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GNDU Question Paper-2024
Bachelor of Computer Application (BCA) (Hons.)
5
th
Semester (Batch 2024-28) (CBGS)
CHEMISTRY
(Physical Chemistry-III)
Time Allowed: Three Hours Max. Marks:35
Note: Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks.
SECTION-A
1. (a) Define Transport number. Write experimental method for the determination of
transport number by Hittorf method.
(b) Derive the relationship between the Gibbs free energy change (AG) and the EMF of an
electrochemical cell.
2. (a) Define specific conductance and equivalent conductance. How are they related ? 1
(b) Using Kohlrausch's Law of Independent Migration of lons, calculate the limiting molar
conductivities of M and Xions.
(c) Discuss how the equivalent conductance of MX varies with dilution and explain this
behaviour using the Debye-Huckel-Onsager equation for strong electrolytes.
SECTION-B
3. (a) Define activity and activity coefficient. How do they relate to the effective
concentration of ions in a solution?
(b) Calculate the mass defect and binding energy for the helium-4 nucleus (H
e4
) given the
following data:
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Mass of a proton = 1.007276 u
Mass of a neutron = 0.008665 u
Mass of a helium-4 nucleus = 4.001506 u
(Use u = 931.5MeV/(c2)).
4. (a) A sparingly soluble salt CaF
2
, is dissolved in water. Write the equilibrium expression
and derive the expression for its solubility
(S) in terms of K
sp
.
5.(b) Define the terms
(i) Mass defect and
(ii) Binding energy.
(c) Discuss the factors that contribute to the nuclear stability of isotopes. Why are certain
isotopes more stable than others?
SECTION-C
5. (a) State the Born-Oppenheimer approximation and explain its significance in molecular
spectroscopy. Why is this approximation valid in most cases?
(b) Define degrees of freedom in the context of molecular motion. How do the degrees of
freedom differ for linear and non-linear molecules?
(c) Derive the expression for the rotational energy levels of a rigid diatomic molecule using
semiclassical principles.
6. (a) Describe the three primary ways in which electromagnetic radiation interacts with
matter: absorption, emission and scattering, Provide an example of a spectroscopic
technique based on each type of interaction.
(b) State the selection rule for rotational transitions in diatomic molecules. Explain why
some transitions are forbidden according to quantum mechanical principles. For a
diatomic molecule with a dipole moment, which transitions are allowed according to the
selection rule ? Calculate the wavenumber of the transition from J=2 to J = 3 if the
rotational constant B = 2.5cm
-1
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SECTION-D
7. (a) Derive the expression for the energy levels of a simple harmonic oscillator and
explain how these energy levels are quantized.
(b) Define polarizability and explain its role in Raman scattering.
(c) State and explain the selection rules for electronic transitions in molecules. What
factors influence whether an electronic transition is allowed or forbidden? 2
8. (a) Define anharmonic motion in the context of molecular vibrations. How does
anharmonicity influence the IR spectrum of a molecule?
(b) Describe the Franck-Condon principle and its significance in determining the intensity
of electronic transitions. How does this principle relate to the shape of the potential
energy curves ?
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GNDU Answer Paper-2024
Bachelor of Computer Application (BCA) (Hons.)
5
th
Semester (Batch 2024-28) (CBGS)
CHEMISTRY
(Physical Chemistry-III)
Time Allowed: Three Hours Max. Marks:35
Note: Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks.
SECTION-A
1. (a) Define Transport number. Write experimental method for the determination of
transport number by Hittorf method.
(b) Derive the relationship between the Gibbs free energy change (AG) and the EMF of an
electrochemical cell.
Ans: Introduction
When electricity passes through an electrolyte solution, both positive ions (cations) and
negative ions (anions) move towards opposite electrodes. However, they do not always
move at the same speed. Some ions move faster than others. The concept of Transport
Number helps us understand how much electric current is carried by each type of ion.
This concept was introduced by the German scientist Johann Wilhelm Hittorf, who
developed an experimental method to determine the transport number of ions.
Definition of Transport Number
Transport Number (or Transference Number) is the fraction of the total electric current
carried by a particular ion in an electrolyte solution.
It is represented by t.
Transport number of cation = t⁺
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Transport number of anion = t⁻
The sum of both transport numbers is always:
t⁺ + t⁻ = 1
This means the entire electric current is shared between the positive and negative ions.
Example
Suppose 100 units of electricity pass through a solution.
If positive ions carry 40 units,
Negative ions carry 60 units.
Then,
t⁺ = 40/100 = 0.40
t⁻ = 60/100 = 0.60
And,
0.40 + 0.60 = 1
Why is Transport Number Important?
Transport number helps us to:
Understand ion movement in electrolytes.
Calculate ionic mobility.
Design batteries and electrochemical cells.
Study electrolysis processes.
Improve industrial chemical processes.
Hittorf Method
The Hittorf Method determines transport number by measuring the change in
concentration of the electrolyte near the electrodes after electrolysis.
Principle
When current passes through an electrolyte:
Positive ions move towards the cathode.
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Negative ions move towards the anode.
Since ions move at different speeds, the concentration of the solution around the electrodes
changes.
By measuring this concentration change, the transport number can be calculated.
Diagram of Hittorf Apparatus
Battery
+-------------+
| |
Platinum Platinum
Anode Cathode
| |
-----------------------------
| Anode Chamber |
|--------------------------|
| Middle Chamber |
|--------------------------|
| Cathode Chamber |
-----------------------------
Electrolyte Solution
The apparatus consists of:
Three chambers
Two platinum electrodes
Electrolyte solution
Battery
Ammeter
The middle chamber prevents mixing of solutions during electrolysis.
Experimental Procedure
Step 1
Fill all three chambers with the same electrolyte solution (for example, AgNO₃ solution).
Step 2
Connect the electrodes to a battery.
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Step 3
Allow a known amount of electricity to pass through the solution.
The quantity of electricity is measured using a coulometer or by measuring current and
time.
Step 4
Stop electrolysis after a fixed time.
Step 5
Carefully remove the solution from the anode and cathode chambers.
Step 6
Analyze chemically how much the concentration has changed.
Step 7
Using the loss or gain in concentration and the quantity of electricity passed, calculate the
transport number.
Observation
If the concentration near the anode decreases more than near the cathode, it indicates that
cations are moving more slowly.
If the concentration near the cathode changes more, it indicates that anions are moving
more slowly.
Thus, the movement of ions can be studied.
Advantages of Hittorf Method
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Gives accurate transport numbers.
Suitable for many electrolytes.
Helps study ionic movement.
Important in electrochemistry.
Limitations
Time-consuming.
Requires careful chemical analysis.
Small experimental errors affect results.
1 (b) Derive the Relationship Between Gibbs Free Energy Change (ΔG) and EMF of an
Electrochemical Cell
Introduction
An electrochemical cell converts chemical energy into electrical energy.
For example:
Batteries
Dry cells
Car batteries
When a chemical reaction occurs inside the cell, electricity is produced.
The relationship between the Gibbs Free Energy (ΔG) and EMF (Electromotive Force) tells
us how much electrical work can be obtained from a chemical reaction.
