CHEM 131 - ADVANCED GENERAL CHEMISTRY I Crystal field theory Question Bank Set 2

CHEM 131 - ADVANCED GENERAL

CHEMISTRY I - Crystal field theory

Question Bank - Set 2

Liberty University

Question 1

Question

What is the crystal field theory and how does it explain the color of transition

metal complexes? Provide a detailed explanation.

Solution

Crystal field theory is a model used to describe the bonding and properties of

transition metal complexes. It focuses on the interaction between the negatively

charged ligands and the positively charged metal center in these complexes. The

theory explains the splitting of the d orbitals in the metal ion in the presence

of ligands, leading to different energy levels and color phenomena.

Step 1: In crystal field theory, ligands approach the central metal ion along

the x, y, and z axes. The electrostatic interaction between the negatively charged

electron pairs on the ligands and the positively charged metal ion causes the

five d orbitals (dxy, dxz, dyz, dx2-y2, and dz2) to split into two groups - the t2g

set (dxy, dxz, dyz) and the egset (dx2-y2, dz2).

Step 2: The energy difference between the two sets of orbitals is called the

crystal field splitting energy (∆). Depending on the geometry of the complex

(e.g., octahedral, tetrahedral), the magnitude of ∆ varies. For example, in an

octahedral complex, the t2gorbitals have lower energy while the egorbitals have

higher energy.

Step 3: When white light passes through the complex, it interacts with

the d-d transition that involves the movement of electrons between the split

d orbitals. Electrons absorb light energy to jump from the lower energy t2g

orbitals to the higher energy egorbitals.

Step 4: The absorbed light corresponds to a specific wavelength or color

based on the energy difference between the two sets of orbitals. The color of the

complex observed by the human eye is the complementary color of the absorbed

light. For example, if a complex absorbs light in the red region of the spectrum,

it appears green to the observer.

This explains how crystal field theory provides a framework for understand-

ing the colors exhibited by transition metal complexes based on the splitting of

d orbitals and the absorption of light energy.

Question 2

Question

Consider a complex with the formula [Co(NH3)6]3+. Determine the crystal field

stabilization energy (CFSE) for this complex using Crystal Field Theory.

Solution

1. The CFSE for an octahedral complex is given by the equation:

CFSE = ∆o·n

Where ∆ois the crystal field splitting energy and nis the number of elec-

trons.

2. In an octahedral complex like [Co(NH3)6]3+, cobalt (Co) has an oxidation

state of +3, meaning it has 6 d-electrons.

3. The crystal field splitting energy for an octahedral field is typically around

0.6∆o

4. For an octahedral field, the d-orbitals split into two sets of energy levels:

eg(higher energy) and t2g(lower energy).

5. In the [Co(NH3)6]3+ complex, all 6 d-electrons are paired up in the t2g

set, resulting in a total CFSE of 0.

6. Therefore, the crystal field stabilization energy (CFSE) for the complex

[Co(NH3)6]3+ is 0 .

Question 3

Question

An octahedral complex is formed from a metal ion with a d4electron configura-

tion. Determine the number of unpaired electrons using Crystal Field Theory.

Solution

Crystal Field Theory (CFT) can be used to determine the number of unpaired

electrons in transition metal complexes based on the splitting of the d orbitals

in the presence of ligands. In an octahedral field, the d orbitals split into two

sets: the lower energy t2gset and the higher energy egset.

Given that the metal ion has a d4electron configuration, all the d orbitals

will be singly occupied. We can use the energy level diagram of an octahedral

field to determine the number of unpaired electrons.

Step 1: Fill the d orbitals with the 4 electrons. - The d orbitals are filled

as: dx2−y2, dz2, dxy , dxz , dyz .

Step 2: Determine the energy splitting in an octahedral field. - In an

octahedral field, the d orbitals split into two sets: t2g(lower energy) and eg

(higher energy).

t2geg

Step 3: Count the number of electrons in the t2gset for the number of

unpaired electrons. - Since all d orbitals are singly occupied, there are 1 electron

in each of the dx2−y2, dz2, dxy orbitals, giving a total of 3 unpaired electrons.

Thus, in an octahedral complex with a d4electron configuration, there are

3 unpaired electrons according to Crystal Field Theory.

Question 4

Question

Explain how crystal field theory accounts for the color of transition metal com-

plexes. Consider a complex with an octahedral geometry.

Solution

Crystal field theory is a model used to describe the electronic structure and

properties of transition metal complexes. It focuses on the interaction between

the metal d orbitals and the ligands surrounding the metal ion.

Step 1: In an octahedral complex, the ligands approach along the axes

between the d orbital lobes. This results in a splitting of the five degenerate d

orbitals into two sets of different energies: the t2gset lower in energy and the

egset higher in energy.

Step 2: When light is absorbed by the complex, an electron from the lower

energy t2gset can be promoted to the higher energy egset. The energy required

for this promotion corresponds to the color of light absorbed.

Step 3: The absorbed light appears as the complementary color to the

absorbed light. For example, if a complex absorbs light in the green region of

the spectrum, it will appear red to the human eye.

Step 4: The magnitude of the energy gap between the t2gand egsets is

determined by the nature of the ligands attached to the metal ion. Strong-field

ligands cause a larger energy gap, resulting in absorption of light in the visible

range and the appearance of complementary colors.

Step 5: Therefore, the color of transition metal complexes can be explained

by crystal field theory based on the splitting of d orbitals and the absorption of

light corresponding to the energy gap between these levels.

Question 5

Question

Consider a complex with the formula [Co(H2O)6]3+. Using Crystal Field The-

ory, determine the number of unpaired electrons in this complex.

Solution

Step 1: Identify the oxidation state of cobalt. Since the complex is [Co(H2O)6]3+,

the oxidation state of cobalt is +3.

Step 2: Determine the electronic configuration of Co3+. Cobalt has 27

electrons. Co3+ will have 24 electrons, which can be filled in the following way:

1s22s22p63s23p64s23d6

Step 3: Determine the number of unpaired electrons using crystal field the-

ory. In an octahedral crystal field, the five d orbitals split into two levels: the

lower energy level (t2g) which contains three orbitals, and the higher energy

level (eg) which contains two orbitals.

Since all six water ligands are weak ligands, the crystal field splitting will

be small. According to Crystal Field Theory, electrons will first fill the t2glevel

before pairing up in the eglevel.

Since there are 6 electrons in the t2glevel, all three orbitals will be filled,

leading to 0 unpaired electrons.

Therefore, the [Co(H2O)6]3+ complex has 0 unpaired electrons.

Question 6

Question

Consider a square planar complex with a metal ion in the +2 oxidation state.

Given that the dx2−y2orbital is most affected by crystal field splitting in this

geometry:

1. Determine the number of unpaired electrons in the complex.

2. Predict the type of magnetic behavior exhibited by the complex.

Solution

1. To determine the number of unpaired electrons in the square planar complex,

we must first consider the electron configurations before and after the crystal

field splitting. In the +2 oxidation state, the metal ion has 2 delectrons.

Step 1: Before crystal field splitting, the electron configuration is (dxy )2(dxz )1(dyz )1(dx2−y2)2(dz2)0.

Step 2: After crystal field splitting, the dx2−y2orbital is most affected. The

electrons will fill the lower energy orbitals first, leading to a splitting of the

dx2−y2orbital.

Step 3: The dx2−y2orbital will gain 1 electron and the dz2orbital will lose 1

electron due to the energy difference. The new electron configuration becomes

(dxy )2(dxz )1(dyz )1(dx2−y2)1(dz2)1.

Step 4: Counting the number of unpaired electrons (those in the dx2−y2and

dz2orbitals), we find that there are 2 unpaired electrons in the complex.

2. The presence of unpaired electrons in a complex can lead to param-

agnetic behavior. Since the complex has 2 unpaired electrons, it will exhibit

paramagnetic behavior.

Question 7

Question

Consider a complex ion with a coordination number of 6 where the ligands form

an octahedral geometry. If the crystal field splitting energy (∆oct) is calculated

to be 800 cm−1, determine the energy difference between the t2gand egorbitals.

Solution

To find the energy difference between the t2gand egorbitals, we can use the

formula:

∆t2g−eg= 2∆oct

Step 1: Calculate the energy difference.

∆t2g−eg= 2∆oct

= 2 ×800 cm−1

= 1600 cm−1

Therefore, the energy difference between the t2gand egorbitals is 1600 cm−1.

Question 8

Question

Consider the complex ion [Co(H2O)6]3+. Determine the splitting pattern of the

d-orbitals when this complex forms and explain why this splitting occurs based

on Crystal Field Theory.

Solution

Step 1: The splitting pattern of the d-orbitals when the complex forms can be

determined using Crystal Field Theory (CFT). In an octahedral crystal field,

the d-orbitals split into two sets, with three orbitals having higher energy (eg)

and two orbitals having lower energy (t2g).

