Dufile Seed SS S.

5 Chemistry, 2024 Term 1

ATOMIC STRUCTURE
An atom is spherical in shape and has a small region in the center called the nucleus. The atom
contains protons and neutrons, and electrons as fundamental particles.
The three sub – atomic particles are found in distinct and separate regions.

Characteristics of a proton, a neutron and an electronic

Particle Symbol Relative mass Relative charge
Particle Symbol Relative mass Relative charge
Proton 𝑝 1 +1
Neutron 𝑛 1 0
Electron 𝑒 1/1837 -1

Although atoms contain electrically charged particles, the atoms themselves are electrically
neutral. This is because a neutral atom contains an equal number of electrons and protons.

Terms associated with the atom

i. Atomic number, Z (proton number): This is the number of protons in the nucleus of an
atom. In a neutral atom, it is equal to the number of electrons around the nucleus.
ii. Mass number, A: This is the number of protons and neutrons in the nucleus of an atom.
Protons and neutrons are collectively called nucleons. Mass number is equal to the
total number of nucleons in an atom.
iii. Nuclides: This is used to describe any atomic species of which proton number and
nucleon number are specified. Nuclides are written as 126𝐶 , 23
11𝑁𝑎 e.t.c

Isotopes
Isotopes are atoms with the same number of protons but different mass number or number of
neutrons.
OR:
Isotopes are atoms of the same element with the same atomic number but different mass
numbers.

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Dufile Seed SS S.5 Chemistry, 2024 Term 1

ELECTRONIC STRUCTURE OF AN ATOM (ATOM SPECTRUM)

A spectrum is a range of electromagnetic energies arranged in order of increasing or decreasing
wavelength or frequency of electromagnetic radiation emitted or absorbed by an atom or
molecule.
Or
A spectrum is a bundle of different lines of different frequencies which may be visible or invisible
when white light is passed through a glass prism or diffracting grating.
Atomic spectrum explains how electrons in atoms of different elements are arranged in their
atoms when heated.
Each element produces a unique set of spectral lines. Since no two elements emit the same
spectral lines, elements can be identified by their line spectrum. Atomic Spectra are produced by
emission or absorption of energy.
Note
An electromagnetic spectrum is the entire range of wave length or frequency of electromagnetic
radiation extending from gamma rays to the longest radio waves and including visible light.

The hydrogen spectrum

This is a series (group) of lines, some in the visible region and others in the invisible region. In each
series, the spacing between the adjacent lines decreases as frequency of the waves giving the lines
increases. (Or as the wavelength decreases). The decrease in the spacing is caused by decreased
nuclear attraction in the electrons resulting into a continuum in each series.
In the visible part of the spectrum, this is observed as a number of lines, each of different colour.
The four prominent colours being red, blue-green, blue and violet.

Features of the spectrum

- The spectrum obtained consists of a number of definite lines.
- Each line corresponds to a definite wavelength of radiation.
- As frequency increases in each of the series of lines, each line becomes closer to the
previous line until the lines converge and the spectrum becomes continuous.

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- In each series, the interval between the sequences of lines becomes smaller and smaller
towards the high frequency end of spectrum until the lines run together or converge to
form a continuum of light.
Why do the lines get closer?
The differences in energy between successive energy levels in atoms become smaller and smaller
with increasing distance of the energy levels from the nucleus until they finally merge to form a
continuum of light

Energy level diagram for the hydrogen atom

How each series is formed.

- The Lyman Series is formed when the electron jumps from any higher energy level to first
energy level 𝑛 = 1.
- The Balmer Series is formed when the electron jumps from any higher energy level to the
second energy level 𝑛 = 2.
- The Paschen Series is formed when the electron jumps from any higher energy level to the
third energy level 𝑛 = 3.
- The Brackett series is formed when the electron jumps from any higher energy level to the
fourth energy level 𝑛 = 4.
- The Pfund series is formed when the electron jumps from any higher energy level to the
fifth energy level 𝑛 = 5.