What is Gibbs Free Energy (ΔG)?
Gibbs Free Energy is the amount of energy available to perform useful work during a
chemical reaction.
It helps predict whether a reaction occurs naturally.
Interpretation
ΔG < 0 → Reaction is spontaneous.
ΔG = 0 → System is in equilibrium.
ΔG > 0 → Reaction is non-spontaneous.
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What is EMF?
EMF (Electromotive Force) is the maximum voltage produced by an electrochemical cell
when no current is flowing.
It is measured in volts (V).
Higher EMF means the cell can do more electrical work.
Electrical Work Produced by the Cell
Electrical work is equal to:
Electrical Work = Charge × Voltage
The total charge transferred is:
Charge = nF
where:
n = number of electrons transferred
F = Faraday constant = 96500 C mol⁻¹
Therefore,
Electrical Work = nFE
Relation with Gibbs Free Energy
Maximum useful work done by the electrochemical cell equals the decrease in Gibbs free
energy.
Hence,
ΔG = Electrical Work
Substituting electrical work:
ΔG = nFE
This is the required relationship.
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Final Equation
where:
ΔG = Gibbs free energy change (J mol⁻¹)
n = Number of electrons transferred
F = Faraday constant (96500 C mol⁻¹)
E = EMF of the cell (V)
Diagram of an Electrochemical Cell
Zn Electrode Cu Electrode
(Anode) (Cathode)
Zn | Zn² || Cu² | Cu
Salt Bridge
Electrons flow through the wire
from Zinc to Copper.
Zn → Zn² + 2e
Cu² + 2e → Cu
Meaning of the Equation
If EMF is Positive
ΔG becomes negative.
The reaction is spontaneous.
The cell works automatically.
If EMF is Zero
ΔG is zero.
No electrical work is produced.
The cell is at equilibrium.
If EMF is Negative
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ΔG becomes positive.
The reaction is not spontaneous.
External energy is needed.
Importance of the Relationship
The equation ΔG = nFE is very useful because it:
Predicts whether a chemical reaction is spontaneous.
Helps calculate the maximum electrical energy obtainable from a cell.
Explains the working of batteries and fuel cells.
Is widely used in electrochemistry, chemical engineering, and industrial processes.
Helps scientists compare the efficiency of different electrochemical cells.
Conclusion
Transport Number tells us what fraction of the total electric current is carried by each ion in
an electrolyte. The Hittorf method determines this value by measuring the change in
electrolyte concentration near the electrodes after electrolysis. This experiment
demonstrates that cations and anions usually move at different speeds.
The relationship ΔG = nFE connects chemistry with electricity. It shows that the decrease in
Gibbs free energy during a spontaneous chemical reaction is converted into electrical
energy. A positive EMF corresponds to a negative ΔG, indicating a spontaneous reaction
capable of performing useful electrical work. These concepts are fundamental for
understanding electrolysis, batteries, fuel cells, and many industrial electrochemical
processes.
2. (a) Define specific conductance and equivalent conductance. How are they related ?
(b) Using Kohlrausch's Law of Independent Migration of lons, calculate the limiting molar
conductivities of M and Xions.
(c) Discuss how the equivalent conductance of MX varies with dilution and explain this
behaviour using the Debye-Huckel-Onsager equation for strong electrolytes.
Ans: Introduction
Imagine you have a glass of water. Pure water hardly conducts electricity. But when you add
common salt (NaCl), acid (HCl), or any electrolyte, it breaks into positively charged ions
(cations) and negatively charged ions (anions). These ions move freely in water and carry
electric current.
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The more easily these ions move, the better the solution conducts electricity. This ability is
measured by conductance.
To understand this topic, we need to learn three important concepts:
1. Specific Conductance (κ)
2. Equivalent Conductance (Λeq)
3. Kohlrausch's Law and Debye-Hückel-Onsager Equation
(a) Specific Conductance (κ)
Definition
Specific conductance is the conductance of a solution that is 1 cm long and has a cross-
sectional area of 1 cm².
In simple words,
It tells us how well one cubic centimetre (1 cm³) of a solution conducts electricity.
Unit
S cm⁻¹ (Siemens per centimetre)
Easy Example
Suppose you have two glasses.
Glass A contains concentrated salt solution.
Glass B contains very dilute salt solution.
Electricity flows much better through Glass A because it has more ions.
Therefore,
Specific conductance of Glass A > Specific conductance of Glass B
Diagram
Battery
(+)
|
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[Solution]
Na+ Cl−
Na+ Cl−
Cl− Na+
More ions
↓
Better flow of electricity
↓
Higher Specific Conductance
Equivalent Conductance (Λeq)
Definition
Equivalent conductance is the conductance of all the ions produced by one gram
equivalent of an electrolyte when dissolved in a solution.
Unlike specific conductance, here we are interested in the conductance produced by a fixed
amount of electrolyte rather than a fixed volume of solution.
Unit
S cm² eq⁻¹
Simple Example
Imagine putting 1 gram equivalent of NaCl into water.
If you dissolve it in
100 mL water
500 mL water
1000 mL water
the amount of salt remains the same.
However, in more water the ions become farther apart and can move more freely.
Therefore,
Equivalent conductance increases.
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Difference between Specific Conductance and Equivalent Conductance
Specific Conductance
Equivalent Conductance
Conductance of 1 cm³ solution
Conductance of one gram equivalent of
electrolyte
Depends on number of ions in one unit
volume
Depends on conductance of fixed quantity of
electrolyte
Decreases on dilution
Increases on dilution
Unit: S cm⁻¹
Unit: S cm² eq⁻¹
Relation between Specific Conductance and Equivalent Conductance
The mathematical relationship is:
Where
Λeq = Equivalent conductance
κ = Specific conductance
N = Normality
Meaning of the Formula
If concentration decreases (solution becomes dilute),
κ decreases
Λeq increases
because ions get more space to move.
(b) Kohlrausch's Law of Independent Migration of Ions
Statement
At infinite dilution, every ion moves independently and contributes its own fixed value to
the total molar conductivity of the electrolyte.
This means that when the solution is extremely dilute, ions do not interfere with one
another. Each ion contributes independently to the conductivity.
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Diagram
Concentrated Solution
Na+ ←→ Cl−
↑ ↓
Lots of collisions
Low mobility
-----------------------------
Very Dilute Solution
Na+ Cl−
Large distance
No interference
High mobility
Mathematical Expression
For an electrolyte MX,
Where
Λ°m = Limiting molar conductivity
λ°M = Ionic conductivity of M⁺ ion
λ°X = Ionic conductivity of X⁻ ion
How to Calculate λ°M and λ°X
Suppose
and
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Then
Similarly,
if λ°M is known, then
This is the method generally used in numerical problems based on Kohlrausch's Law.
Importance of Kohlrausch's Law
It helps us
Calculate limiting molar conductivity.
Find conductivity of weak electrolytes.
Calculate degree of dissociation.
Determine solubility of sparingly soluble salts.
(c) Variation of Equivalent Conductance with Dilution
When an electrolyte MX is dissolved in water,
it splits into
MX
↓
M⁺ + X⁻
These ions carry electricity.