Step 2: The splitting arises because the d-orbitals experience a net repulsion

due to the negatively charged ligands surrounding the central metal ion. This

repulsion leads to an energy difference between the higher energy (eg) and lower

energy (t2g) sets of d-orbitals.

Step 3: The egorbitals experience greater repulsion because they point

directly along the axes towards the ligands, resulting in higher energy compared

to the t2gorbitals, which experience less repulsion.

Step 4: In the case of the [Co(H2O)6]3+ complex, the central metal ion is

Co(III) surrounded by six water ligands. This arrangement causes the d-orbitals

to split into eg(higher energy) and t2g(lower energy) sets.

Step 5: Therefore, when the [Co(H2O)6]3+ complex forms, the d-orbitals

split into two sets as per Crystal Field Theory: egwith higher energy and t2g

with lower energy.

Question 9

Question

Consider a complex with the formula [Co(H2O)6]3+. Determine the number of

unpaired electrons in this complex using Crystal Field Theory.

Solution

Crystal Field Theory helps us predict the number of unpaired electrons in a

transition metal complex by considering the splitting of d orbitals in the presence

of ligands.

Step 1: Determine the oxidation state of the central metal ion The

oxidation state of the cobalt ion (Co) in this complex is +3.

Step 2: Identify the electronic configuration of the cobalt ion The

electronic configuration of Co3+ is [Ar]3d6.

Step 3: Determine the electronic configuration after ligand field

splitting In an octahedral crystal field, the d orbitals split into two energy

levels. The d orbitals dz2and dx2−y2are at a higher energy level denoted as eg,

while the dxy ,dxz , and dyz orbitals are at a lower energy level denoted as t2g.

Step 4: Fill the orbitals with electrons according to the Aufbau

principle The electrons from the Co3+ ion will occupy the lower energy orbitals

first before pairing up:

eg:↑↑ t2g:↑↑↑

Step 5: Determine the number of unpaired electrons In this case,

there are 4 electrons unpaired in the t2gorbitals, resulting in 4 unpaired

electrons in the complex.

Question 10

Question

Consider a tetrahedral complex with four ligands. Calculate the crystal field

splitting energy (∆oct) for this complex. Assume the ligands are point charges

located at the corners of a regular tetrahedron centered at the metal ion.

Solution

To calculate the crystal field splitting energy (∆oct ) for a tetrahedral complex,

we need to consider the repulsion between the metal ion and the ligands, which

are point charges located at the corners of a regular tetrahedron centered at the

metal ion.

Step 1: Find the distance between the metal ion and the ligands.

Let rbe the distance between the metal ion and the ligands. For a regular

tetrahedron, the distance from the center to the corner is given by the formula:

r=a√3

where ais the length of a side of the tetrahedron.

Step 2: Calculate the interaction energy. The interaction energy be-

tween the metal ion and the ligands can be calculated using Coulomb’s law:

U=k·q1·q2

where kis Coulomb’s constant, q1and q2are the charges on the metal ion and

the ligands (respectively), and ris the distance between them.

For a tetrahedral complex, the metal ion has a charge of −4e(where eis the

charge of an electron) and each ligand has a charge of −e. Substituting these

values into the Coulomb’s law equation gives:

U=k·(−4e)·(−e)

Step 3: Calculate the crystal field splitting energy (∆oct). The

crystal field splitting energy (∆oct) is defined as the energy difference between

the energy of the system with the ligands approaching the metal ion along the

axes (high energy) and the system with the ligands approaching the metal ion

between the axes (low energy). For a tetrahedral complex, the crystal field

splitting energy can be calculated as:

∆oct =4U

Substitute the expression for Uinto the formula for ∆oct:

∆oct =4

3·k·(−4e)·(−e)

Now, plug in the value of rinto the equation to find ∆oct.

Question 11

Question

Consider a coordination complex with the formula [Co(NH3)6]3+. Assume that

the ligands are weak-field ligands. Determine the number of unpaired electrons

in the complex using crystal field theory.

Solution

Crystal field theory can be used to determine the number of unpaired electrons

in a coordination complex. In this theory, the ligand field splits the energy

levels of the metal d-orbitals. For weak-field ligands, the energy ordering of the

d-orbitals is as follows: t2g¡eg.

Step 1: Determine the electronic configuration of Co3+.Cobalt (Co)

has 27 electrons, so Co3+ has 24 electrons. The electronic configuration of Co3+

can be represented as [Ar]3d6.

Step 2: Determine the splitting of the d-orbitals by the ligand

field. In an octahedral field, the d-orbitals split into two sets of different energy

levels: t2g(lower energy) and eg(higher energy). For weak-field ligands, the

degenerate dorbitals are filled before any pairing occurs.

Step 3: Determine the distribution of electrons in the t2gand eg

orbitals. Since Co3+ has 6 electrons in the dsubshell, they will occupy the

t2gorbitals first. The distribution of electrons in the t2gand egorbitals is

(t2g)6(eg)0.

Step 4: Determine the number of unpaired electrons. To determine

the total number of unpaired electrons, we subtract the total number of electrons

in the t2gorbitals from the total number of electrons in the dsubshell. In this

case, 6−0 = 6. Therefore, the complex [Co(N H3)6]3+ has 6unpaired electrons.

Question 12

Question

Consider a complex with the formula [Co(H2O)6]3+. Using Crystal Field The-

ory, determine the number of unpaired electrons in this complex.

Solution

Step 1: Write the electron configuration of a neutral cobalt atom.

Co : [Ar]4s23d7

Step 2: Determine the electron configuration of the cobalt ion in the complex.

Since the complex is [Co(H2O)6]3+, the cobalt ion has a +3 charge. This means

it loses 3 electrons from its neutral state.

Co3+ : [Ar]3d6

Step 3: Determine the splitting of the d orbitals in the presence of the

ligands (H2O molecules). In an octahedral field, the d orbitals split into two

sets - t2gand eg. The three lower energy orbitals are t2gand the two higher

energy orbitals are eg.

Step 4: Fill in the electrons to the orbitals. Since cobalt has 6 electrons in

its 3d orbitals, they will fill the lower energy t2gset of orbitals first.

Step 5: Calculate the number of unpaired electrons. In the t2gset, there

are 3 orbitals available for electrons to occupy. Since each orbital can hold 2

electrons (one spin up and one spin down), the maximum number of electrons

that can occupy these orbitals is 6. Since all 6 t2gorbitals are filled, there are

0 unpaired electrons in this complex.

Question 13

Question

Consider a complex ion with the formula [CoCl4]2−.

1. Determine the oxidation state of the cobalt ion in this complex.

2. Draw the crystal field splitting diagram for this complex ion and label the

energy levels.

Solution

1. To determine the oxidation state of the cobalt ion, we need to consider the

charge of the complex ion as a whole and the charge carried by each chloride

ion. Since each chloride ion has a charge of −1, the total negative charge from

the four chloride ions is −4. Since the complex ion [CoCl4]2−has a charge of

−2, the cobalt ion must have a charge of +2 to balance the charges. Thus, the

oxidation state of the cobalt ion is +2.

2. Drawing the crystal field splitting diagram involves considering the in-

teraction between the d orbitals of the cobalt ion and the surrounding chloride

ligands. In an octahedral crystal field, the d orbitals split into two groups - the

lower energy group containing t2gorbitals and the higher energy group contain-

ing egorbitals.

The crystal field splitting diagram for [CoCl4]2−complex ion is as follows:

t2gt2g

The energy of the t2gorbitals is lower than that of the egorbitals due to the

interaction with the ligands. The labeling of the energy levels is as follows:



t2gorbitals: dxy ,dyz ,dzx



egorbitals: dx2−y2,dz2

This completes the crystal field splitting diagram for the [CoCl4]2−complex

ion.

Question 14

Question

Consider a tetrahedral coordination complex with a metal ion at the center.

Given that the crystal field splitting energy (∆0) is 800 cm−1, calculate the

energy gap between the t2gand eglevels in terms of ∆0.

Solution

To calculate the energy gap between the t2gand eglevels, we need to find the

difference in energy between the highest energy orbital in t2gand the lowest

energy orbital in eg.

Step 1: The energy ordering of the d-orbitals in a tetrahedral field is eg>

t2g. Thus, the t2gorbitals are lower in energy than the egorbitals by ∆0.

Step 2: The energy gap between the highest energy orbital in t2gand the

lowest energy orbital in egis equal to ∆0.

Therefore, the energy gap between the t2gand eglevels in terms of ∆0is

800 cm−1.

Question 15

Question

Consider a complex ion with a metal center in a tetrahedral ligand field. If the

crystal field splitting energy (∆t) is 0.6 times the crystal field splitting energy

in an octahedral field (∆o), calculate the wavelength of light absorbed when an

electron transitions from the t2gorbitals to the egorbitals within this complex.