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Types of Spectra
a) Hydrogen emission spectrum
This is formed when electricity (or electric discharge) is passed through a discharge tube
containing hydrogen gas at low pressure. The hydrogen molecules break up into single
hydrogen atoms.
The single electrons of each hydrogen atom absorbs energy and is promoted (or jumps)
from the ground state (energy level 𝑛 = 1) to higher energy level (excited state) where it
is unstable.
When the electron of each hydrogen atom falls back to the ground state (lower energy
level), it emits out light (or radiation) of a definite amount of energy giving rise to the
emission spectrum recorded on the photographic plate.
A description of the hydrogen emission spectrum

b) Absorption spectrum
This is formed when an atoms absorbs light of a certain wave length, black lines appear in
the spectrum where light of some wave length has been absorbed. The absorbed energy
by the atoms causes energy changes from lower to higher energy levels.

BOHR'S MODEL OF THE ATOM

Rutherford’s model of an atom could not explain that facts as to why the atomic spectra consisted
of separate lines. The model did not explain why atoms absorb or emit light of certain frequencies.
In his model too, Rutherford could not explain why the spectral lines converge to form a
continuum.
In 1913, Niels Bohr proposed a new model of atom which explained some of these things and also
the emission spectrum of hydrogen. Bohr's theory was based on Planck's quantum theory and was
built on the following postulates.

Postulates of Bohr's theory

- The single hydrogen electron could travel around the nucleus in various possible orbits but
only certain orbits in which the electron possessed a whole number of quanta of energy
were permissible.
- No energy was radiated while the electron was rotating in a permissible orbit.

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- The permissible orbits were called stationary states or energy levels.

- The atom could be excited by an electric discharge or strong heat.
- The electron could absorb energy and jump to one of the higher energy levels.
- The excited state was however unstable and the electron could fall back to the ground state.
- As the electron falls back to the lower energy level, it gives out some of its energy in the
form of radiation. The wavelength of the radiation emitted was determined by the energy
difference of the electron in the two energy levels.
From
ℎ𝑐
Planck's equation, 𝐸2 − 𝐸1 = ℎ𝑣 = where h is Planck's constant.𝜂 the frequency of a
𝛾
photon emitted or absorbed energy.

The weaknesses of Bohr’s atomic model

- Bohr’s theory cannot explain the atomic spectra of atoms with more than one electron.
- Bohr’s model could not explain the hydrogen atomic spectrum under the influence of
external magnetic field i.e. the Zeeman effect, when the spectral line is split into several
components in the presence of a magnetic field.
- It could not explain the hydrogen spectrum under the influence of an external electric field.
i.e. the Stark effect, when the spectral line gets split into fine lines in the presence of an
electric field.

Bohr's explanation of hydrogen spectrum

Hydrogen atom has only one electron occupying energy level, 𝑛 = 1 where it is stable. When
energy is supplied to hydrogen in the discharge tube, the electrons jump to higher energy level
(𝑛 = 2,3,4,5,6, . . . ) depending on the quantity of energy absorbed where it becomes unstable,
then drops back to the ground level where it is stable due to nuclear attraction.
The process of returning to energy level (ground state) may occur in steps and whichever energy
level it returns to, if emits radiation of different wavelength which gives rise to different lines in
the emission spectrum.
The Lyman series in the emission spectrum arises when the electrons move to the 𝑛 = 1 level, the
Balmer, Paschen, Bracket and Pfund series arise from transitions to 𝑛 = 2, 𝑛 = 3, 𝑛 = 4 and 𝑛 =
5 from the higher orbitals respectively.
Series N M Region of the series
Lyman 1 2,3,4 Ultraviolet
Balmer 2 3,4,5 Visible
Paschen 3 4,5,6 Infrared
Bracket 4 5,6,7 Infrared
Pfund 5 6,7,8 Infrared

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The wavelength of the lines and series to which they belong are related to the equation;
1 1 1
= 𝑅𝐻 ( 2 − 2 )
λ 𝑛1 𝑛2
Where 𝜆 = wavelength of a particular line
𝑅𝐻 = Rydeberg's constant (10967800𝑚−1 )
𝑛1 and 𝑛2 are the quantum numbers for the final and initial energy levels respectively.

Worked example
1. If an electron falls from an energy level 𝑛 = 1 to 𝑛 = 3 an energy level . Calculate the;
i) Wavelength in nanometers of the emitted radiation.
ii) The frequency
iii) Change in energy of the radiation.
(Rydberg constant = 109678𝑐𝑚−1 , velocity of light = 3 × 108 𝑚𝑠 −1 , Plank’s constant =
6.6256 × 10−34 𝐽𝑠)

Evidence for the existence of energy levels in an atom

The hydrogen spectrum provides evidence for the existence of energy levels in an atom. This is
because;
- Hydrogen has one electron yet it produces a spectrum containing many lines separated
from each other.
- There are several series of lines.eg. Lyman series, Balmer series, e.t.c each series
representing a particular energy level to which an electron returns.
- The spacing between adjacent lines in each series differs.
- There is a continuum in each series of lines.
- The light giving each line is of definite energy or frequency.
- Give out light of different colours.
- Each of the big lines comprises if two small lines.