In Concentrated Solution
M⁺ X⁻ M⁺ X⁻
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Very close together
↓
Strong attraction
↓
Less movement
↓
Lower conductance
After Dilution
M⁺ X⁻
M⁺
X⁻
Large distance
↓
Free movement
↓
Higher conductance
Therefore
As dilution increases,
Equivalent conductance also increases.
This increase is especially noticeable for weak electrolytes, while strong electrolytes show a
smaller increase because they are already almost completely dissociated.
Debye-Hückel-Onsager Equation
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This equation explains why the conductivity of strong electrolytes changes with
concentration.
It is written as:
Where
Λ = Molar (or equivalent) conductance at concentration C
Λ° = Conductance at infinite dilution
A and B = Constants (depend on solvent and temperature)
C = Concentration
Simple Explanation of the Equation
The equation tells us that:
As concentration (C) increases, the term √C becomes larger.
A larger value is subtracted from Λ°, so the conductance (Λ) becomes smaller.
As the solution is diluted (C decreases), the subtraction becomes smaller, and Λ
approaches Λ°.
This happens because ions experience less attraction and fewer collisions, allowing them to
move more freely.
Why Conductance Increases on Dilution
1. Ions move farther apart.
2. Electrostatic attraction between ions decreases.
3. Collisions become fewer.
4. Ion mobility increases.
5. Electrical current flows more easily.
Conclusion
Specific conductance measures the ability of a unit volume of solution to conduct
electricity, while equivalent conductance measures the conductance produced by one gram
equivalent of an electrolyte. They are related through the expression Λeq = 1000κ/N.
Kohlrausch's Law states that at infinite dilution each ion contributes independently to the
total conductivity, making it possible to calculate limiting molar conductivities of ions. As a
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solution of MX is diluted, the ions move more freely and equivalent conductance increases.
The Debye-Hückel-Onsager equation explains this behavior quantitatively by showing that
conductance decreases with increasing concentration because of stronger interionic
interactions and increases on dilution as these interactions become weaker.
SECTION-B
3. (a) Define activity and activity coefficient. How do they relate to the effective
concentration of ions in a solution?
(b) Calculate the mass defect and binding energy for the helium-4 nucleus (H
e4
) given the
following data:
Mass of a proton = 1.007276 u
Mass of a neutron = 0.008665 u
Mass of a helium-4 nucleus = 4.001506 u
(Use u = 931.5MeV/(c2)).
Ans: 3(a) Define Activity and Activity Coefficient. How do they relate to the effective
concentration of ions in a solution?
Introduction
When we prepare a solution, we usually measure the amount of a substance by its
concentration (moles per litre). However, in real solutions, especially those containing ions,
the ions continuously attract and repel each other. Because of these interactions, the ions
do not behave as if they are completely free. This is why chemists use the concept of
activity instead of just concentration.
Think of it like this:
Imagine a classroom with only 5 students. Every student has enough space to move around
freely. Now imagine another classroom with 100 students. Although there are 100 students
present, they cannot move as freely because they keep bumping into each other.
Similarly, in a solution, ions are present in a certain concentration, but because they interact
with neighbouring ions, their "effective" concentration becomes different from the actual
concentration.
Activity
Activity is the effective concentration of an ion or molecule in a solution.
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It tells us how much of the substance is actually available to participate in a chemical
reaction.
In an ideal solution, activity and concentration are almost the same. But in real solutions,
activity is usually lower because ions influence one another.
Definition:
Activity is the effective concentration of an ion that determines its actual chemical
behaviour in a solution.
Activity Coefficient
The Activity Coefficient is represented by the Greek letter γ (gamma).
It tells us how much the actual behaviour of an ion differs from ideal behaviour.
It is defined as
Activity (a) Concentration (C)
Where:
a = Activity
γ = Activity coefficient
C = Concentration
Relationship Between Activity and Concentration
There are three important cases.
Case 1: Ideal Solution
If
then
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This means the ions behave perfectly, and activity equals concentration.
Case 2: Real Solution
Usually,
Therefore,
The effective concentration becomes smaller because ions attract or repel one another.
Case 3: Very Dilute Solution
When the solution becomes extremely dilute, ions remain far apart.
Then,
Hence,
Simple Diagram
Dilute Solution
+ - + -
Lots of space between ions
γ ≈ 1
Activity ≈ Concentration
Concentrated Solution
+ - + - + - + - + -
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Ions are very close together
Strong attraction/repulsion
γ < 1
Activity < Concentration
Why is Activity Important?
Activity is used because many chemical calculations depend on the effective concentration
rather than the actual concentration.
It is important in:
Electrochemistry
Electrochemical cells
Equilibrium calculations
Solubility calculations
Acid-base chemistry
Without using activity, calculated values may differ from experimental results.
Key Points
Activity = Effective concentration of ions.
Activity Coefficient (γ) = Measures deviation from ideal behaviour.
Formula:
For ideal solutions:
For real solutions:
As concentration increases, activity becomes smaller than concentration because
ions interact with each other.
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3(b) Calculate the Mass Defect and Binding Energy of Helium-4 Nucleus
Introduction
Every atom has a nucleus made of protons and neutrons.
One might expect that the mass of the nucleus should simply equal the sum of the masses
of all its protons and neutrons.
Surprisingly, it does not.
The nucleus is always slightly lighter than the total mass of its separate particles.
This missing mass is called the Mass Defect.
Where does this missing mass go?
It is converted into Binding Energy, which holds the nucleus together according to Einstein's
famous equation:
This binding energy acts like a very strong "glue" that keeps the protons and neutrons
bound inside the nucleus.
Given Data
Mass of one proton
Mass of one neutron
Mass of helium nucleus
Helium-4 contains:
2 protons
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2 neutrons
Conversion factor:
Step 1: Calculate Total Mass of Separate Nucleons
Mass of 2 protons
Mass of 2 neutrons
Total mass
Step 2: Calculate Mass Defect
Mass Defect
Step 3: Calculate Binding Energy
Binding Energy
Simple Diagram
Before nucleus is formed
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Proton Proton
● ●
Neutron Neutron
○ ○
Total mass = 4.031882 u
After nucleus forms
_________
/ \
| He-4 |
| ● ○ ● ○ |
\_________/
Mass = 4.001506 u
Missing mass
4.031882 − 4.001506
= 0.030376 u
↓
Converted into
Binding Energy
≈ 28.29 MeV
Why is Binding Energy Important?
Binding energy tells us how strongly the nucleus is held together.
A larger binding energy means a more stable nucleus.
If energy greater than the binding energy is supplied, the nucleus can break apart.
Nuclear reactions such as fusion (in the Sun) and fission (in nuclear reactors) involve
changes in binding energy, releasing enormous amounts of energy.
Final Answer
(a)
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Activity: The effective concentration of ions that actually participate in chemical
reactions.
Activity Coefficient (γ): A factor that shows how much an ion's behaviour deviates
from ideal behaviour.
Relation: , where is activity and is concentration. For ideal solutions,
; for real solutions, , so activity is less than concentration.