Given: ∆o= 800 cm−1

Solution

Step 1: Calculate the crystal field splitting energy in the tetrahedral field.

∆t=4

9×∆o=4

9×800 = 355.56 cm−1

Step 2: Convert the energy difference to wavelength using the formula:

E=hc

where Eis the energy difference in Joules, his Planck’s constant (6.626 x 10−34

J s), cis the speed of light (3.00 x 108m/s), and λis the wavelength.

Step 3: Convert the crystal field splitting energy to Joules:

E= ∆t×100 ×1.986 ×10−23 J cm−1

E= 355.56 ×100 ×1.986 ×10−23 = 7.04 ×10−21 J

Step 4: Calculate the wavelength of light absorbed:

λ=hc

λ=6.626 ×10−34 ×3.00 ×108

7.04 ×10−21 = 2.82 ×10−5m

Therefore, the wavelength of light absorbed when an electron transitions

from the t2gorbitals to the egorbitals is 2.82 ×10−5m.

Question 16

Question

Consider a transition metal complex with the formula [Mn(H2O)6]2+. Deter-

mine the energy difference between the t2gand egorbitals in this complex using

Crystal Field Theory.

Solution

Crystal field theory describes the interaction between transition metal ions and

ligands by considering the splitting of the d orbitals. In an octahedral field, the

d orbitals split into two sets: the lower energy t2gset and the higher energy eg

set.

Step 1: Assigning the number of electrons and determining the

oxidation state of the metal For [Mn(H2O)6]2+, we know that the complex

has an overall charge of +2. Each water molecule is neutral and Mn has an

oxidation state of x, so we’ll have:

x+ 6(0) = +2

x= +2

This means Mn is in the +2 oxidation state with 5 d electrons.

Step 2: Drawing the splitting diagram In an octahedral crystal field,

the d orbitals split with the t2gset (dxy, dxz, dyz) lower in energy and the egset

(dx2−y2, dz2)higherinenergy.Sincewehave5delectrons, we′llfilltheorbitalsfollowingHund′srule.

Step 3: Calculating the energy difference The energy difference be-

tween the t2gand egorbitals is the energy required to promote an electron from

the lower energy t2glevel to the higher energy eglevel. This energy difference

(∆o) can be calculated using the crystal field splitting parameter, ∆:

∆o= 10Dq =4

9∆

where Dq is the splitting energy per ligand field.

Step 4: Calculating the crystal field stabilization energy (CF SE)

The crystal field stabilization energy is the energy difference between filling the

d orbitals in a complex and the energy when there are no ligands present. For

an octahedral complex, the CF SE is given by:

CF SE =−0.4∆o×n

where nis the total number of d electrons (5 in this case). This energy

difference corresponds to the stability gained from electron-ligand interactions.

Question 17

Question

An octahedral complex is formed between a metal ion and ligands in a crystal

field. Calculate the crystal field stabilization energy (CFSE) for the complex

[Co(H2O)6]3+. Given: ∆o= 9600 cm−1.

Solution

Step 1: Determine the number of electrons and their configuration in the com-

plex.



Cobalt (Co) has an electronic configuration of [Ar] 3d74s2.



In the complex [Co(H2O)6]3+, cobalt is in the +3 oxidation state, so

it loses three electrons. The electronic configuration of the complex ion

becomes [Co(H2O)6]3+ = [Ar] 3d6.

Step 2: Determine the crystal field splitting energy (∆o) and the CFSE.



The crystal field splitting energy (∆o) = 9600 cm−1.



The CFSE for an octahedral complex is given by: CFSE = −0.4×∆o×n,

where nis the number of electrons in the t2gorbitals.



In an octahedral complex with 6 ligands, the complex ion has 6 electrons

distributed in the t2gorbitals.



Therefore, the CFSE = −0.4×9600 ×6 = −23040 cm−1.

Final Answer: The crystal field stabilization energy (CFSE) for the com-

plex [Co(H2O)6]3+ is −23040 cm−1.

Question 18

Question

Consider a transition metal complex with an octahedral geometry. Determine

the crystal field splitting energy (∆o) in terms of the ligand field stabilization

energy (LF SE) and the pairing energy (P).

Solution

To find the crystal field splitting energy (∆o) in terms of the ligand field stabi-

lization energy (LF SE) and the pairing energy (P), we can use the relationship

between these quantities for an octahedral field.

Step 1: The ligand field stabilization energy (LF SE) is given by:

LF SE =−0.4∆oNc−0.6P(Nc −2n)

where: ∆o= Crystal field splitting energy between t2gand egorbitals Nc=

Total number of ligands n= Number of electrons in t2gorbitals P= Pairing

energy

Step 2: For an octahedral field, Nc= 6. Substituting Nc= 6 into the

equation, we get:

LF SE =−2.4∆o−0.6P(6 −2n)

Step 3: The total number of electrons in an octahedral field is 6. If pairing

occurs, n= 3 (2 in t2g, 1 in eg). In this case, the LF SE becomes:

LF SE =−2.4∆o−0.6P(6 −2∗3) = −2.4∆o−1.2P

Step 4: Equate the LF SE to the experimentally determined value for oc-

tahedral complexes, which is usually known:

LF SE =−2.4∆o−1.2P

Step 5: Solve the equation for ∆o:

−2.4∆o−1.2P= Experimental LFSE

∆o=Experimental LFSE + 1.2P

−2.4

Therefore, the crystal field splitting energy (∆o) in terms of the ligand field

stabilization energy (LF SE) and the pairing energy (P) is given by:

∆o=Experimental LFSE + 1.2P

−2.4

Question 19

Question

Consider a transition metal complex with the formula [Fe(NH3)5(NO2)]Br2.

a. Determine the oxidation state of iron in this complex.

b. Use crystal field theory to predict the number of unpaired electrons in

the Fe(II) complex.

Solution

a. To determine the oxidation state of iron in the complex, we need to consider

the charges of the ligands. The NO2 ligand carries a charge of -1, so the total

charge from the ligands is -1 (from NO2) + 2−1 (from two bromide ions) = -3.

Therefore, the oxidation state of iron can be calculated using the formula:

oxidation state of Fe = charge of complex −total charge from ligands

Given that the overall complex is neutral, the oxidation state of iron is:

oxidation state of Fe = 0 −(−3) = +3

b. To predict the number of unpaired electrons in the Fe(III) complex using

crystal field theory, we will consider the electronic configuration of Fe(III) and

the number of electrons affected by the crystal field splitting.

The electronic configuration of Fe(III) is [Ar]3d5. In an octahedral crystal

field, the five d-orbitals split into three lower energy orbitals (t2gset) and two

higher energy orbitals (egset).

Step 1: Assign electrons to the d-orbitals before splitting.

Since Fe(III) has 5 electrons, we fill the d-orbitals as follows: - t2gset: ↑↓,

↑↓↑Step 2: Consider the crystal field splitting in octahedral coordination.

In an octahedral field, the t2gorbitals are lower in energy and will be filled

before the egorbitals. Since all the electrons pair up in the t2gset, there are no

unpaired electrons in this complex.

Therefore, the Fe(III) complex has zero unpaired electrons.

Question 20

Question

Consider a square planar complex with a central metal ion in the +3 oxidation

state. The complex is diamagnetic and has a high-spin electronic configuration.

Given this information, draw the d-orbital splitting diagram for the complex

and identify the ligand responsible for the high-spin configuration.

Solution

To draw the d-orbital splitting diagram for a square planar complex with a

central metal ion in the +3 oxidation state, we first need to identify the d-

electron configuration. For a +3 oxidation state, the metal ion has 3 fewer

electrons than the neutral atom. Since we are dealing with a high-spin complex,

all d-orbitals will be fully filled with electrons before pairing, resulting in a d3

configuration.

Step 1: Draw the initial d-orbital energy levels without any ligand influence:

Orbital Energy Level

dx2−y2

dz2(Higher energy)

dxy

dyz (Lower energy)

dxz

Step 2: Introduce the ligand field splitting by considering the effect of

ligands on the d-orbitals. In a square planar geometry, the dz2and dx2−y2

orbitals experience greater repulsion than the dxy , dyz , and dxz orbitals due to

their orientation along the axes pointing towards the ligands.

The splitting results in a higher energy level containing dz2and dx2−y2or-

bitals, and a lower energy level containing dxy , dyz , and dxz orbitals.

Orbital Energy Level

dz2, dx2−y2(Higher energy)

dxy , dyz , dxz (Lower energy)

Step 3: Identify the ligands responsible for the high-spin configuration.

Ligands that cause a small ligand field splitting lead to high-spin complexes.

Tangentially-bonding ligands, such as weak field ligands, generally cause small

splitting, resulting in high-spin configurations. Some examples of weak field

ligands include Cl−, F−, and H2O.

Therefore, the ligand responsible for the high-spin configuration in the square

planar complex is most likely a weak field ligand.