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Using the spectrum to find ionization energy of hydrogen

If enough energy is supplied to move the electron up to the infinity level, the atom will be ionized.
The energy gap between the ground state and the point at which the electron leaves the atom can
𝐶
be determined by 𝛥𝐸 = ℎ𝑓 where 𝑓 = where 𝑓 is the frequency, C is speed of light, 𝝀 is the
𝜆
wavelength, 𝒉 is Planck's constant.

Significance or application of the line spectrum

- It shows that within an atom of elements, there are different energy levels (sub-energy or
orbitals) within which the elections are distributed around the nucleus.
- It also shows that the energy difference between these energy levels diminishes as the
distance from the nucleus increases.
- Spectral lines are used in the determination of first ionization energy of different elements.
- Spectral lines are also used in the identification of different metal ions.ie.by the flame test.

Information obtained from separate lines in the spectrum

- The lines give confirmation about the electronic structure of the hydrogen atom.
- The separate lines indicate that there are discrete or definite energy levels in the hydrogen
atom.
- The energy possessed by an electron is quantized.
- Electrons can only exist in specific regions called orbits or energy levels.

Worked example
a. State any two characteristics of the emission spectrum of hydrogen.
- The spacing between the lines in each series decreases until they convergence to a
continuous spectrum (continuum)
- The lines in the visible spectrum have different colours.
- There are bright lines on a dark background.
- The hydrogen spectrum consists of a series of lines in both visible and invisible
region of the electromagnetic spectrum.
b. Explain why hydrogen has one electron yet produces a spectrum containing many lines
Solution
When light is passed through hydrogen atom get excited to lighter energy levels where it
is unstable by absorption of energy as it moves from energy level to the next one in stages.
Each amount of energy absorbed corresponds to a specific have length.
The energy absorbed in each stage gives rise to a clear line in the absorption spectrum.
c. Calculate the frequency of radiation emitted when electron falls from 𝑛 = 4 to 𝑛 = 1
d. Find the wave length of the line formed in the Balmer series when an electron falls for third
energy level.

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Dufile Seed SS S.5 Chemistry, 2024 Term 1

ELECTRONIC DISTRIBUTION IN ATOMS

An orbital is the volume of space around the nucleus where there is a high probability of finding
electrons. There are four orbitals (sub-energy levels) in which electrons are found. These include
𝑠, 𝑝, 𝑑 and 𝑓.
An orbit is a path traced by an electron revolving around the nucleus of an atom

Orbitals and their maximum electrons

Sub-energy levels Maximum
(orbitals) number of electrons
𝑠 2
𝑝 6
𝑑 10
𝑓 14

The table below shows the energy levels and sub-energy levels for the first four principal quantum
numbers.
Energy, n Sub-energy levels Total number of electrons
1 1𝑠 2=2
2 2𝑠2𝑝 2+6=8
3 3𝑠3𝑝3𝑑 2 + 6 + 10 = 18
4 4𝑠4𝑝4𝑑4𝑓 2 + 6 + 10 + 14 = 32

Note: The orbitals (sub-energy levels) differ in energy. Each orbital takes a maximum of two
electrons.
The table below shows how electrons fill the orbitals in each sub-energy level.
Sub-energy level Number of orbitals Electrons
𝑠 1 1×2= 2
𝑝 3 3×2= 6
𝑑 5 5 × 2 = 10
𝑓 7 7 × 2 = 14

Chemist also often represent an orbital as a box that can hold up to 2 electrons. Each electron is
shown as a half arrow indicating different spin. With an orbital, electrons must have opposite
spins.

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ELECTRONIC CONFIGURATION
The electronic configuration of an atom is the distribution of the available electrons of an atom or
ion in the various atomic extra nuclear region.
The following rules are followed in writing electronic configuration.