(b)
Mass of separate nucleons:
Mass of helium-4 nucleus:
Mass Defect:
Binding Energy:
Thus, the missing mass of 0.030376 u is converted into approximately 28.29 MeV of binding
energy, which is the energy that holds the helium-4 nucleus tightly together.
4. (a) A sparingly soluble salt CaF
2
, is dissolved in water. Write the equilibrium expression
and derive the expression for its solubility
(S) in terms of K
sp
.
(b) Define the terms
(i) Mass defect and
(ii) Binding energy.
(c) Discuss the factors that contribute to the nuclear stability of isotopes. Why are certain
isotopes more stable than others?
Ans: 4(a) A Sparingly Soluble Salt (CaF₂), Solubility Product (Ksp), and Solubility (S)
Many salts dissolve easily in water, but some dissolve only a very small amount. Such salts
are called sparingly soluble salts. Calcium fluoride (CaF₂) is one such example. Even though
only a tiny amount dissolves, that small amount is enough to establish an equilibrium
between the dissolved ions and the undissolved solid.
Think of it like adding sugar to cold water. At first, sugar dissolves, but after some time, no
more sugar dissolves because the solution becomes saturated. Similarly, when CaF₂ is added
to water, only a small portion dissolves while the remaining solid stays at the bottom.
Dissolution of Calcium Fluoride
When calcium fluoride dissolves in water, it breaks into calcium and fluoride ions.
Chemical Equation:
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Here:
CaF₂(s) = Solid calcium fluoride
Ca²⁺(aq) = Calcium ions in solution
F⁻(aq) = Fluoride ions in solution
The double arrow () indicates equilibrium, meaning dissolution and crystallization
occur at the same rate.
Equilibrium Diagram
Water
CaF (Solid)
│
│ Dissolves
▼
Ca² + 2F
▲
│ Re-forms solid
At equilibrium:
Some CaF₂ dissolves.
Some dissolved ions combine again to form solid CaF₂.
Both processes occur continuously at the same rate.
Equilibrium Expression (Ksp)
Since CaF₂ is a solid, it is not included in the equilibrium expression.
Therefore,
This equation is called the Solubility Product Expression.
Derivation of Solubility (S)
Suppose the solubility of CaF₂ is S mol/L.
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From the equation,
If S moles dissolve,
Calcium ion concentration = S
Fluoride ion concentration = 2S
Substitute these values into the Ksp equation:
Therefore,
Final Formula
This formula helps us calculate the solubility of calcium fluoride if its Ksp is known.
What is Ksp?
The Solubility Product Constant (Ksp) tells us how much of a sparingly soluble salt can
dissolve in water.
High Ksp → More soluble
Low Ksp → Less soluble
Since CaF₂ has a small Ksp value, only a little amount dissolves.
4(b) Define the Terms
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(i) Mass Defect
Atoms are made of protons and neutrons, which together form the nucleus.
One might expect the mass of the nucleus to be exactly equal to the sum of the masses of
all protons and neutrons.
However, scientists discovered that the nucleus is actually slightly lighter than the total
mass of its individual particles.
This missing mass is called the Mass Defect.
Formula
Mass Defect Sum of masses of nucleons Actual nuclear mass
Simple Example
Suppose,
Total mass of protons and neutrons = 20.050 u
Actual nuclear mass = 20.000 u
Then,
Mass Defect =
This missing mass has been converted into energy according to Einstein's famous equation:
where
E = Energy
m = Mass defect
c = Speed of light
Thus, the lost mass is not destroyedit is transformed into energy.
Diagram
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Separate Nucleons
Protons + Neutrons
│
│ Join Together
▼
Nucleus
Actual Mass
<
Total Mass
Difference = Mass Defect
(ii) Binding Energy
The energy released when protons and neutrons combine to form a nucleus is called
Binding Energy.
It is also the energy required to break the nucleus apart into individual protons and
neutrons.
In simple words,
Binding energy is the "glue" that holds the nucleus together.
Without binding energy, the positively charged protons would strongly repel each other and
the nucleus would fall apart.
Formula
where
Δm = Mass defect
c = Speed of light
Everyday Analogy
Imagine several magnets joined together.
To separate them, you must apply force.
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Similarly,
More force needed → Stronger binding
Less force needed → Weaker binding
The same idea applies to nuclei.
Difference Between Mass Defect and Binding Energy
Mass Defect
Binding Energy
Difference in mass
Energy corresponding to that missing mass
Measured in atomic mass units (u)
Measured in MeV or Joules
Causes binding energy
Holds the nucleus together
4(c) Factors Contributing to Nuclear Stability
Not every nucleus is stable.
Some atoms remain unchanged for billions of years, while others break apart by emitting
radiation. This process is called radioactive decay.
Whether a nucleus is stable depends on several important factors.
1. Correct Neutron-to-Proton Ratio (N/Z Ratio)
Protons repel each other because they all carry positive charge.
Neutrons help reduce this repulsion by providing the strong nuclear force, which acts like a
powerful glue holding the nucleus together.
Light elements are most stable when neutrons ≈ protons.
Heavy elements need more neutrons than protons to remain stable because they
have greater protonproton repulsion.
If the neutron-to-proton ratio is too high or too low, the nucleus becomes unstable and may
undergo radioactive decay.
Stable Nucleus
Protons (+ + +)
Neutrons (○ ○ ○)
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Balanced Ratio
↓
Stable Atom
2. Strong Nuclear Force
The strong nuclear force is one of the strongest forces in nature.
It acts over very short distances and binds protons and neutrons together.
If this attractive force is stronger than the electrical repulsion between protons, the nucleus
remains stable.
3. High Binding Energy per Nucleon
A nucleus with high binding energy per nucleon is more tightly bound.
This means much more energy would be needed to break it apart.
Such nuclei are generally more stable.
Iron-56 is a famous example because it has one of the highest binding energies per nucleon.
4. Magic Numbers
Scientists discovered that nuclei containing certain numbers of protons or neutrons are
especially stable.
These are called magic numbers:
2, 8, 20, 28, 50, 82, 126
These numbers correspond to completely filled nuclear shells, similar to the way filled
electron shells make noble gases chemically stable.
5. Size of the Nucleus
Very heavy nuclei (such as uranium) contain many protons.
As the number of protons increases, the electrical repulsion between them also increases.
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Eventually, the strong nuclear force cannot completely overcome this repulsion, making the
nucleus unstable and radioactive.
Why Are Certain Isotopes More Stable Than Others?
Isotopes are atoms of the same element having the same number of protons but different
numbers of neutrons.
The extra or fewer neutrons affect the neutron-to-proton ratio and, therefore, the stability
of the nucleus.
Example
Carbon has three common isotopes:
Carbon-12 → Stable
Carbon-13 → Stable
Carbon-14 → Radioactive (unstable)
Carbon-14 contains extra neutrons, giving it an unfavorable neutron-to-proton ratio. As a
result, it undergoes radioactive decay to become more stable.
Summary
CaF₂ is a sparingly soluble salt that establishes an equilibrium in water:
Its solubility product is:
and if the solubility is S, then:
Mass defect is the difference between the combined mass of free nucleons and the
actual mass of the nucleus.
