THE RADIAL
DISTRIBUTION
CURVES
By NAFESA MEHEZABIN
DEPARTMENT OF CIVIL ENGINEERING
ROLL 2400163
1
INTRODUCTION
❏
In Bohr’s model, electrons moved in
fixed circular orbits.
Fig: Bohr’s circular orbit of Hydrogen
❏
Unlike Bohr’s fixed orbits, quantum
mechanics says, electrons exist in regions
called orbitals.
Fig: Quantum Mechanical Orbital Cloud of Hydrogen
The radial distribution Function(RDF) shows the probability of finding an electron at
a distance r from the nucleus. RDF helps us understand atomic size, penetration, and
shielding.
2
Atomic Orbitals
❖ What is an Atomic Orbital?
○ A region around the nucleus where an
electron is most likely to be found.
orbital has a round shape
In Short, Orbitals = regions with high
probability of finding an electron
Shapes:
s → spherical shape
p → dumbbell shape
d → cloverleaf shape
f → complex multi-lobed shape
Fig: Shape of Atomic Orbitals
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Quantum Numbers
❖
❖
❖
❖
❖
Quantum numbers are a set of
numbers used to describe the
properties of electrons in atoms There
are four main quantum numbers:
Principal Quantum Number (n): Defines the
energy level and size of the orbital (n = 1, 2,
3...).
Azimuthal Quantum Number (l): Defines the
shape of the orbital (l = 0 for s, 1 for p, 2 for
d… etc).
Magnetic Quantum Number (m_l): Defines
the orientation of the orbital in space.
mₛ (Spin Quantum Number): spin of
electron (±½).
Fig: Quantum Numbers and Orbitals
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Wavefunction & Probability
Schrödinger’s wave function is a mathematical
description of the quantum state of a particle
or system. It is usually denoted by the Greek
letter Ψ (psi).
❏
❏
In short:
Schrödinger’s wave function is a mathematical
function whose square gives the probability of
finding a quantum particle at a given place and
time.
The wave function contains all the
information about a particle, such as
where is an electron in an atom.
The solution of the Schrödinger equation
is the fundamental equation of quantum
mechanics like Newton’s laws in classical
mechanics.
Fig: Schrödinger’s Equation
2
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Understanding Probability Density (ψ )
❖
Wave Function (ψ): A mathematical
description of an electron's state in an
atom. It has no direct physical meaning.
❖ Probability Density (ψ2):
○ Represents the probability of finding
an electron at a given point in space
around the nucleus.
○ Higher ψ2 indicates higher chance of
finding the electron at that specific point.
○ For example take s-orbitals, ψ2 is
spherical and decreases as you move
further from the nucleus.
|ψ|² = probability density at a point.
For atoms,
ψ = R(r)Yₗᵐ(θ,φ)
Radial part R(r) shows how probability
changes with distance r.
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The Need for Radial Probability Function
Definition: The radial probability distribution
function gives the probability of finding an
electron within a thin spherical shell at a
distance 'r' from the nucleus
● Formula: P(r)=4πr2ψ2
○ 4πr2: Represents the surface area of
a sphere at radius r. This accounts for
the increasing volume of space at
greater distances.
○ ψ2: The probability density at that
specific point.
● This function allows us to plot Radial
Probability Distribution Curves.
Imagine a series of thin, concentric spherical onion shells
around the nucleus. The radial probability considers the
probability of finding an electron within each of these
shells.
Fig:probability of finding an electron within each of these imaginary onion
shells
Radial Probability Distribution Curves
**From |ψ|² to Radial Probability:
A thin spherical shell at radius r has area 4πr².
Probability in shell: P(r) dr = 4πr² |R(r)|² dr.
So, P(r) = 4πr² |R(r)|².
RDF is P(r) vs r
●
●
●
X-axis: Distance from the nucleus (r).
Y-axis: Radial Probability (P(r) or 4πr2ψ2).
Maximal Probability: The peaks in these
curves indicate the most probable
distance at which an electron will be
found from the nucleus.