General laws in writing electronic configuration

a. Aufuban's principle
It states that electrons enter orbitals of lowest energy first. The electrons are added to
atomic orbitals starting with the lowest energy orbitals and building up to higher energy
orbitals
The order is as follows:

1𝑠 < 2𝑠 < 2𝑝 < 3𝑠 < 3𝑝 < 4𝑠 < 3𝑑 < 4𝑝 < 5𝑠 < 4𝑑 < 5𝑝 < 6𝑠 < 4𝑓 < 5𝑑 < 6𝑝 <
7𝑠 < 5𝑓 < 6𝑑 < 7𝑝

b. Pauli exclusion principle

It states that all the electrons in any atom must be distinguishable or that no two electrons
in a single atom can have all their quantum numbers alike. Only two electrons can occupy
an orbital and must have opposite spins.
So the maximum number of electrons in s-orbitals is 2𝑠 2
The maximum number of electrons in p-orbitals is 6𝑝6
The maximum number of electrons in d-orbital is 10𝑑10
The maximum number of electrons in f-orbital is 14𝑓 14
Principal quantum number, n represents a group of energy levels (orbits) indicating energy
possessed by an element.

c) Hund's rule of degenerate orbitals (or maximum multiplicity)

According to the rule, when electrons occupy orbitals of equal energy, they do not pair up in
an orbital until all the other orbitals in the sub energy level have been occupied by a single
electron. Eg.

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The table below shows the electronic configuration of some elements

Questions
a. What is the difference between the emission line spectrum and the absorption line
spectrum.
b. What is meant by the term quantum number?
c. Explain;
i. Why hydrogen has one electron yet it produces a spectrum containing many lines.
ii. The spacing between adjacent lines in each series differ.
iii. There is a continuum in each series of lines.
d. What is the significance of the line spectrum?

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RADIOACTIVITY
Radioactivity is the spontaneous disintegration of unstable atoms of an element by giving new
stable and fresh nuclides with emission of either alpha particles, Beta or Gamma rays. The
elements whose atoms disintegrate and emit radiations are called radioactive elements.

Types of radioactivity
a) Natural radioactivity. Is when a substance emits radiations by itself.
b) Induced or artificial radioactivity. Occurs when a substance starts emitting radiations on
exposure to rays from some natural radioactive substance.

Nature and characteristics of radioactive emissions

The nature of the radiations emitted from a radioactive substance was investigated by Rutherford
(1904) by applying electric and magnetic fields. When these radiation were subjected to electric
or magnetic field, these were split into three types 𝛼, 𝛽 and 𝛾 rays as shown below.

Characteristics of radioactive rays

Nature Radiations
𝛼 − 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝛽 – 𝑟𝑎𝑦𝑠 𝛾 − 𝑟𝑎𝑦𝑠
Charge and mass It positively charge and It’s negatively It has no charge and
4 unit mass. charge and no negligible mass.
mass.
Action of magnetic Deflected towards the Deflected to anode. Not deflected.
field cathode.

Ionizing power Very high ionization Moderate ionization Very low.

power power
Effect on 𝑍𝑛𝑆 plate They cause Very little effect. Very little effect.
luminescence.
Penetrating power Low Moderate Highest penetrating
penetrating power power
than 𝛼 −particles.

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Range Travels shorter Faster than 𝛼 - The fastest

distance particles.

Theory of radioactive disintegration

Rutherford and Soddy, in 1903, postulated that radioactivity is a nuclear phenomenon and all the
radioactive changes are taking place in the nucleus of the atom. They presented an interpretation
of the radioactive processes and the origin of radiations in the form of a theory known as theory
of radioactive disintegration. During radioactivity;-
- The atomic nuclei of the radioactive elements are unstable and liable to disintegrate any
moment.
- The disintegration is spontaneous, i.e. constantly breaking. The rate of breaking is not
affected by external factors like temperature, pressure, chemical combination etc.
During disintegration, atoms of new elements called daughter elements having different
physical and chemical properties than the parent elements come into existence.
- During disintegration, either alpha or beta particles are emitted from the nucleus.
Balancing radioactive equations:
In balancing radioactive equations, the
i. sum of the atomic numbers on the left should equal to the sum of atomic numbers
on the right hand side.
ii. sum of the mass numbers on the left hand side should be equal to that on the right
hand side.