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Binding energy is the energy released when the nucleus forms (or required to break
it apart) and is given by
.
Nuclear stability depends on the neutron-to-proton ratio, strong nuclear force, high
binding energy per nucleon, magic numbers, and nuclear size.
Certain isotopes are more stable because they have the right balance of protons and
neutrons and a more tightly bound nucleus.
SECTION-C
5. (a) State the Born-Oppenheimer approximation and explain its significance in molecular
spectroscopy. Why is this approximation valid in most cases?
(b) Define degrees of freedom in the context of molecular motion. How do the degrees of
freedom differ for linear and non-linear molecules?
(c) Derive the expression for the rotational energy levels of a rigid diatomic molecule using
semiclassical principles.
Ans: 5(a) Born-Oppenheimer Approximation, Degrees of Freedom, and Rotational Energy
of a Rigid Diatomic Molecule
(a) Born-Oppenheimer Approximation
Simple Explanation
Imagine two people dancing together:
One person is very light and fast (electron).
The other person is very heavy and slow (nucleus).
The lighter person can move around many times while the heavier person hardly changes its
position.
This is exactly what happens inside a molecule.
Electrons are extremely light and move very quickly.
Atomic nuclei are thousands of times heavier and therefore move much more
slowly.
The Born-Oppenheimer Approximation assumes that while electrons are moving, the nuclei
remain almost fixed. After calculating the motion of electrons, the movement of nuclei is
studied separately.
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This approximation makes solving molecular problems much easier because we do not have
to calculate the motion of electrons and nuclei at the same time.
Simple Diagram
Without Approximation
Electron
Nucleus ↔ ↕
(Both moving together)
Very difficult to solve
Born-Oppenheimer Approximation
Step 1:
e
(+) Fixed Nucleus
Step 2:
Now study the slow movement of the nucleus separately.
Why is this Approximation Valid?
The approximation works because of the huge difference in mass between electrons and
nuclei.
Particle
Relative Mass
Speed
Electron
Very small
Very fast
Nucleus
Very large
Slow
For example,
A proton is about 1836 times heavier than an electron.
Since heavier particles move much more slowly, electrons adjust almost instantly to
any small movement of the nuclei.
Therefore, in most molecules, treating nuclei as fixed while studying electrons gives highly
accurate results.
Significance in Molecular Spectroscopy
Molecular spectroscopy studies how molecules absorb or emit light.
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A molecule has three main types of motion:
Electronic motion
Vibrational motion
Rotational motion
Because of the Born-Oppenheimer approximation:
Electronic energy can be calculated separately.
Vibrational energy can be calculated separately.
Rotational energy can also be calculated separately.
This greatly simplifies the interpretation of molecular spectra.
Without this approximation, molecular spectroscopy would become extremely complicated.
(b) Degrees of Freedom
What are Degrees of Freedom?
A degree of freedom is an independent way in which a molecule can move.
Think of a toy airplane.
It can:
Move forward
Move backward
Move left
Move right
Rotate
Tilt
Each independent movement is called a degree of freedom.
Similarly, molecules also possess different independent motions.
Total Degrees of Freedom
If a molecule contains N atoms, then each atom has three coordinates:
x
y
z
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Therefore,
Total Degrees of Freedom = 3N
Types of Molecular Motion
The total degrees of freedom are divided into:
1. Translational motion
2. Rotational motion
3. Vibrational motion
(i) Translational Motion
The whole molecule moves from one place to another.
It always has 3 translational degrees of freedom.
↑
|
← Molecule →
|
↓
Movement along:
X-axis
Y-axis
Z-axis
(ii) Rotational Motion
The molecule rotates about its axes.
Linear Molecule
Example:
O = C = O
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A linear molecule rotates about only two useful axes.
Rotation around its own bond axis is almost negligible because its moment of inertia is
extremely small.
Therefore,
Rotational Degrees of Freedom = 2
Non-Linear Molecule
Example:
H
|
H O
A bent (non-linear) molecule can rotate about all three perpendicular axes.
Therefore,
Rotational Degrees of Freedom = 3
(iii) Vibrational Motion
Atoms vibrate about their equilibrium positions.
The remaining degrees of freedom become vibrational.
Formula
Linear Molecule
Vibrational Degrees of Freedom
= 3N − 5
Non-linear Molecule
Vibrational Degrees of Freedom
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= 3N − 6
Comparison Table
Property
Linear Molecule
Non-linear Molecule
Total Degrees of Freedom
3N
3N
Translational
3
3
Rotational
2
3
Vibrational
3N − 5
3N − 6
Example
Carbon Dioxide (CO₂)
Number of atoms = 3
Total Degrees of Freedom
= 3 × 3 = 9
Translation = 3
Rotation = 2
Vibration = 9 − 5 = 4
Water (H₂O)
Number of atoms = 3
Total Degrees of Freedom
= 9
Translation = 3
Rotation = 3
Vibration = 9 − 6 = 3
(c) Rotational Energy Levels of a Rigid Diatomic Molecule
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What is a Rigid Diatomic Molecule?
A diatomic molecule consists of two atoms joined together.
Examples:
H₂
O₂
N₂
CO
A rigid molecule means that the distance between the two atoms (bond length) remains
constant while rotating.
Imagine a dumbbell rotating.
●────────●
Rotating
The two balls stay at the same distance from each other.
Semiclassical Model
According to the semiclassical model:
The molecule rotates like a tiny rigid object.
Its rotational kinetic energy is
Where:
= Rotational energy
= Angular momentum
= Moment of inertia
Angular Momentum Quantization
Quantum mechanics states that angular momentum is not continuous. It is quantized.
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where:
is the rotational quantum number.
Substituting this into the energy equation gives:
This is the expression for the rotational energy levels of a rigid diatomic molecule.
Energy Level Diagram
Energy
↑
J = 4 ───────────────
J = 3 ───────────
J = 2 ────────
J = 1 ─────
J = 0 ──
──────────────────────────→ Rotational Quantum Number (J)
As the value of J increases:
Angular momentum increases.
Rotational energy increases.
The spacing between successive levels also becomes larger.
Importance of Rotational Energy Levels
They help explain the rotational spectra of molecules observed in the microwave
region.
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The spectra provide information about bond length, moment of inertia, and
molecular structure.
They are useful in identifying unknown molecules and studying molecular properties.
Conclusion
The BornOppenheimer approximation is based on the fact that electrons move much
faster than nuclei, allowing their motions to be treated separately. This greatly simplifies
molecular spectroscopy by separating electronic, vibrational, and rotational motions.
The degrees of freedom describe all the possible independent motions of a molecule. Every
molecule has 3 translational degrees of freedom, while linear molecules have 2 rotational
and vibrational degrees of freedom, and non-linear molecules have 3 rotational
and vibrational degrees of freedom.
Finally, a rigid diatomic molecule rotates with a fixed bond length, and its rotational energy
is quantized. The rotational energy levels are given by:
where is the rotational quantum number and is the moment of inertia. These quantized
energy levels form the basis of rotational spectroscopy and help scientists determine
important molecular properties.
6. (a) Describe the three primary ways in which electromagnetic radiation interacts with
matter: absorption, emission and scattering, Provide an example of a spectroscopic
technique based on each type of interaction.
