating a ratio of each element relative to the element with the smallest mole quantity. This is done to find the simplest ratio in which the elements combine to form the compound.

This simplification is based on the principle that chemical compounds form in set proportions according to the Law of Definite Proportions. By dividing the mole quantity of every element by the smallest mole quantity, we achieve normalized ratios, which ideally should be whole numbers. If they aren't, they may be multiplied by an appropriate factor to convert them to whole numbers, as we see with the mole ratios of potassium, carbon, and oxygen in the manganese complex problem.

It's essential to approach this methodically because these normalized ratios correspond directly to the subscript numbers in the empirical formula. Through careful calculation and rounding when necessary (while considering significant figures), we arrive at the empirical formula—a crucial piece of information for chemists in understanding the makeup of a substance.

Coordination Chemistry

Coordination chemistry involves the study of compounds characterized by coordination compounds, in which a central atom, usually a metal, binds to a group of molecules or anions called ligands. These ligands possess 'lone pairs', which they can donate to an empty orbital of a central metal atom or ion, forming coordinate covalent bonds.

In the case of the manganese complex in the exercise, the central metal ion is manganese (Mn), and its ligands are oxalate ions (C2O4) and bromine atoms (Br). The coordination sphere consists of those atoms or molecules bonded directly to the central metal ion, and in this example, the oxalate ions and bromine atoms belong to manganese’s coordination sphere.

In notation, the coordination sphere is enclosed within square brackets to differentiate the complex from ions that are not directly attached to the metal ion, like the potassium ions (K) in our problem. The coordination sphere's charge determines the charge balance with the other ions present. The exercise specifies that the complex behaves similarly in conductivity to potassium ferrocyanide, K4[Fe(CN)6], suggesting the overall charge is balanced with an equal number of cations (K+) and the manganese complex anion carrying a charge resulting from the central ion and its ligands.

Coordination compounds like this manganese complex are ubiquitous in biochemistry, catalysis, and materials science due to the unique properties arising from the metal-ligand interactions. The nuanced understanding of these interactions is fundamental to manipulating and applying these compounds in various technological and scientific contexts.