The disintegration process may proceed in one of the following two ways,
a. 𝛼 − particle emission ( 42𝐻𝑒). When an is emitted from the nucleus of an atom of the
parent element, the nucleus of the new element, called daughter element
Its atomic mass or atomic mass number less by four units and nuclear charge or atomic
number less by 2 units because 𝛼 -particle has mass of 4 units and nuclear charge of
two units.
Characteristics
- 𝛼 –particle are positively charged, i.e helium atom
- They are deflected towards negative plate in an electric field.
- 𝛼 –particle has lowest penetrating power.
- 𝛼 -particle are heavier than 𝛽 –particle and 𝛾 –rays
Examples
235 234 4
92𝑈 → 90𝑇ℎ + 2𝐻𝑒
226 222 4
88𝑅𝑎 → 86𝑅𝑛 + 2𝐻𝑒

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b. 𝛽 − particle emission ( −10𝑒).

Whenever a beta particle is emitted from the nucleus of a radioactive atom, the
nucleus of product have the same atomic mass but nuclear charge or atomic number
is increased by 1 unit than the parent element.
Beta particle emission is due to the result of decay of neutron into proton and electron.
Characteristics
- 𝛽 –particle are negatively charged
- They are deflected towards positive plate in an electric field.
- 𝛽 –particle has moderate penetrating power.

NUCLEAR FISSION AND NUCLEAR FUSION

Definition

Nuclear fission: The splitting of a heavier atom like that of uranium – 235 into a number of
fragments of much smaller mass, by suitable bombardment with sub-atomic particles with
liberation of huge amount of energy
235 140 93
92𝑈 + 10𝑛 → 56𝐵𝑎 + 36𝐾𝑟 + 3 10𝑛 + energy

Application of Nuclear fission

i. Manufacture of atomic bomb: An atomic bomb is based upon the process of that
nuclear fission in which no secondary neutron escapes the lump of a fissile material.
There is accordingly a sudden release of a tremendous amount of energy, which
represents an explosive force much greater than that of the most powerful TNT bomb
ii. Atomic pile or Nuclear reactor: It is a device to obtain the nuclear energy in a controlled
way to be used for peaceful purposes. The most common reactor consists of a large
assembly of graphite (an allotropic form of carbon) blocks having rods of uranium metal
(fuel).

Nuclear fusion: Is defined as a process in which lighter nuclei fuse together to form a heavier nuclei.
The energy released in the process is sufficient to maintain the temperature and to keep the
process going on.
4
2𝐻𝑒 + 2 01𝑛 → 4 11𝐻

Hydrogen bomb: Hydrogen bomb is based on the fusion of hydrogen nuclei into heavier ones by
the thermonuclear reactions with release of enormous energy.

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Difference between Nuclear fission and fusion

Nuclear fission Nuclear fusion

The process occurs only in the nuclei of heavy The process occurs only in the nuclei of light
elements. elements.
The process involves the fission of the heavy The process involves the fission of the lighter
nucleus to the lighter nuclei. nuclei to heavy nucleus.
The process can take place at ordinary The process takes place at higher temperature
temperature.
The energy liberated during this process is high
The energy liberated during the process is
(200 MeV per fission) comparatively low
(3 to 24 MeV per fusion)
Percentage efficiency of the energy conversion Percentage efficiency of the energy conversion
is comparatively less. is high (four times to that of the fission
process).
The process can be controlled for useful The process cannot be controlled.
purposes.

Exercise
1. Complete each of the following equations
226 4
i. 88𝑅𝑎 → ……………+ 2𝐻𝑒

ii. 16
7𝑁 → ……………+ −10𝑒
14
iii. 6𝐶 → 𝛽 + ……………
73 73
iv. 33𝐴𝑠 + ……………→ 32𝐺𝑒
60
v. 27𝐶𝑜 → 𝛽 + ……………
238
vi. 92𝑈 → ……………+ 42𝐻𝑒
238
vii. 92𝑈 → ……………+ −10𝑒
96 2
viii. 42𝑀𝑜 + 1𝐻 → ……………+ 10𝑛
223
ix. 91𝑃𝑎 → 𝛽 + …………… → 42𝐻𝑒 + ……………
226 219
x. 88𝑅𝑎 → ……………+ 86𝑅𝑛 → …………… + 215
84𝑃𝑜
238 95
xi. 92𝑈 + 10𝑛 → 42𝑀𝑜 + ……………+ 2 10𝑛 + 7𝛽