(b) State the selection rule for rotational transitions in diatomic molecules. Explain why
some transitions are forbidden according to quantum mechanical principles. For a
diatomic molecule with a dipole moment, which transitions are allowed according to the
selection rule ? Calculate the wavenumber of the transition from J=2 to J = 3 if the
rotational constant B = 2.5cm
-1
Ans: 6. (a) Interaction of Electromagnetic Radiation with Matter: Absorption, Emission and
Scattering
Electromagnetic radiation (EMR) includes radio waves, microwaves, infrared (IR), visible
light, ultraviolet (UV), X-rays, and gamma rays. When this radiation falls on a substance, it
does not always behave in the same way. Depending on the energy of the radiation and the
nature of the substance, the radiation may be absorbed, emitted, or scattered.
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Think of sunlight falling on a tree. Some light is absorbed by the leaves, some is reflected or
scattered in different directions, and sometimes the leaves release energy in another form.
The same thing happens at the atomic and molecular level.
1. Absorption
Absorption means that a molecule or atom takes in (absorbs) energy from electromagnetic
radiation.
Imagine a student climbing stairs. The student needs energy to move from one step to the
next. Similarly, molecules absorb energy to move from a lower energy level to a higher
energy level.
Simple Diagram
Higher Energy Level
↑
│ Absorbs Energy
│
Lower Energy Level
When light of the correct energy falls on a molecule, the molecule absorbs it and becomes
excited.
Everyday Example
A black shirt becomes warmer than a white shirt in sunlight because it absorbs more light
energy.
Spectroscopic Technique Based on Absorption
Infrared (IR) Spectroscopy
Molecules absorb infrared radiation.
This causes their chemical bonds to vibrate.
Scientists use IR spectroscopy to identify different functional groups in compounds.
Other examples include:
UV-Visible Spectroscopy
Microwave Spectroscopy
2. Emission
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Emission is the opposite of absorption.
Here, an excited atom or molecule releases energy and returns to a lower energy state.
Imagine climbing to the top of a staircase and then coming back down. While coming down,
you release the energy that was used to climb.
Simple Diagram
Higher Energy Level
│
│ Releases Energy
↓
Lower Energy Level
The released energy appears as light or electromagnetic radiation.
Everyday Example
Neon sign boards glow because excited gas atoms emit light.
Fireworks produce different colours because excited atoms emit different
wavelengths.
Spectroscopic Technique Based on Emission
Atomic Emission Spectroscopy (AES)
Atoms are heated.
Electrons become excited.
When they return to lower energy levels, they emit characteristic light.
This helps identify different elements.
3. Scattering
Scattering occurs when light strikes a particle and changes its direction.
Instead of being absorbed, the light simply bounces off or changes direction.
Imagine throwing a tennis ball against a wall. Instead of sticking to the wall, it bounces away
in another direction.
Simple Diagram
Incoming Light
→
○ Molecule
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↗ ↘
Scattered Light
Some scattered light has the same energy, while some has slightly different energy.
Everyday Example
The sky appears blue because sunlight is scattered by tiny air molecules.
Clouds appear white because water droplets scatter all colours of light.
Spectroscopic Technique Based on Scattering
Raman Spectroscopy
A laser beam is directed onto a sample.
Most light is scattered unchanged.
A very small amount changes energy.
This energy change provides information about molecular vibrations.
Summary Table
Interaction
What Happens?
Energy Change
Spectroscopic
Technique
Absorption
Molecule absorbs
light
Gains energy
Infrared (IR)
Spectroscopy
Emission
Molecule
releases light
Loses energy
Atomic Emission
Spectroscopy
Scattering
Light changes
direction
Usually no change (or slight
change in Raman)
Raman Spectroscopy
(b) Selection Rule for Rotational Transitions
Before understanding the selection rule, let's first understand rotational motion.
A diatomic molecule consists of two atoms joined together, such as HCl, CO, or NO.
These molecules rotate continuously.
H ●────● Cl
Rotation
However, molecules cannot rotate with any random energy.
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According to Quantum Mechanics, rotational energy is quantized, meaning molecules can
rotate only at specific allowed energy levels.
These rotational energy levels are represented by the rotational quantum number J.
J = 0
J = 1
J = 2
J = 3
J = 4
The molecule jumps only between these fixed levels.
Selection Rule
For rotational spectroscopy, the selection rule is
ΔJ = ±1
This means the rotational quantum number can change by only one unit.
Allowed Transitions
J = 0 → J = 1 ✓
J = 1 → J = 2 ✓
J = 2 → J = 3 ✓
J = 3 → J = 4 ✓
or in reverse,
J = 3 → J = 2 ✓
J = 2 → J = 1 ✓
Forbidden Transitions
These transitions are not allowed:
J = 0 → J = 2 ✗
J = 1 → J = 3 ✗
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J = 2 → J = 4 ✗
because ΔJ ≠ ±1.
Quantum mechanics allows only transitions where one unit of angular momentum is
exchanged with the electromagnetic radiation.
Why Are Some Transitions Forbidden?
Quantum mechanics states that molecules interact with electromagnetic radiation only
under certain conditions.
If a transition does not satisfy the selection rule (ΔJ = ±1), the probability of that transition
occurring is essentially zero. Such transitions are called forbidden transitions.
Another important condition is that the molecule must have a permanent dipole moment.
Molecules with Dipole Moment
Examples:
HCl ✓
CO ✓
NO ✓
These molecules can absorb microwave radiation because they have a permanent dipole
moment.
Molecules without Dipole Moment
Examples:
H₂
N₂
O₂
These molecules do not show pure rotational spectra because they have no permanent
dipole moment, even though they rotate.
Allowed Transition for a Diatomic Molecule with Dipole Moment
Since the molecule has a permanent dipole moment, the allowed rotational transitions are
those satisfying
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ΔJ = +1 or ΔJ = −1
Examples:
J = 0 → 1 ✓
J = 1 → 2 ✓
J = 2 → 3 ✓
J = 4 → 3 ✓
Numerical Problem
Given
Rotational constant
B = 2.5 cm⁻¹
Transition
J = 2 → J = 3
For rotational spectroscopy, the transition wavenumber is
Here,
cm
Substitute the values:
cm
Answer
Wavenumber = 15 cm⁻¹
Key Points to Remember
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Electromagnetic radiation interacts with matter in three ways: absorption, emission,
and scattering.
Absorption increases the energy of molecules and is used in IR spectroscopy.
Emission occurs when excited atoms or molecules release energy, forming the basis
of Atomic Emission Spectroscopy.
Scattering changes the direction of light and is used in Raman spectroscopy.
Rotational energy levels are quantized and are represented by the rotational
quantum number J.
The selection rule for rotational transitions is ΔJ = ±1.
Only diatomic molecules with a permanent dipole moment (such as HCl or CO)
show pure rotational spectra.
Transitions like J = 2 → J = 3 are allowed because ΔJ = +1, while transitions like J = 2
→ J = 4 are forbidden because they violate the selection rule.
For B = 2.5 cm⁻¹, the transition J = 2 → J = 3 occurs at a wavenumber of 15 cm⁻¹.