● Nodes:
○ Regions where the probability of finding an
electron is zero.More nodes = more
oscillations in RDF
○ The number of radial nodes = n−l−1.
○ For a 1s orbital, n=1,l=0, so 1−0−1=0 radial
nodes.
○ For a 2s orbital, n=2,l=0, so 2−0−1=1 radial
node.
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Radial Probability Curve for 1s Orbital
●
Characteristics:
○ One peak.
○
○
○
Most probable distance: 0.529A˚
(Bohr radius).
0 radial node.
Probability is highest close to the
nucleus and falls off.
Interpretation: For a 1s electron, you are most
likely to find it at the Bohr radius, but it can be
found at other distances too.
Here,
●
●
●
●
n = 2, l = 0.
Radial nodes = 1.
Two peaks (inner small, outer large).
Inner peak = penetration. Outer peak
= most probable.
Fig:Radial Probability Curve for 1s Orbital
Radial Probability Curve for 2s Orbital
● Characteristics:
○ Two peaks.
○ One radial node (where P(r)=0).
○ The second peak is further from the
nucleus and generally higher.
Here,
●
Interpretation: The 2s electron has a
non-zero probability of being very close to
the nucleus, then a region where it won't be
found (node), and then a higher probability
further out.
n = 2, l = 0.
Radial nodes = 1.
Inner peak = penetration. Outer peak =
most probable.
Fig:Radial Probability Curve for 2s Orbital
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Radial Probability Curve for 2p Orbital
● Characteristics:
●
○ One peak.
○ No radial nodes (n−l−1=2−1−1=0).
○ Probability is zero at the nucleus, then
increases to a maximum, and decreases.
Interpretation: Unlike s-orbitals, p-orbitals have
zero probability of finding the electron at the
nucleus.
Here,
n = 2, l = 1.
Radial nodes = 0.
One broad peak, starts at zero near nucleus.
More diffuse than 2s; less penetration.
Fig: Radial Probability Curve for 2p Orbital
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Comparing 1s, 2s, and 2p Curves
●
Size: As 'n' increases, the most
probable distance from the nucleus
increases, and the orbital is larger.
●
Nodes: The number of radial nodes
increases with n for a given l.
●
Penetration: S-orbitals show greater
probability near the nucleus
compared to p-orbitals, even at the
same principal quantum number.
Fig: Radial Probability Curve of 1s, 2s, 2p Orbitals
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Radial Probability Curve of 3s, 3p, 3d
3s: 2 radial nodes → 3 peaks.
3p: 1 radial node → 2 peaks.
3d: 0 radial nodes → 1 peak
General rule: higher n → more peaks.
Fig: Radial Probability Curve of 3s, 3p, 3d
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Significance Of Radial Probability
Penetration & Shielding
1. Penetration: ability of orbital to get close to
nucleus. Order: s > p > d > f.
2.Shielding: inner electrons block nuclear
charge.
3. Effective nuclear charge (Z_eff): reduced
attraction felt by outer electrons.
4. RDF explains penetration & shielding clearly
Chemical Significance
1. Atomic size: outermost RDF peak ≈ atomic
radius.
2.Ionization energy: stronger penetration →
higher energy needed to remove electron.
3.Reactivity: diffuse orbitals = weakly held
electrons (e.g., alkali metals).
4.RDF links atomic structure to periodic
properties
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Summary and Conclusion
Radial Probability Distribution Curves
graphically represent the probability of finding
an electron at a specific distance from the
nucleus.
● They are derived from the square of the wave
function (ψ2) and the volume element (4πr2).
● These curves reveal crucial information about:
○ The most probable distance of an electron.
○ The presence and number of radial nodes.
○ The size and shape characteristics of
different orbitals.
● Understanding these curves is fundamental to
comprehending atomic structure and
chemical bonding.
Reference
●
1.Griffiths, D.J., Introduction to Quantum Mechanics
2.Atkins, P., Molecular Quantum Mechanics
3.Chemistry LibreTexts
4.Khan Academy – Quantum Numbers
THANK YOU