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Nuclear stability
Nuclear stability is the ability of the nucleus of an atom to resist spontaneous decay that would
result into formation of other nuclei and emission of radiations.
The stability of a nucleus is related to the ratio of the number of neutrons to the number of
𝑛
protons. This ratio is represented as (𝑝)
Comparison between stable and unstable nuclei
Stable nucleus Unstable nucleus
Number of protons is approximately The number of neutrons is much greater than the
equal to the number of neutrons number of protons or has a high number of protons
than neutrons
Does not undergo radioactive Undergoes radioactive decay emitting alpha and
disintegration. beta particles to attain stability
Those with odd numbers of both protons Those with even numbers of both protons and
and neutrons are least stable neutrons are most stable.

Note:
𝑛
- Lighter elements with mass number up to 20 have the 𝑝 ≅ 1.
- Element with atomic number > 20, addition of protons is not favoured because of the
creation of more repulsive forces in the nucleus. Addition of neutrons is favoured up to
𝑛
= 1.6
𝑝
- Beyond this value, atomic nuclei become unstable due an increased number of protons
that lead to increased repulsive forces within the nuclei.
The nuclei therefore emit radiations to become stable and get converted to new nuclei.

Factors affecting nuclear stability

𝑛 𝑛
i. Nuclear stability depends on the value. Nuclei that have value equal to one are
𝑝 𝑝
𝑛 𝑛
stable. Those with 𝑝 > 1 have more neutrons than protons and those with 𝑝 < 1 have
more protons than neutrons hence unstable.
ii. Atomic number
Stability of the nucleus decreases as the atomic number increases. Atoms with atomic
number 83 are have unstable nuclei.
iii. Mass number
As mass number increases, the number of protons in the nucleus also increases,
repulsive forces increase much more rapidly than the attractive forces making the
nucleus unstable.

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iv. Half-life
The higher half-life of a radioactive isotope, the slower the decay rate and the nucleus
becomes more stable and the lower the half-life, the higher the decay rate and the
nucleus is less stable. For example iodine-131 with half-life about 8 days is less stable
than Uranium-238 with half-life of about 4.5 billion years.
v. Binding energy
This is the energy given out when a nucleus is formed from its constituent neutrons and
protons or The energy required to separate the nucleus into its constituent neutrons
and protons.
The greater the binding energy, the more stable the nucleus and the lower the binding
energy, the less stable the nucleus.

Variation of number of neutrons and protons

- The dotted line is a line along which number or protons is equal to number of neutrons. It
𝑛
is called the stability line and along this line 𝑝 = 1
- The stability belt or stability region or band of nuclear stability is the area on the graph
where all stable nuclei lie. Any nuclei out of this band are unstable and undergo
radioactivity to achieve stability.
- Region X
In this region, number of neutrons is greater than number of protons. The neutron-proton
ratio is greater than one. To stabilize the nucleus, the neutron–proton ratio can be reduced
by reducing the number of neutrons and increasing the number of protons.
This happens by beta emission which reduces the number of neutrons by one and increases
the number of protons by one.

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- Region Y
In this region, number of protons is greater than number of neutrons. The neutron-proton
ratio is less than one. To stabilize the nucleus, the neutron–proton ratio can be increased
by reducing the number of protons and increasing the number of neutrons. This happens
by electron capture or positron emission.
- Region Z
In this region, the nuclides have atomic number greater than 83 and are heavy. To stabilize
the nucleus, both the number of protons and the number of neutrons must be reduced. This
happens by a series of alpha emissions.

Rate of radioactive decay

“According to the law of radioactive decay, the quantity of a radio-element which disappears in
unit time (rate of disintegration) is directly proportional to the amount present.”

Decay law
It states that the rate of decay is directly proportional to the number of un decayed atom or un
decayed mass of a given sample of an element.