SECTION-D
7. (a) Derive the expression for the energy levels of a simple harmonic oscillator and
explain how these energy levels are quantized.
(b) Define polarizability and explain its role in Raman scattering.
(c) State and explain the selection rules for electronic transitions in molecules. What
factors influence whether an electronic transition is allowed or forbidden?
Ans: 7(a) Derive the expression for the energy levels of a Simple Harmonic Oscillator (SHO)
and explain how these energy levels are quantized.
Introduction
A Simple Harmonic Oscillator (SHO) is one of the most important models in quantum
mechanics. It describes the motion of particles that move back and forth about a fixed
position under a restoring force. Examples include the vibration of atoms in a molecule, a
spring with a mass attached to it, and molecular vibrations.
In classical physics, a vibrating particle can have any amount of energy. However, quantum
mechanics tells us something very different: the energy of a vibrating particle can only
have certain fixed values. This is called energy quantization.
Energy Levels of a Quantum Harmonic Oscillator
The allowed energy levels of a quantum harmonic oscillator are given by:
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Where:
= Energy of the particle
n = Quantum number (0, 1, 2, 3, ...)
h = Planck's constant
ν (nu) = Frequency of vibration
This formula shows that the particle cannot possess random energy values. It can only exist
in these fixed energy states.
What is Quantization?
Quantization means that energy is available only in discrete (fixed) amounts rather than
continuously.
Imagine climbing a staircase.
You can stand on the 1st, 2nd, or 3rd step.
You cannot stand halfway between two steps.
Similarly, an electron or vibrating molecule can occupy only certain energy levels.
Zero-Point Energy
When n = 0,
This is called the Zero-Point Energy.
It means that even at absolute zero temperature (0 K), the particle still possesses some
energy. It never becomes completely motionless, which is a unique prediction of quantum
mechanics.
Energy Level Diagram
Higher Energy
│
│ n = 4 -------------------
│
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│ n = 3 -------------------
│
│ n = 2 -------------------
│
│ n = 1 -------------------
│
│ n = 0 ------------------- (Zero-point energy = ½hν)
│
└──────────────────────────────
Notice that the spacing between successive energy levels is equal to hν.
Key Points
Energy is not continuous.
Only fixed energy values are allowed.
Lowest energy is ½hν, not zero.
Vibrational energy is quantized.
7(b) Define Polarizability and explain its role in Raman Scattering.
What is Polarizability?
Polarizability is the ability of a molecule's electron cloud to become distorted when an
external electric field (such as light) is applied.
Think of the electron cloud as a soft balloon around the nucleus.
If the balloon changes shape easily → High polarizability.
If it hardly changes shape → Low polarizability.
Simple Example
Imagine holding a sponge.
A soft sponge changes shape easily.
A hard stone does not.
Similarly, some molecules allow their electron clouds to deform easily, while others do not.
Raman Scattering
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When light falls on a molecule, most light is scattered without changing its energy. This is
called Rayleigh scattering.
A very small portion of the light exchanges energy with the vibrating molecule and is
scattered with a different frequency. This phenomenon is called Raman scattering.
Role of Polarizability
Raman scattering occurs only if the vibration of a molecule changes its polarizability.
If the vibration changes the electron cloud significantly:
Raman scattering is strong.
A Raman spectral line appears.
If there is no change in polarizability, Raman scattering does not occur.
Diagram
Incident Light
↓
__________
/ \
| Molecule | Electron cloud changes shape
\__________/
↓
Scattered Light
(Frequency changes)
Importance
Raman spectroscopy helps scientists:
Identify molecules
Study chemical bonds
Analyze unknown compounds
Investigate crystal structures
Detect impurities
Key Points
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Polarizability = ease of electron cloud distortion.
Raman effect depends on the change in polarizability.
Greater change → Stronger Raman signal.
7(c) State and explain the selection rules for electronic transitions in molecules. What
factors influence whether an electronic transition is allowed or forbidden?
What is an Electronic Transition?
An electronic transition occurs when a molecule absorbs energy (usually ultraviolet or
visible light) and an electron moves from a lower-energy orbital to a higher-energy orbital.
Higher Energy Orbital
↑
│ Absorbs Light
│
Lower Energy Orbital
However, not every transition is possible. Nature follows certain rules called selection
rules.
Selection Rules
1. Spin Selection Rule
The electron's spin must remain unchanged during the transition.
Allowed:
↑ → ↑
Forbidden:
↑ → ↓
Mathematically,
where ΔS is the change in total spin.
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2. Laporte Selection Rule
This rule mainly applies to centrosymmetric molecules.
Allowed transitions occur between orbitals of different symmetry:
g → u ✔
u → g ✔
Forbidden transitions:
g → g ✖
u → u ✖
Here:
g (gerade) = symmetric
u (ungerade) = antisymmetric
Why Are Some Transitions Forbidden?
A transition is called forbidden when it has a very low probability of occurring. It is not
absolutely impossible, but it happens very weakly because it violates one or more selection
rules.
Factors Affecting Electronic Transitions
1. Spin of the Electron
If the spin changes, the transition becomes forbidden.
2. Molecular Symmetry
The symmetry of the orbitals determines whether the transition follows the Laporte rule.
3. Change in Dipole Moment
A transition is more likely if the interaction with light produces a change in the molecule's
electric dipole moment.
4. Type of Molecular Orbital
Transitions such as π → π* and n → π* have different probabilities depending on the
orbitals involved.
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5. Molecular Environment
The surrounding solvent, neighboring atoms, temperature, and molecular structure can
influence the intensity of electronic transitions.
Simple Analogy
Think of a school with rules for moving between classrooms.
If you follow the school's rules, you are allowed to enter the next classroom easily.
If you break the rules, entry is difficult or restricted.
Similarly, electrons can move to higher energy levels only if the selection rules are satisfied.
If the rules are violated, the transition becomes forbidden or very weak.
Conclusion
The Simple Harmonic Oscillator demonstrates that vibrational energy is quantized, meaning
molecules can occupy only fixed energy levels given by
, with a non-zero
zero-point energy even at 0 K. Polarizability describes how easily a molecule's electron
cloud is distorted, and a change in polarizability during vibration is the essential condition
for Raman scattering, making Raman spectroscopy a valuable tool for studying molecular
structure. Finally, electronic transitions occur when electrons absorb light and move to
higher-energy orbitals, but these transitions are governed by selection rules such as the
spin selection rule () and the Laporte rule. Whether a transition is allowed or
forbidden depends on factors like electron spin, orbital symmetry, dipole moment
changes, orbital type, and the molecular environment. Together, these concepts explain
how quantum mechanics controls molecular vibrations, light scattering, and the interaction
of molecules with electromagnetic radiation.
8. (a) Define anharmonic motion in the context of molecular vibrations. How does
anharmonicity influence the IR spectrum of a molecule?
(b) Describe the Franck-Condon principle and its significance in determining the intensity
of electronic transitions. How does this principle relate to the shape of the potential
energy curves ?
Ans: 8. (a) Define Anharmonic Motion in the Context of Molecular Vibrations. How Does
Anharmonicity Influence the IR Spectrum of a Molecule?