Question 21

Question

Explain how crystal field theory can be used to predict the relative energies of

the d orbitals in a transition metal complex. Use the example of an octahedral

coordination complex to illustrate your explanation.

Solution

Crystal field theory is a model used to describe the splitting of d orbitals in

transition metal complexes based on the interactions between the metal ion

and surrounding ligands. In an octahedral coordination complex, six ligands

surround the central metal ion, leading to the splitting of the five d orbitals into

two different energy levels.

Step 1: Metal-ligand interactions In an octahedral coordination com-

plex, the ligands approach along the x, y, and z axes, towards the metal ion

located at the origin. The repulsion between the ligands and the metal’s d

orbitals causes the d orbitals to split into two sets: the t2g set and the eg set.

Step 2: Energy difference The t2g set consists of the lower energy dxy,

dyz, and dxz orbitals, which experience less repulsion from the ligands due to

their orientation along the axes where the ligands are not located. The eg set

consists of the higher energy dx2−y2anddz2orbitals, whichexperiencegreaterrepulsionastheypointdirectlytowardstheligandsalongtheaxes.

Step 3: Relative energies The splitting of the d orbitals in an octahedral

field leads to the t2g set being lower in energy compared to the eg set. The

energy separation between the t2g and eg sets is denoted as o, with o being the

crystal field splitting energy.

Therefore, crystal field theory predicts that the relative energies of the d

orbitals in an octahedral coordination complex are as follows: dxy, dyz, dxz ¡

dx2−y2, dz2.

This understanding of the relative energy levels of the d orbitals is crucial for

interpreting the electronic spectra and magnetic properties of transition metal

complexes.

Question 22

Question

Consider a complex with the formula [Co(N H3)6]3+.

1. Using crystal field theory, determine the number of unpaired electrons in

the complex.

2. Identify the geometry of the complex.

Solution

1. To determine the number of unpaired electrons in the complex, we need to

consider the splitting of the dorbitals in the presence of the ligands. For an

octahedral field, the dorbitals split into two groups: t2g(dxy ,dxz ,dyz ) and eg

(dx2−y2,dz2).

The number of unpaired electrons can be calculated using the formula:

number of unpaired electrons = (neg−nt2g)

where negis the number of electrons in the egorbitals and nt2gis the number

of electrons in the t2gorbitals.

In the [Co(N H3)6]3+ complex, cobalt (Co) has a 3+ charge, meaning it loses

3 electrons. Cobalt is in the +3 oxidation state, so it has the electronic configu-

ration 3d6. Co(III) is in a high spin configuration, meaning all the electrons go

into separate orbitals before pairing up.

Therefore, in an octahedral field, the electron configuration will be t6

2ge0

Calculating the number of unpaired electrons:

number of unpaired electrons = (0 −6) = 6

2. The geometry of the complex can be determined based on the number of

unpaired electrons. Since there are 6 unpaired electrons, the complex is in an

octahedral geometry.

Question 23

Question

A transition metal complex has a certain number of unpaired electrons in its

d orbitals. Using crystal field theory, explain how the number of unpaired

electrons is affected when the ligands are changed from weak-field ligands to

strong-field ligands.

Solution

1. Weak-field Ligands: In the presence of weak-field ligands, the d orbitals

of the transition metal ion are stabilized due to repulsion from ligand electrons.

This leads to a small energy gap between the t2g and eg orbitals and results in

a high-spin configuration with more unpaired electrons.

2. Strong-field Ligands: When strong-field ligands are introduced, the

d orbitals are split to a greater extent due to the stronger interactions with

ligand electrons. This results in a larger energy gap between the t2g and eg

orbitals, promoting the pairing of electrons to achieve a lower overall energy.

As a result, the complex typically adopts a low-spin configuration with fewer

unpaired electrons.

Therefore, when transitioning from weak-field ligands to strong-field ligands,

the number of unpaired electrons in a transition metal complex is generally

reduced due to the promotion of electron pairing in the stronger crystal field

environment.

Question 24

Question

Consider a complex ion with the formula [Cr(HO)]3+ in an octahedral crystal

field environment. Calculate the crystal field splitting energy (∆oct) in joules

per mole given that the experimental value for the energy of absorption of the

complex in a spectrophotometer is 15,000 cm−1.

Solution

To calculate the crystal field splitting energy (∆oct), we need to convert the

given value from cm−1to joules per mole. The relationship between the energy

in cm−1and the energy in joules is given by:

E=hc¯ν

where: E= energy in joules, h= Planck’s constant (6.62607015×10−34 J s),

c= speed of light (2.998 ×108m/s), ¯ν= frequency in Hz.

We are given that the experimental value for the absorption energy is 15,000

cm−1. To convert this to Hz, we use the conversion factor 1 cm−1= 2.998 ×

1010 Hz. Thus, the frequency in Hz is:

¯ν= 15,000 cm−1×2.998 ×1010 Hz/cm = 4.494 ×1014 Hz

Now, we can calculate the energy in joules:

E= (6.62607015×10−34 J s)(2.998×108m/s)(4.494×1014 Hz) = 8.998×10−19 J

Next, we convert this energy to joules per mole by multiplying by Avogadro’s

number:

8.998 ×10−19 J×6.022 ×1023 molecules/mol = 5.419 ×105J/mol

Therefore, the crystal field splitting energy (∆oct) for the complex ion is

5.419 ×105J/mol.

Question 25

Question

Consider a square planar complex of the type [M(A)2(B)X], where A and B

are monodentate ligands and X is a bidentate ligand. Determine the number

of unpaired electrons in the d-orbitals of the central metal ion based on crystal

field theory.

Solution

1. Identify the geometry of the complex: Given that the complex is square

planar, the d-orbitals of the central metal ion will split into two sets depending

on their orientation relative to the ligand field.

2. Determine the electron configuration of the metal ion: Assuming the

central metal ion is in the +2 oxidation state, the electron configuration could

be written as [Xe]4dn−2, where n is the number of d-electrons.

3. Analyze the splitting of d-orbitals: In a square planar complex, the d-

orbitals split into two sets: the lower energy t2g set and the higher energy eg

set. The t2g set contains three orbitals (dxy, dyz, dxz) while the eg set contains

two orbitals (dx2−y2, dz2).

4. Determine the electron configuration after splitting: In a square planar

complex, the d-electron configuration will distribute among the t2g and eg sets

depending on the ligand field strength.

5. Count the number of unpaired electrons: Based on the electron configu-

ration distribution, count the number of unpaired electrons in the d-orbitals to

determine the magnetic properties of the complex.

Question 26

Question

Consider a coordination complex with an octahedral geometry. The complex is

formed by the ligand CN−and the metal ion Fe3+. Calculate the crystal field

stabilization energy (CFSE) for this complex. Given: ∆o= 70 kJ/mol.

Solution

Step 1: Determine the number of electrons in the metal ion Fe3+. Since Fe has

an atomic number of 26, Fe3+ has 23 electrons.

Step 2: Determine the electron configuration of Fe3+. The electron configu-

ration of Fe is [Ar] 3d64s2. With 3 electrons removed to form Fe3+ , the electron

configuration becomes [Ar] 3d5.

Step 3: Determine the number of d electrons that participate in crystal field

splitting. In an octahedral field, the 5 d electrons of Fe3+ will split into two

groups: 3 in the lower energy t2gorbitals and 2 in the higher energy egorbitals.

Step 4: Calculate the crystal field stabilization energy (CFSE) using the

formula: CFSE = -0.4 ∆o(neg- nt2g) where ∆ois the splitting energy and neg

and nt2gare the number of electrons in the egand t2gorbitals, respectively.

Step 5: Substitute the values into the formula. CFSE = -0.4 * 70 kJ/mol *

(2 - 3) = -0.4 * (-70) kJ/mol = 28 kJ/mol

Therefore, the crystal field stabilization energy (CFSE) for the Fe3+ complex

with CN−ligands is 28 kJ/mol.

Question 27

Question

Consider a square planar complex with a metal ion in the +2 oxidation state. If

this complex absorbs light in the visible region, would you expect the absorption

band to be broad or sharp? Justify your answer.

Solution

To determine whether the absorption band of a square planar complex with a

metal ion in the +2 oxidation state would be broad or sharp, we need to consider

the Crystal Field Theory.

Step 1: Understand Crystal Field Theory Crystal Field Theory (CFT)

is used to describe the bonding and properties of coordination complexes. In

square planar complexes, the metal ion is surrounded by four ligands in a plane.

Step 2: Electronic Transition in Square Planar Complexes In square

planar complexes, the d orbitals split into two sets of different energy levels: the

t2g set (lower in energy) and the eg set (higher in energy) due to the effect of

ligands. Electronic transitions between these energy levels give rise to absorption

bands in the visible region.