Let N be the number of atoms of the radioactive element present at time 𝑡 = 0, and after time 𝑡,
the number of atoms remaining after 𝑡 = 𝑡 is 𝑁𝑡 . The rate of disintegration
𝑑[𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠]
Using 𝑟𝑎𝑡𝑒 = 𝑑[𝑡𝑖𝑚𝑒]
𝑑𝑁
= 𝒌[𝐴]
𝑑𝑡
𝑑𝑁
= 𝒌𝑁𝑡
𝑑𝑡
𝑑𝑁 = 𝒌𝑁𝑡 𝑑𝑡
𝑑𝑁
= 𝒌𝑑𝑡
𝑁𝑡
Integrating on both sides
𝑁=𝑵 𝑡=𝒕
𝑑𝑁
∫ = 𝒌 ∫ 𝑑𝑡
𝑁𝑡
𝑁=𝟎 𝑡=𝟎

ln 𝑁𝑡 ⋮0𝑥 = −𝒌𝑡

ln 𝑁𝑡 − ln 𝑁0 = −𝒌𝑡
𝑁
ln 𝑁𝑡 = −𝒌𝑡
0
𝑁𝑡
= 𝑒 −𝒌𝑡
𝑁0
𝑁𝑡 = 𝑁0 𝑒 −𝒌𝑡 OR 𝑁𝑜 = 𝑁𝑡 𝑒 𝒌𝑡
Let 𝑁𝑡 be the concentration at any time, 𝑡 as 𝑵 and 𝑵𝟎 be the initial concentration of the
reactant, the

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Dufile Seed SS S.5 Chemistry, 2024 Term 1

𝑵 = 𝑵𝟎 𝒆−𝒌𝒕
Or
Taking 𝑙𝑜𝑔10 on both side
𝑁
log10 ( 𝑁𝑜 ) = log10 𝑒 𝑘𝑡
𝑡
𝑁𝑜 1
log10 ( 𝑁 ) = 𝑘𝑡 (log )
𝑡 10
𝑁𝑜 𝑘𝑡
log10 ( 𝑁 ) = log
𝑡 10
𝑁𝑜
log10 × log10 ( 𝑁 ) = 𝑘𝑡
𝑡
𝑁0
2.303𝑙𝑜𝑔 ( 𝑁 ) = 𝒌𝑡
𝑡

When a graph of is plotted against time, a straight line graph from the origin but with a
negative gradient is obtained as shown below
𝑁
However for log10 (𝑁𝑡 ) against 𝒕, it gives
𝑜
𝒕
log10 𝑁𝑜

−𝒌
Slope, 𝑘 = 𝟐.𝟑𝟎𝟑
log10 𝑁𝑡

𝑁
2.303𝑙𝑜𝑔 ( 𝑁0 ) = 𝒌𝑡
𝑡
𝑁0 𝒌
𝑙𝑜𝑔 ( 𝑁 ) = (𝟐.𝟑𝟎𝟑) 𝑡
𝑡
𝑁
However when 𝑙𝑜𝑔 ( 𝑁0 ) is plotted against time, 𝒕 then it gives
𝑡

𝑁
𝑙𝑜𝑔 ( 𝑁0 )
𝑡

𝒌
𝑘= 𝟐.𝟑𝟎𝟑

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Dufile Seed SS S.5 Chemistry, 2024 Term 1

HALF-LIFE
This is the time taken for a radioactive material to disintegrate to half of its original amount,
it’s denoted as (𝑡1⁄ )
2
𝑁0
At half life, 𝑁𝑡 = 2
𝑁
Then 2.303𝑙𝑜𝑔 ( 𝑁00 ) = 𝒌𝑡1⁄
2
2
2.303𝑙𝑜𝑔(2) = 𝒌𝑡1⁄
2
2.303𝑙𝑜𝑔(2)
𝑡1⁄ =
2 𝒌

0.693
𝑡1⁄ =
2 𝒌

From the integrated rate equation for a first order reaction, a plot of 𝑁𝑡 against 𝒕 gives

𝒕𝟎 𝒕𝟏 𝒕𝟐

𝒕
For a first order reaction, 𝒕𝟎 = 𝒕𝟏 = 𝒕𝟐 = 𝒕𝟏⁄
𝟐

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Worked examples

1. A radioactive isotope decays from an initial count of 160 counts per minute to 20 counts
per minute in 27 days. Calculate its half-life.

2. The nuclide of carbon-14 has a half-life of 5600 years. Calculate the age of a wood from an
ancient tomb, if this wood gave 10 counts per minute per gram of carbon compared with
the 15 counts that are given by the carbon from new wood.

3. If the decay constant for radium is 1.356 × 10−11 𝑠 −1 . Calculate the time required for 90%
of a sample of radium to disintegrate.

4. The half-life of Radium is 1590 years. How long will it take for a sample of Radium to decay
to 25% of its original amount?

