Question

For each of the following metals, write the electronic configuration of the atom and its \(3+\) ion: (a) \(\mathrm{Ru}\), (b) Mo, (c) Co. Draw the crystal-field energy-level diagram for the \(d\) orbitals of an octahedral complex, and show the placement of the \(d\) electrons for each \(3+\) ion, assuming a weak-field complex. How many unpaired electrons are there in each case?

Short answer

The electronic configurations for the metal atoms and their 3+ ions are as follows: (a) Ru: \([Kr]4d^7 5s^1\) and Ru³⁺: \([Kr]4d^5\), (b) Mo: \([Kr]4d^5 5s^1\) and Mo³⁺: \([Kr]4d^3\), (c) Co: \([Ar]3d^7 4s^2\) and Co³⁺: \([Ar]3d^6\). For weak-field octahedral complexes, the number of unpaired electrons are: (a) Ru³⁺ has two unpaired electrons, (b) Mo³⁺ has one unpaired electron, and (c) Co³⁺ has two unpaired electrons.

Step-by-step solution

Find the atomic numbers

Using the periodic table, look for the atomic numbers of the given metals: - Ru (Ruthenium) has an atomic number of 44. - Mo (Molybdenum) has an atomic number of 42. - Co (Cobalt) has an atomic number of 27.

Determine the electron configurations

Using the atomic numbers from step 1, write the electron configurations for the metal atoms: (a) Ru (44) has an electron configuration of \([Kr]4d^7 5s^1\). (b) Mo (42) has an electron configuration of \([Kr]4d^5 5s^1\). (c) Co (27) has an electron configuration of \([Ar]3d^7 4s^2\). Now, let's determine the electron configurations for the 3+ ions by removing three electrons: (a) Ru³⁺ has an electron configuration of \([Kr]4d^5\). (b) Mo³⁺ has an electron configuration of \([Kr]4d^3\). (c) Co³⁺ has an electron configuration of \([Ar]3d^6\).

Draw the crystal-field energy-level diagrams

Draw the crystal-field energy-level diagram for the d orbitals of an octahedral complex for each 3+ ion. In weak-field complexes, the energy difference between the d orbitals is small, and electrons fill the orbitals according to Hund's rule: (a) Ru³⁺ (\([Kr]4d^5\)): 1. Start with three low-energy, doubly-degenerate orbitals and two high-energy, doubly-degenerate orbitals. 2. Place five electrons in the orbitals, following Hund's rule. 3. There are two unpaired electrons. (b) Mo³⁺ (\([Kr]4d^3\)): 1. Start with three low-energy, doubly-degenerate orbitals and two high-energy, doubly-degenerate orbitals. 2. Place three electrons in the orbitals, following Hund's rule. 3. There is one unpaired electron. (c) Co³⁺ (\([Ar]3d^6\)): 1. Start with three low-energy, doubly-degenerate orbitals and two high-energy, doubly-degenerate orbitals. 2. Place six electrons in the orbitals, following Hund's rule. 3. There are two unpaired electrons.

Count the unpaired electrons

Using the crystal-field energy-level diagrams, count the number of unpaired electrons for each 3+ ion. (a) Ru³⁺ has two unpaired electrons. (b) Mo³⁺ has one unpaired electron. (c) Co³⁺ has two unpaired electrons.

Key concepts

Electronic Configuration

Understanding the electronic configuration of an atom is essential for predicting its chemical behavior, including its bonding and magnetic properties. The electronic configuration depicts the arrangement of electrons in an atom's orbitals, which are various energy levels. Electrons fill these orbitals in a manner that follows specific rules, such as the Aufbau principle, Pauli exclusion principle, and Hund's rule.

Take, for example, the transition metal Ruthenium (Ru) with an atomic number of 44. Its ground state configuration is [Kr]4d^7 5s^1, which shows that it has seven electrons in the 4d subshell and one in the 5s subshell. However, when Ruthenium forms a 3+ ion, three electrons are removed, generally starting with the outermost s orbital and then from the d orbitals, leading to a configuration of [Kr]4d^5.

Similarly, other metals will also lose electrons from their outermost orbitals when they form positive ions. This loss of electrons and the resulting electronic configurations are not just intellectual exercises; they provide the necessary foundation for building crystal-field theory models for complex ions.

Octahedral Complex

In crystal-field theory, the geometry of a complex plays a crucial role in the distribution of electron energy levels. An octahedral complex, for instance, is a molecule or ion where a central metal atom is surrounded by six ligands at equal distances, positioned at the corners of an octahedron.

In such complexes, the metal's d orbitals split into two different energy levels due to the electrostatic interactions with the ligands. There are three lower-energy orbitals (t2g) and two higher-energy ones (eg). For a weak-field complex, where the difference in energy between eg and t2g orbitals is relatively small, electrons will occupy the t2g orbitals first, spreading out to maximize their spins in accordance with Hund's rule. The number of electrons that the central metal ion contributes will dictate how these orbitals are populated.

Drawing crystal-field energy-level diagrams for octahedral complexes is pivotal for visualizing electronic transitions, predicting magnetic properties, and understanding the color and reactivity of the complex.

Unpaired Electrons

Unpaired electrons are those that occupy an orbital alone, without a paired electron with opposite spin. These unpaired electrons have crucial implications for the magnetic properties of the atom or ion; the presence of unpaired electrons usually leads to paramagnetism, while the absence (fully paired electrons) results in diamagnetism.

In the context of octahedral complexes, counting the number of unpaired electrons allows us to predict the magnetic behavior of the complex. For example, a Ru³⁺ ion in a weak-field octahedral complex has two unpaired electrons. This is determined by arranging the five d electrons into the three lower-energy t2g orbitals and the two eg orbitals, according to Hund's rule. Thus, Ru³⁺ would exhibit magnetic properties.

Through crystal-field theory, we can assess the magnetic properties of complexes, an invaluable tool for material scientists and inorganic chemists. Understanding the interplay between the geometry of the complex, electronic configurations, and the subsequent arrangement of unpaired electrons, allows us to interpret and predict the behavior of complex ions in various chemical processes and technological applications.