Step 3: Energy Difference Between t2g and eg Orbitals In square

planar complexes, the energy difference between the t2g and eg orbitals is rel-

atively small compared to octahedral complexes. This leads to a lower energy

gap and broader absorption bands in the visible region.

Step 4: Absorption Band in Square Planar Complexes Due to the

smaller energy gap between the t2g and eg orbitals in square planar complexes,

the absorption band is expected to be broad rather than sharp. This is because a

broader range of wavelengths can be absorbed by electrons transitioning between

these energy levels.

Therefore, in a square planar complex with a metal ion in the +2 oxidation

state, we would expect the absorption band to be broad.

Question 28

Question

In a crystal field theory study, a transition metal complex is formed with a

metal ion in a coordination sphere of six ligands. The complex shows a high-

spin configuration in an octahedral field. What can you infer about the energy

ordering of the d-orbitals in this complex based on crystal field theory principles?

Solution

To determine the energy ordering of the d-orbitals in an octahedral crystal field,

we need to consider the splitting of the d-orbitals by the surrounding ligands.

In an octahedral crystal field, the d-orbitals are split into two sets; the lower

energy set (t2g) containing the dxy, dyz, and dzx orbitals, and the higher energy

set (eg) containing the dx2-y2 and dz2 orbitals.

Step 1: In a high-spin octahedral complex, the ligand field is weak enough

that the electrons occupy the low energy t2g set before the higher energy eg set.

Step 2: Since the complex is in a high-spin configuration, all six ligand

orbitals are singly occupied with parallel spins before any pairing occurs. This

indicates that the energy gap between the t2g and eg sets is relatively small.

Step 3: Therefore, based on crystal field theory principles, we can infer that

in this high-spin octahedral complex, the energy of the t2g set is lower than that

of the eg set. The dxy, dyz, and dzx orbitals are lower in energy and fill before

the dx2-y2 and dz2 orbitals.

Thus, in this complex, the energy ordering of the d-orbitals based on crystal

field theory principles is dxy, dyz, dzx (t2g) ¡ dx2-y2, dz2 (eg).

Question 29

Question

For a transition metal complex with the coordination number of 6, determine

the number of unpaired electrons based on Crystal Field Theory if it has the

following splitting pattern: E=−0.4∆o.

Solution

Crystal Field Theory (CFT) can be used to predict the number of unpaired

electrons in transition metal complexes based on the splitting pattern of d-

orbitals in a crystal field environment.

Step 1: Identify the Splitting Pattern The splitting pattern given in

the question is E=−0.4∆o, which corresponds to an octahedral field.

Step 2: Determine the Number of Unpaired Electrons In an octa-

hedral field, the energy difference between the t2g and eg sets is ∆o.

According to CFT, for an octahedral field, the number of unpaired electrons

(n) can be calculated using the formula: n=1

2(5 −x), where xis the number

of electrons in the t2g set.

Given that the splitting pattern is E=−0.4∆o, we can deduce that x= 4

(since the t2g set is lower in energy).

Therefore, the number of unpaired electrons is: n=1

2(5 −4) = 1

2(1) = 0.5.

However, since electrons are not fractional, we round to the nearest whole

number. Therefore, the complex has 1 unpaired electron according to CFT

in an octahedral field with E=−0.4∆o.

Question 30

Question

An octahedral complex is formed when four chloride ions and two water molecules

coordinate with a central metal ion. Calculate the crystal field stabilization en-

ergy (CFSE) in this complex if the energy required to pair two electrons in the

t2gorbitals is −0.4 times the energy required to split the egorbitals. Assume

the pairing energy is Pand the splitting energy for egis ∆o.

Solution

Step 1: Calculate the CFSE for the complex. The CFSE can be calculated using

the formula:

CFSE = −0.4P×2∆o

Step 2: Substitute the given values into the formula. Given that the energy

required to pair electrons in t2gorbitals is −0.4P, and the energy required to

split egorbitals is ∆o, we have:

CFSE = −0.4P×2∆o

Step 3: Simplify the expression.

CFSE = −0.8P∆o

Therefore, the crystal field stabilization energy (CFSE) in this complex is

−0.8P∆o.

Question 31

Question

Consider a coordination complex with the formula [Co(H2O)6]3+ in an octahe-

dral crystal field. Determine the number of unpaired electrons in the complex

using Crystal Field Theory.

Solution

Crystal Field Theory (CFT) describes the splitting of d orbitals in coordination

complexes due to the electrostatic interaction with ligands.

Step 1: Identify the number of d-electrons The cobalt ion in the

complex is in the +3 oxidation state, which gives it a total of 7 d-electrons

(Co3+ is [Ar] 3d6).

Step 2: Determine the electronic configuration in the octahedral

crystal field In an octahedral crystal field, the d-orbitals split into two sets:

one with lower energy (t2g) and one with higher energy (eg). The electrons fill

the lower energy orbitals first before pairing up in the higher energy orbitals.

For a complex with 6 ligands (e.g., H2O), the d-orbitals split as follows: -

t2g: dxy , dxz , dyz - eg: dz2, dx2−y2

Step 3: Fill the d-electrons into the energy levels Since we have 7

d-electrons in Co3+, we distribute them among the t2gand eglevels as follows:

- t2g(lower energy level): ↑↓↑↑ (dxy , dxz , dyz ; 3 electrons) - eg(higher energy

level): ↑↑↑ (dz2, dx2−y2; 3 electrons)

Step 4: Count the number of unpaired electrons In the complex

[Co(H2O)6]3+, there are 3 unpaired electrons in the egset of d-orbitals.

Therefore, the number of unpaired electrons in the complex is 3.

Question 32

Question

Consider a coordination complex with the formula [Co(NH3)6]Cl3. Determine

the number of unpaired electrons in the complex using crystal field theory.

Solution

Crystal field theory describes the splitting of d orbitals in a transition metal com-

plex due to the presence of ligands. In octahedral complexes like [Co(N H3)6]Cl3,

the d orbitals split into two sets of different energy levels: three lower energy

orbitals (t2g) and two higher energy orbitals (eg).

Step 1: Determine the oxidation state of Co. To find the oxidation

state of Co, we can set up an equation using the charges of the ligands and the

overall charge of the complex. Let X be the oxidation state of Co.

X+ 6(0) + 3(−1) = 0

X−3=0

X= +3

Step 2: Count the number of d-electrons in Co3+.Cobalt in the +3

oxidation state has 7 d-electrons.

Step 3: Determine the number of unpaired electrons. In an oc-

tahedral field, the d electrons will fill the lower energy t2gorbitals first. The

distribution of electrons will be as follows:

t2g:↑↑↑↓↓↓

Therefore, there are 3 unpaired electrons in the t2gset.

Answer: The coordination complex [Co(NH3)6]Cl3has 3 unpaired elec-

trons based on crystal field theory.

Question 33

Question

Consider a complex with the molecular formula [Co(N H3)6]3+. Determine the

crystal field splitting energy (∆) in units of eV given that the absorption wave-

length for the complex is 645 nm. Assume the absorption is due to a T2g→Eg

transition.

Solution

Step 1: Convert absorption wavelength to energy in eV . Step 2: Use the rela-

tionship between energy and ∆ to find the crystal field splitting energy.

Step 1: The energy of a photon can be calculated using the equation:

E=hc

where his the Planck constant (6.626×10−34 J·s), cis the speed of light (3.00×

108m/s), and λis the wavelength of the absorbed light in meters. Converting

the absorption wavelength to meters:

λ= 645 nm = 645 ×10−9m= 6.45 ×10−7m

Calculating the energy:

E=(6.626 ×10−34 J·s)×(3.00 ×108m/s)

6.45 ×10−7m≈9.74 ×10−19 J

Converting energy to electron volts:

1eV = 1.602 ×10−19 J

E≈9.74 ×10−19 J

1.602 ×10−19 J/eV ≈6.08 eV

Step 2: The crystal field splitting energy (∆) relates to the energy of the

absorbed light (E) according to the equation:

E=4

3∆

Substitute E= 6.08 eV into the equation to solve for ∆:

6.08 eV =4

3∆

∆ = 3

4×6.08 eV

∆≈4.56 eV

Therefore, the crystal field splitting energy (∆) for the complex [Co(NH3)6]3+

is approximately 4.56 eV .

Question 34

Question

Consider a complex with an octahedral geometry where the ligands are in the

xy-plane. If a metal ion with a d3configuration forms this complex, what will

be the ground-state electron configuration according to Crystal field theory?

Solution

To determine the ground-state electron configuration for a metal ion with a d3

configuration in an octahedral field where the ligands are in the xy-plane, we

need to consider how the crystal field splits the dorbitals.

Step 1: Identify the delectrons The metal ion has 3 delectrons in this

case.

Step 2: Split the dorbitals In an octahedral crystal field with ligands in

the xy-plane, the dorbitals will split into two sets: eg(which includes dx2−y2

and dz2) and t2g(which includes dxy ,dyz , and dxz ).