Introduction
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Atoms in a molecule are not fixed in one place. They are always vibrating because they
possess energy. Imagine two atoms connected by a spring. As they move closer and farther
from each other, the bond stretches and compresses. This vibration is called molecular
vibration.
Scientists often use the harmonic oscillator model, where the bond behaves like a perfect
spring. However, in reality, chemical bonds are not perfect springs. Their behavior is slightly
different, especially when stretched too much. This real behavior is called anharmonic
motion.
What is Harmonic Motion?
In harmonic motion:
The bond acts like a perfect spring.
Stretching and compressing require equal energy.
The vibration is perfectly regular.
The molecule can never break, no matter how much the bond is stretched (which is
not true in reality).
Diagram: Harmonic Motion
Atom O ===== Spring ===== Atom O
← Stretch → ← Compress →
The atoms move back and forth equally, just like a perfectly elastic spring.
What is Anharmonic Motion?
Anharmonic motion is the real type of molecular vibration where the bond does not behave
like a perfect spring.
When the bond is stretched too much:
It becomes easier to stretch further.
The bond can eventually break.
The restoring force decreases.
The energy levels are no longer equally spaced.
Simple Example
Think of a rubber band.
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At first, it stretches easily.
As you keep pulling, it becomes weaker.
Finally, it snaps.
A chemical bond behaves in a similar way. This is anharmonic motion.
Diagram: Anharmonic Potential Energy Curve
Energy
^
|
| ________
| ___/
| ___/
| ___/
| ___/
|_____/__________________________> Bond Length
Stable Bond Stretching Bond Breaks
Unlike the harmonic curve, this curve becomes flatter as the bond stretches, showing that
bond breaking is possible.
Difference Between Harmonic and Anharmonic Motion
Harmonic Motion
Anharmonic Motion
Ideal model
Real molecular behavior
Perfect spring
Imperfect bond
Equal spacing of energy levels
Unequal spacing of energy levels
Bond never breaks
Bond can break
Simplified theory
Practical and realistic
Effect of Anharmonicity on the IR Spectrum
IR (Infrared) Spectroscopy studies molecular vibrations by measuring the absorption of
infrared radiation.
In Harmonic Motion
Only one vibration rule is allowed:
Δv = ±1
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This means a molecule can move only from one vibrational level to the next.
Example:
v = 0 → v = 1
Only one absorption peak appears.
In Anharmonic Motion
Since the energy levels are unequal, additional transitions become possible.
These include:
Δv = ±2
Δv = ±3
These extra absorption bands are called overtones.
Example:
v = 0 → v = 2
v = 0 → v = 3
Diagram of Vibrational Energy Levels
Harmonic
v=3 ----------------
v=2 ----------------
v=1 ----------------
v=0 ----------------
(Equal spacing)
Anharmonic
v=3 --------------
(smaller gap)
v=2 -------------------
v=1 ------------------------
v=0 -----------------------------
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(Unequal spacing)
Influence on IR Spectrum
Because of anharmonicity:
Extra peaks (overtones) appear.
Combination bands may appear.
Peak positions shift slightly.
The spectrum becomes more realistic and detailed.
Scientists obtain more accurate information about molecular structure.
Importance of Anharmonic Motion
Anharmonic motion helps scientists:
Study real molecular vibrations.
Predict bond breaking.
Understand molecular energy accurately.
Interpret IR spectra correctly.
Analyze chemical reactions involving bond stretching.
(b) Describe the FranckCondon Principle and Its Significance in Determining the Intensity
of Electronic Transitions. How Does This Principle Relate to the Shape of the Potential
Energy Curves?
Introduction
When a molecule absorbs ultraviolet (UV) or visible light, one of its electrons jumps from a
lower energy level to a higher energy level. This process is called an electronic transition.
However, this jump happens so quickly that the atomic nuclei do not have enough time to
move. This important idea is explained by the FranckCondon Principle.
What is the FranckCondon Principle?
The FranckCondon Principle states:
During an electronic transition, electrons move so quickly that the nuclei remain almost
fixed in their positions.
The electron changes its energy instantly, but the nuclei are much heavier and move much
more slowly.
Simple Everyday Example
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Imagine taking a photograph of a person jumping.
The camera captures the person instantly before they can change position.
Similarly:
The electron changes energy almost instantly.
The nuclei do not move during that instant.
Why Does This Happen?
Electrons are extremely light.
Nuclei are thousands of times heavier.
Electron movement occurs in about 10⁻¹⁵ seconds.
Nuclear vibration takes about 10⁻¹² seconds.
Therefore, electrons move first while nuclei stay almost stationary.
Vertical Transition
Since nuclei do not move, the electronic transition is shown as a vertical line on a potential
energy diagram.
Diagram
Energy
^
Excited State
_________
/ \
/ \
| ↑
| │ Electronic Transition
| │ (Vertical)
| │
\______│________
Ground State
_________
/ \
/ \
_/______________\____________> Bond Length
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The upward vertical arrow shows that the electron changes energy while the bond length
remains unchanged.
Significance in Electronic Transition Intensity
Not all electronic transitions absorb light equally.
The intensity of an absorption band depends on how well the vibrational wavefunctions of
the initial and final states overlap.
Good Overlap
Ground: (~~~~)
Excited: (~~~~)
Large overlap means:
Strong absorption
High intensity peak
Poor Overlap
Ground: (~~~~)
Excited: (~~~~)
Small overlap means:
Weak absorption
Low intensity peak
Relation to Potential Energy Curves
Each electronic state has its own potential energy curve.
These curves represent how the energy of a molecule changes as the bond length changes.
If the Curves are Similar
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Excited
/\
Ground
/\
Maximum overlap occurs.
Strong electronic transition.
Intense absorption peak.
If the Curves are Shifted
Excited
/\
Ground
/\
Less overlap occurs.
Lower intensity.
Several vibrational transitions may appear, producing multiple closely spaced peaks.
Importance of the FranckCondon Principle
The FranckCondon Principle helps scientists:
Explain why some electronic transitions are strong while others are weak.
Predict the intensity of absorption and emission spectra.
Understand the vibrational structure seen in UVVisible spectra.
Relate spectral features to changes in bond length and molecular geometry.
Interpret molecular behavior in photochemistry and spectroscopy.
Conclusion
Anharmonic motion describes the real behavior of vibrating chemical bonds. Unlike the
ideal harmonic model, real bonds have unequal vibrational energy levels and can eventually
break. Because of anharmonicity, IR spectra contain additional features such as overtones,
combination bands, and slight peak shifts, making the spectra more realistic and
informative.
The FranckCondon Principle explains that electronic transitions occur so rapidly that the
nuclei remain essentially fixed during the process. As a result, transitions are represented as
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vertical transitions on potential energy curves. The degree of overlap between the
vibrational wavefunctions of the initial and final states determines the intensity of
electronic transitions. Together, anharmonicity and the FranckCondon Principle provide a
deeper understanding of molecular vibrations, infrared spectroscopy, and electronic
spectroscopy.
This paper has been carefully prepared for educational purposes. If you notice any mistakes or
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