5. The half-life of radioisotope bromine-82 is 36 hours. Calculate the fraction of a sample of

the isotope will remain after one day.

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Dufile Seed SS S.5 Chemistry, 2024 Term 1

6. The half-life of a radioactive element is 150 seconds. What percentage of the isotope will
remain after 600 seconds?

7. If a sample of a radioactive isotope has a half-life of 3.11 hours and an activity of 1000 at a
certain time, calculate the activity one hour later.

8. It takes 1620 years for 0.03 moles of Radium-226 to decay to 0.015 moles. Calculate the
number of moles of Radium-226 left when 26g decayed for 6 years?

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Dufile Seed SS S.5 Chemistry, 2024 Term 1

Use of radioactive isotopes

- Destruction of cancer tissue e.g cobalt 60 destroys cancerous cells by emitting gamma
particles.
- For sterilizing surgical instruments.
- Helps to detect underground leakages in water pipes. This is done when a short-lived
radioactive isotope is introduced into such pipes, the presence of the radioactivity on the
surface indicates escape of water or fuel from the pipelines.
- Carbon-14 radioactivity is used to calculate the age of plants or animals which have lived
long ago.
- Radioactive isotopes are used to detect/trace path of an element through the body.
- Detect engine wear; they can also be detected by measuring the rate of radioactivity I the
engine oil. Normally piston rings that are made up of radioactive are used.
- Investigating the mechanism of the reaction

Difference between radioactive reactions and ordinary reactions

- Radioactivity reaction are affected by physical factors e.g pressure, temperature etc while
chemical reactions are not affected by them.
- Radioactive reactions involve the nucleus which splits to give flesh nuclides while chemical
reactions involve valence shell electrons.
- In nuclear reactions a lot of heat is given off while in chemical reactions give less heat.

Application of radioactivity

Radioisotopes find numerous applications in a variety of areas such as medicine, agriculture,

biology, chemistry, archeology, engineering and industry.

Hazards of radiations

- It destroys body cells

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Dufile Seed SS S.5 Chemistry, 2024 Term 1

Exercise

1. The activity of 234 234

90𝑇ℎ was reduced to 25% in 50days. Determine the half-life of 90𝑇ℎ
2. The sample of wood has initial activity of 260 counts per second on Geiger counter. After
30 minutes the activity declined to 250 counts per second. Calculate the half-life of the
sample of the wood.
3. A certain radioactive phosphorus decay is found to be 30% at the end of 50 minutes.
Calculative the;-
i. The decay constant
ii. The time taken for the decay to be 90% completive.

4. A compound A is converted to B and C as according to the equation;-

𝐴→𝐵+𝐶
The data below was obtained for the reaction above
Time/min 0.0 7.2 18.0 36.0 72.0 108.0
[𝐴]/𝑚𝑜𝑙 −1
100 91 79 63 40 25
a. Plot a graph of concentration of [𝐴] against time.
b. Using your graph, complete the table below
Change in concentration of A Time taken/min
From To
80 40 56
60 30 54
c. Determine;-
i. The half-life of the reaction
ii. The order of the reaction
iii. The rate constant of the reaction and its unit.

5. The rate constant for the conversion of a radioactive sample of phosphorus into silicon is
2.82 × 10−3 . Calculate the fraction of its original sample that was left after 200 seconds.

6. A dose of 1.0 × 10−4 of Astatine 211 is given to a patient of cancer of thyroid gland. If the
half-life of Astatine is 7.21hrs. Calculate how much remained in the body remains in the
body after 24hrs.

7. The table below shows data for radioactive decay of element W.

Time (hours) 0.0 5.0 10.0 15.0 20.0 25.0 30.0
Activity 25.00 23.00 21.25 19.50 18.00 16.50 15.25
(counts per minute)

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Dufile Seed SS S.5 Chemistry, 2024 Term 1

a) Plot a graph of activity against time

b) Determine the half-life of element W.
c) Determine the decay constant and state its units.

8. A sample of bromine was irradiated in a nuclear reactor. The table below shows the
radioactivity count rates at various times
Time (hours) 0 1.0 2.0 5.0 10 25 50 75 100
Count rate 500 268 242 225 204 154 95 55 35
a) Plot a graph of count rate against time
b) Use your graph to determine;
i) half-life.
ii) the rate constant in 𝑠 −1
iii) order of the reaction.

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