Step 3: Assign the delectrons According to Hund’s rule, the electrons

will fill the egorbitals before filling the t2gorbitals.

Step 4: Determine the ground-state electron configuration Since

we have 3 electrons, they will fill the egorbitals first. Thus, the ground-state

electron configuration will be d3with 3 electrons in the egorbitals (dx2−y2and

dz2).

Therefore, the ground-state electron configuration according to Crystal field

theory for a metal ion with a d3configuration in an octahedral field with ligands

in the xy-plane is t3

2g.

Question 35

Question

Consider a complex with the central metal ion Co2+ in a coordination environ-

ment with octahedral symmetry. The ligands are arranged in such a way that

the magnitude of the crystal field splitting parameter, ∆oct, is given by 14000

cm−1. Calculate the wavelength (in nm) of light absorbed when an electron

transitions from the t2glevel to the eglevel in this complex.

Solution

Step 1: Calculate the energy difference between the t2gand eglevels. This

energy difference is equal to ∆oct. Step 2: Use the relationship E=hc/λ to

find the wavelength of light absorbed, where Eis the energy difference, his

Planck’s constant, cis the speed of light, and λis the wavelength. Step 3:

Substitute the calculated energy difference into the equation from Step 2 to find

the wavelength in nm.

complex observed by the human eye is the complementary color of the absorbed

light. For example, if a complex absorbs light in the red region of the spectrum,

it appears green to the observer.

This explains how crystal field theory provides a framework for understand-

ing the colors exhibited by transition metal complexes based on the splitting of

d orbitals and the absorption of light energy.

Question 2

Question

Consider a complex with the formula [Co(NH3)6]3+. Determine the crystal field

stabilization energy (CFSE) for this complex using Crystal Field Theory.

Solution

1. The CFSE for an octahedral complex is given by the equation:

CFSE = ∆o·n

Where ∆ois the crystal field splitting energy and nis the number of elec-

trons.

2. In an octahedral complex like [Co(NH3)6]3+, cobalt (Co) has an oxidation

state of +3, meaning it has 6 d-electrons.

3. The crystal field splitting energy for an octahedral field is typically around

0.6∆o

4. For an octahedral field, the d-orbitals split into two sets of energy levels:

eg(higher energy) and t2g(lower energy).

5. In the [Co(NH3)6]3+ complex, all 6 d-electrons are paired up in the t2g

set, resulting in a total CFSE of 0.

6. Therefore, the crystal field stabilization energy (CFSE) for the complex

[Co(NH3)6]3+ is 0 .

Question 3

Question

An octahedral complex is formed from a metal ion with a d4electron configura-

tion. Determine the number of unpaired electrons using Crystal Field Theory.

Solution

Crystal Field Theory (CFT) can be used to determine the number of unpaired

electrons in transition metal complexes based on the splitting of the d orbitals

in the presence of ligands. In an octahedral field, the d orbitals split into two

sets: the lower energy t2gset and the higher energy egset.

Given that the metal ion has a d4electron configuration, all the d orbitals

will be singly occupied. We can use the energy level diagram of an octahedral

field to determine the number of unpaired electrons.

Step 1: Fill the d orbitals with the 4 electrons. - The d orbitals are filled

as: dx2−y2, dz2, dxy , dxz , dyz .

Step 2: Determine the energy splitting in an octahedral field. - In an

octahedral field, the d orbitals split into two sets: t2g(lower energy) and eg

(higher energy).

t2geg

Step 3: Count the number of electrons in the t2gset for the number of

unpaired electrons. - Since all d orbitals are singly occupied, there are 1 electron

in each of the dx2−y2, dz2, dxy orbitals, giving a total of 3 unpaired electrons.

Thus, in an octahedral complex with a d4electron configuration, there are

3 unpaired electrons according to Crystal Field Theory.

Question 4

Question

Explain how crystal field theory accounts for the color of transition metal com-

plexes. Consider a complex with an octahedral geometry.

Solution

Crystal field theory is a model used to describe the electronic structure and

properties of transition metal complexes. It focuses on the interaction between

the metal d orbitals and the ligands surrounding the metal ion.

Step 1: In an octahedral complex, the ligands approach along the axes

between the d orbital lobes. This results in a splitting of the five degenerate d

orbitals into two sets of different energies: the t2gset lower in energy and the

egset higher in energy.

Step 2: When light is absorbed by the complex, an electron from the lower

energy t2gset can be promoted to the higher energy egset. The energy required

for this promotion corresponds to the color of light absorbed.

Step 3: The absorbed light appears as the complementary color to the

absorbed light. For example, if a complex absorbs light in the green region of

the spectrum, it will appear red to the human eye.

Step 4: The magnitude of the energy gap between the t2gand egsets is

determined by the nature of the ligands attached to the metal ion. Strong-field

ligands cause a larger energy gap, resulting in absorption of light in the visible

range and the appearance of complementary colors.

Step 5: Therefore, the color of transition metal complexes can be explained

by crystal field theory based on the splitting of d orbitals and the absorption of

light corresponding to the energy gap between these levels.

Question 5

Question

Consider a complex with the formula [Co(H2O)6]3+. Using Crystal Field The-

ory, determine the number of unpaired electrons in this complex.

Solution

Step 1: Identify the oxidation state of cobalt. Since the complex is [Co(H2O)6]3+,

the oxidation state of cobalt is +3.

Step 2: Determine the electronic configuration of Co3+. Cobalt has 27

electrons. Co3+ will have 24 electrons, which can be filled in the following way:

1s22s22p63s23p64s23d6

Step 3: Determine the number of unpaired electrons using crystal field the-

ory. In an octahedral crystal field, the five d orbitals split into two levels: the

lower energy level (t2g) which contains three orbitals, and the higher energy

level (eg) which contains two orbitals.

Since all six water ligands are weak ligands, the crystal field splitting will

be small. According to Crystal Field Theory, electrons will first fill the t2glevel

before pairing up in the eglevel.

Since there are 6 electrons in the t2glevel, all three orbitals will be filled,

leading to 0 unpaired electrons.

Therefore, the [Co(H2O)6]3+ complex has 0 unpaired electrons.

Question 6

Question

Consider a square planar complex with a metal ion in the +2 oxidation state.

Given that the dx2−y2orbital is most affected by crystal field splitting in this

geometry:

1. Determine the number of unpaired electrons in the complex.

2. Predict the type of magnetic behavior exhibited by the complex.

Solution

1. To determine the number of unpaired electrons in the square planar complex,

we must first consider the electron configurations before and after the crystal

field splitting. In the +2 oxidation state, the metal ion has 2 delectrons.

Step 1: Before crystal field splitting, the electron configuration is (dxy )2(dxz )1(dyz )1(dx2−y2)2(dz2)0.

Step 2: After crystal field splitting, the dx2−y2orbital is most affected. The

electrons will fill the lower energy orbitals first, leading to a splitting of the

dx2−y2orbital.

Step 3: The dx2−y2orbital will gain 1 electron and the dz2orbital will lose 1

electron due to the energy difference. The new electron configuration becomes

(dxy )2(dxz )1(dyz )1(dx2−y2)1(dz2)1.

Step 4: Counting the number of unpaired electrons (those in the dx2−y2and

dz2orbitals), we find that there are 2 unpaired electrons in the complex.

2. The presence of unpaired electrons in a complex can lead to param-

agnetic behavior. Since the complex has 2 unpaired electrons, it will exhibit

paramagnetic behavior.

Question 7

Question

Consider a complex ion with a coordination number of 6 where the ligands form

an octahedral geometry. If the crystal field splitting energy (∆oct) is calculated

to be 800 cm−1, determine the energy difference between the t2gand egorbitals.

Solution

To find the energy difference between the t2gand egorbitals, we can use the

formula:

∆t2g−eg= 2∆oct

Step 1: Calculate the energy difference.

∆t2g−eg= 2∆oct

= 2 ×800 cm−1

= 1600 cm−1

Therefore, the energy difference between the t2gand egorbitals is 1600 cm−1.

Question 8

Question

Consider the complex ion [Co(H2O)6]3+. Determine the splitting pattern of the

d-orbitals when this complex forms and explain why this splitting occurs based

on Crystal Field Theory.

Solution

Step 1: The splitting pattern of the d-orbitals when the complex forms can be

determined using Crystal Field Theory (CFT). In an octahedral crystal field,

the d-orbitals split into two sets, with three orbitals having higher energy (eg)

and two orbitals having lower energy (t2g).

Step 2: The splitting arises because the d-orbitals experience a net repulsion

due to the negatively charged ligands surrounding the central metal ion. This

repulsion leads to an energy difference between the higher energy (eg) and lower

energy (t2g) sets of d-orbitals.

Step 3: The egorbitals experience greater repulsion because they point

directly along the axes towards the ligands, resulting in higher energy compared

to the t2gorbitals, which experience less repulsion.

Step 4: In the case of the [Co(H2O)6]3+ complex, the central metal ion is

Co(III) surrounded by six water ligands. This arrangement causes the d-orbitals

to split into eg(higher energy) and t2g(lower energy) sets.

Step 5: Therefore, when the [Co(H2O)6]3+ complex forms, the d-orbitals

split into two sets as per Crystal Field Theory: egwith higher energy and t2g

with lower energy.

Question 9

Question

Consider a complex with the formula [Co(H2O)6]3+. Determine the number of

unpaired electrons in this complex using Crystal Field Theory.

Solution

Crystal Field Theory helps us predict the number of unpaired electrons in a

transition metal complex by considering the splitting of d orbitals in the presence

of ligands.

Step 1: Determine the oxidation state of the central metal ion The

oxidation state of the cobalt ion (Co) in this complex is +3.

Step 2: Identify the electronic configuration of the cobalt ion The

electronic configuration of Co3+ is [Ar]3d6.

Step 3: Determine the electronic configuration after ligand field

splitting In an octahedral crystal field, the d orbitals split into two energy

levels. The d orbitals dz2and dx2−y2are at a higher energy level denoted as eg,

while the dxy ,dxz , and dyz orbitals are at a lower energy level denoted as t2g.

Step 4: Fill the orbitals with electrons according to the Aufbau

principle The electrons from the Co3+ ion will occupy the lower energy orbitals

first before pairing up:

eg:↑↑ t2g:↑↑↑

Step 5: Determine the number of unpaired electrons In this case,

there are 4 electrons unpaired in the t2gorbitals, resulting in 4 unpaired

electrons in the complex.

Question 10

Question

Consider a tetrahedral complex with four ligands. Calculate the crystal field

splitting energy (∆oct) for this complex. Assume the ligands are point charges

located at the corners of a regular tetrahedron centered at the metal ion.

Solution

To calculate the crystal field splitting energy (∆oct ) for a tetrahedral complex,

we need to consider the repulsion between the metal ion and the ligands, which

are point charges located at the corners of a regular tetrahedron centered at the

metal ion.

Step 1: Find the distance between the metal ion and the ligands.

Let rbe the distance between the metal ion and the ligands. For a regular

tetrahedron, the distance from the center to the corner is given by the formula:

r=a√3

where ais the length of a side of the tetrahedron.

Step 2: Calculate the interaction energy. The interaction energy be-

tween the metal ion and the ligands can be calculated using Coulomb’s law:

U=k·q1·q2

where kis Coulomb’s constant, q1and q2are the charges on the metal ion and

the ligands (respectively), and ris the distance between them.

For a tetrahedral complex, the metal ion has a charge of −4e(where eis the

charge of an electron) and each ligand has a charge of −e. Substituting these

values into the Coulomb’s law equation gives:

U=k·(−4e)·(−e)

Step 3: Calculate the crystal field splitting energy (∆oct). The

crystal field splitting energy (∆oct) is defined as the energy difference between

the energy of the system with the ligands approaching the metal ion along the

axes (high energy) and the system with the ligands approaching the metal ion

between the axes (low energy). For a tetrahedral complex, the crystal field

splitting energy can be calculated as:

∆oct =4U

Substitute the expression for Uinto the formula for ∆oct:

∆oct =4

3·k·(−4e)·(−e)

Now, plug in the value of rinto the equation to find ∆oct.

Question 11

Question

Consider a coordination complex with the formula [Co(NH3)6]3+. Assume that

the ligands are weak-field ligands. Determine the number of unpaired electrons

in the complex using crystal field theory.

Solution

Crystal field theory can be used to determine the number of unpaired electrons

in a coordination complex. In this theory, the ligand field splits the energy

levels of the metal d-orbitals. For weak-field ligands, the energy ordering of the

d-orbitals is as follows: t2g¡eg.

Step 1: Determine the electronic configuration of Co3+.Cobalt (Co)

has 27 electrons, so Co3+ has 24 electrons. The electronic configuration of Co3+

can be represented as [Ar]3d6.

Step 2: Determine the splitting of the d-orbitals by the ligand

field. In an octahedral field, the d-orbitals split into two sets of different energy

levels: t2g(lower energy) and eg(higher energy). For weak-field ligands, the

degenerate dorbitals are filled before any pairing occurs.

Step 3: Determine the distribution of electrons in the t2gand eg

orbitals. Since Co3+ has 6 electrons in the dsubshell, they will occupy the

t2gorbitals first. The distribution of electrons in the t2gand egorbitals is

(t2g)6(eg)0.

Step 4: Determine the number of unpaired electrons. To determine

the total number of unpaired electrons, we subtract the total number of electrons

in the t2gorbitals from the total number of electrons in the dsubshell. In this

case, 6−0 = 6. Therefore, the complex [Co(N H3)6]3+ has 6unpaired electrons.

Question 12

Question

Consider a complex with the formula [Co(H2O)6]3+. Using Crystal Field The-

ory, determine the number of unpaired electrons in this complex.

Solution

Step 1: Write the electron configuration of a neutral cobalt atom.

Co : [Ar]4s23d7

Step 2: Determine the electron configuration of the cobalt ion in the complex.

Since the complex is [Co(H2O)6]3+, the cobalt ion has a +3 charge. This means

it loses 3 electrons from its neutral state.

Co3+ : [Ar]3d6

Step 3: Determine the splitting of the d orbitals in the presence of the

ligands (H2O molecules). In an octahedral field, the d orbitals split into two

sets - t2gand eg. The three lower energy orbitals are t2gand the two higher

energy orbitals are eg.

Step 4: Fill in the electrons to the orbitals. Since cobalt has 6 electrons in

its 3d orbitals, they will fill the lower energy t2gset of orbitals first.

Step 5: Calculate the number of unpaired electrons. In the t2gset, there

are 3 orbitals available for electrons to occupy. Since each orbital can hold 2

electrons (one spin up and one spin down), the maximum number of electrons

that can occupy these orbitals is 6. Since all 6 t2gorbitals are filled, there are

0 unpaired electrons in this complex.

Question 13

Question

Consider a complex ion with the formula [CoCl4]2−.

1. Determine the oxidation state of the cobalt ion in this complex.

2. Draw the crystal field splitting diagram for this complex ion and label the

energy levels.

Solution

1. To determine the oxidation state of the cobalt ion, we need to consider the

charge of the complex ion as a whole and the charge carried by each chloride

ion. Since each chloride ion has a charge of −1, the total negative charge from

the four chloride ions is −4. Since the complex ion [CoCl4]2−has a charge of

−2, the cobalt ion must have a charge of +2 to balance the charges. Thus, the

oxidation state of the cobalt ion is +2.

2. Drawing the crystal field splitting diagram involves considering the in-

teraction between the d orbitals of the cobalt ion and the surrounding chloride

ligands. In an octahedral crystal field, the d orbitals split into two groups - the

lower energy group containing t2gorbitals and the higher energy group contain-

ing egorbitals.

The crystal field splitting diagram for [CoCl4]2−complex ion is as follows:

t2gt2g

The energy of the t2gorbitals is lower than that of the egorbitals due to the

interaction with the ligands. The labeling of the energy levels is as follows:



t2gorbitals: dxy ,dyz ,dzx



egorbitals: dx2−y2,dz2

This completes the crystal field splitting diagram for the [CoCl4]2−complex

ion.

Question 14

Question

Consider a tetrahedral coordination complex with a metal ion at the center.

Given that the crystal field splitting energy (∆0) is 800 cm−1, calculate the

energy gap between the t2gand eglevels in terms of ∆0.

Solution

To calculate the energy gap between the t2gand eglevels, we need to find the

difference in energy between the highest energy orbital in t2gand the lowest

energy orbital in eg.

Step 1: The energy ordering of the d-orbitals in a tetrahedral field is eg>

t2g. Thus, the t2gorbitals are lower in energy than the egorbitals by ∆0.

Step 2: The energy gap between the highest energy orbital in t2gand the

lowest energy orbital in egis equal to ∆0.

Therefore, the energy gap between the t2gand eglevels in terms of ∆0is

800 cm−1.

Question 15

Question

Consider a complex ion with a metal center in a tetrahedral ligand field. If the

crystal field splitting energy (∆t) is 0.6 times the crystal field splitting energy

in an octahedral field (∆o), calculate the wavelength of light absorbed when an

electron transitions from the t2gorbitals to the egorbitals within this complex.

Given: ∆o= 800 cm−1

Solution

Step 1: Calculate the crystal field splitting energy in the tetrahedral field.

∆t=4

9×∆o=4

9×800 = 355.56 cm−1

Step 2: Convert the energy difference to wavelength using the formula:

E=hc

where Eis the energy difference in Joules, his Planck’s constant (6.626 x 10−34

J s), cis the speed of light (3.00 x 108m/s), and λis the wavelength.

Step 3: Convert the crystal field splitting energy to Joules:

E= ∆t×100 ×1.986 ×10−23 J cm−1

E= 355.56 ×100 ×1.986 ×10−23 = 7.04 ×10−21 J

Step 4: Calculate the wavelength of light absorbed:

λ=hc

λ=6.626 ×10−34 ×3.00 ×108

7.04 ×10−21 = 2.82 ×10−5m

Therefore, the wavelength of light absorbed when an electron transitions

from the t2gorbitals to the egorbitals is 2.82 ×10−5m.

Question 16

Question

Consider a transition metal complex with the formula [Mn(H2O)6]2+. Deter-

mine the energy difference between the t2gand egorbitals in this complex using

Crystal Field Theory.

Solution

Crystal field theory describes the interaction between transition metal ions and

ligands by considering the splitting of the d orbitals. In an octahedral field, the

d orbitals split into two sets: the lower energy t2gset and the higher energy eg

set.

Step 1: Assigning the number of electrons and determining the

oxidation state of the metal For [Mn(H2O)6]2+, we know that the complex

has an overall charge of +2. Each water molecule is neutral and Mn has an

oxidation state of x, so we’ll have:

x+ 6(0) = +2

x= +2

This means Mn is in the +2 oxidation state with 5 d electrons.

Step 2: Drawing the splitting diagram In an octahedral crystal field,

the d orbitals split with the t2gset (dxy, dxz, dyz) lower in energy and the egset

(dx2−y2, dz2)higherinenergy.Sincewehave5delectrons, we′llfilltheorbitalsfollowingHund′srule.

Step 3: Calculating the energy difference The energy difference be-

tween the t2gand egorbitals is the energy required to promote an electron from

the lower energy t2glevel to the higher energy eglevel. This energy difference

(∆o) can be calculated using the crystal field splitting parameter, ∆:

∆o= 10Dq =4

9∆

where Dq is the splitting energy per ligand field.

Step 4: Calculating the crystal field stabilization energy (CF SE)

The crystal field stabilization energy is the energy difference between filling the

d orbitals in a complex and the energy when there are no ligands present. For

an octahedral complex, the CF SE is given by:

CF SE =−0.4∆o×n

where nis the total number of d electrons (5 in this case). This energy

difference corresponds to the stability gained from electron-ligand interactions.

Question 17

Question

An octahedral complex is formed between a metal ion and ligands in a crystal

field. Calculate the crystal field stabilization energy (CFSE) for the complex

[Co(H2O)6]3+. Given: ∆o= 9600 cm−1.

Solution

Step 1: Determine the number of electrons and their configuration in the com-

plex.



Cobalt (Co) has an electronic configuration of [Ar] 3d74s2.



In the complex [Co(H2O)6]3+, cobalt is in the +3 oxidation state, so

it loses three electrons. The electronic configuration of the complex ion

becomes [Co(H2O)6]3+ = [Ar] 3d6.

Step 2: Determine the crystal field splitting energy (∆o) and the CFSE.



The crystal field splitting energy (∆o) = 9600 cm−1.



The CFSE for an octahedral complex is given by: CFSE = −0.4×∆o×n,

where nis the number of electrons in the t2gorbitals.



In an octahedral complex with 6 ligands, the complex ion has 6 electrons

distributed in the t2gorbitals.



Therefore, the CFSE = −0.4×9600 ×6 = −23040 cm−1.

Final Answer: The crystal field stabilization energy (CFSE) for the com-

plex [Co(H2O)6]3+ is −23040 cm−1.

Question 18

Question

Consider a transition metal complex with an octahedral geometry. Determine

the crystal field splitting energy (∆o) in terms of the ligand field stabilization

energy (LF SE) and the pairing energy (P).

Solution

To find the crystal field splitting energy (∆o) in terms of the ligand field stabi-

lization energy (LF SE) and the pairing energy (P), we can use the relationship

between these quantities for an octahedral field.

Step 1: The ligand field stabilization energy (LF SE) is given by:

LF SE =−0.4∆oNc−0.6P(Nc −2n)

where: ∆o= Crystal field splitting energy between t2gand egorbitals Nc=

Total number of ligands n= Number of electrons in t2gorbitals P= Pairing

energy

Step 2: For an octahedral field, Nc= 6. Substituting Nc= 6 into the

equation, we get:

LF SE =−2.4∆o−0.6P(6 −2n)

Step 3: The total number of electrons in an octahedral field is 6. If pairing

occurs, n= 3 (2 in t2g, 1 in eg). In this case, the LF SE becomes:

LF SE =−2.4∆o−0.6P(6 −2∗3) = −2.4∆o−1.2P

Step 4: Equate the LF SE to the experimentally determined value for oc-

tahedral complexes, which is usually known:

LF SE =−2.4∆o−1.2P

Step 5: Solve the equation for ∆o:

−2.4∆o−1.2P= Experimental LFSE

∆o=Experimental LFSE + 1.2P

−2.4

Therefore, the crystal field splitting energy (∆o) in terms of the ligand field

stabilization energy (LF SE) and the pairing energy (P) is given by:

∆o=Experimental LFSE + 1.2P

−2.4

Question 19

Question

Consider a transition metal complex with the formula [Fe(NH3)5(NO2)]Br2.

a. Determine the oxidation state of iron in this complex.

b. Use crystal field theory to predict the number of unpaired electrons in

the Fe(II) complex.

Solution

a. To determine the oxidation state of iron in the complex, we need to consider

the charges of the ligands. The NO2 ligand carries a charge of -1, so the total

charge from the ligands is -1 (from NO2) + 2−1 (from two bromide ions) = -3.

Therefore, the oxidation state of iron can be calculated using the formula:

oxidation state of Fe = charge of complex −total charge from ligands

Given that the overall complex is neutral, the oxidation state of iron is:

oxidation state of Fe = 0 −(−3) = +3

b. To predict the number of unpaired electrons in the Fe(III) complex using

crystal field theory, we will consider the electronic configuration of Fe(III) and

the number of electrons affected by the crystal field splitting.

The electronic configuration of Fe(III) is [Ar]3d5. In an octahedral crystal

field, the five d-orbitals split into three lower energy orbitals (t2gset) and two

higher energy orbitals (egset).

Step 1: Assign electrons to the d-orbitals before splitting.

Since Fe(III) has 5 electrons, we fill the d-orbitals as follows: - t2gset: ↑↓,

↑↓↑Step 2: Consider the crystal field splitting in octahedral coordination.

In an octahedral field, the t2gorbitals are lower in energy and will be filled

before the egorbitals. Since all the electrons pair up in the t2gset, there are no

unpaired electrons in this complex.

Therefore, the Fe(III) complex has zero unpaired electrons.

Question 20

Question

Consider a square planar complex with a central metal ion in the +3 oxidation

state. The complex is diamagnetic and has a high-spin electronic configuration.

Given this information, draw the d-orbital splitting diagram for the complex

and identify the ligand responsible for the high-spin configuration.

Solution

To draw the d-orbital splitting diagram for a square planar complex with a

central metal ion in the +3 oxidation state, we first need to identify the d-

electron configuration. For a +3 oxidation state, the metal ion has 3 fewer

electrons than the neutral atom. Since we are dealing with a high-spin complex,

all d-orbitals will be fully filled with electrons before pairing, resulting in a d3

configuration.

Step 1: Draw the initial d-orbital energy levels without any ligand influence:

Orbital Energy Level

dx2−y2

dz2(Higher energy)

dxy

dyz (Lower energy)

dxz

Step 2: Introduce the ligand field splitting by considering the effect of

ligands on the d-orbitals. In a square planar geometry, the dz2and dx2−y2

orbitals experience greater repulsion than the dxy , dyz , and dxz orbitals due to

their orientation along the axes pointing towards the ligands.

The splitting results in a higher energy level containing dz2and dx2−y2or-

bitals, and a lower energy level containing dxy , dyz , and dxz orbitals.

Orbital Energy Level

dz2, dx2−y2(Higher energy)

dxy , dyz , dxz (Lower energy)

Step 3: Identify the ligands responsible for the high-spin configuration.

Ligands that cause a small ligand field splitting lead to high-spin complexes.

Tangentially-bonding ligands, such as weak field ligands, generally cause small

splitting, resulting in high-spin configurations. Some examples of weak field

ligands include Cl−, F−, and H2O.

Therefore, the ligand responsible for the high-spin configuration in the square

planar complex is most likely a weak field ligand.

Question 21

Question

Explain how crystal field theory can be used to predict the relative energies of

the d orbitals in a transition metal complex. Use the example of an octahedral

coordination complex to illustrate your explanation.

Solution

Crystal field theory is a model used to describe the splitting of d orbitals in

transition metal complexes based on the interactions between the metal ion