Shapes of Orbitals

Visualize and understand the 3D shapes of s, p, and d orbitals including nodes and probability distributions.

5 min read

Introduction

Orbitals are 3D regions in space where there’s a high probability of finding an electron. Understanding their shapes is crucial for predicting chemical bonding and molecular geometry.

Orbital Shapes Diagram

Explore the shapes of s, p, and d orbitals with their phases and nodal planes:

Shapes of Atomic Orbitalss OrbitalSpherical (l = 0)xynucleusNo angular nodesRadial nodes = n-1p_x OrbitalDumbbell along x-axisxy-+nodal plane (yz)p_y OrbitalDumbbell along y-axisxy-+nodal plane (xz)p_z OrbitalDumbbell along z-axisxzy-+nodal plane (xy)d Orbitals (l = 2, two angular nodes)d_xy4 lobes between x,y axesxy++--d_xz4 lobes between x,z axesxz++--d_yz4 lobes between y,z axesyz++--d_x2-y24 lobes along x,y axesxy++--d_z2Dumbbell + torusxz++-Key Concepts+Positive phase-Negative phaseNucleusNodal plane (probability = 0)Angular nodes = lRadial nodes = n - l - 1Total nodes = n - 1Phase important for bonding!

Interactive Demo: Orbital Shapes

Explore the 3D shapes of different atomic orbitals:

s Orbitals

Shape

Spherical - same probability in all directions from the nucleus.

Characteristics

OrbitalRadial nodesSize
1s0Smallest
2s1Larger
3s2Largest

Key Points

  1. Spherically symmetric: probability depends only on distance from nucleus
  2. No angular nodes: l = 0, so no directional dependence
  3. Number of radial nodes = n - l - 1 = n - 1

1s vs 2s

The 2s orbital:

  • Has a larger average radius
  • Has one radial node (spherical surface where ψ = 0)
  • Has a small inner lobe closer to nucleus

p Orbitals

Shape

Dumbbell (or figure-8) shaped with two lobes on opposite sides of the nucleus.

Three Orientations

OrbitalOrientationNodal plane
pₓAlong x-axisyz plane
pᵧAlong y-axisxz plane
p_zAlong z-axisxy plane

Characteristics

  1. Start from n = 2 (no 1p orbital)
  2. One angular node (nodal plane through nucleus)
  3. All three p orbitals are degenerate (same energy in absence of field)
  4. Radial nodes = n - 2

Phase Signs

The two lobes have opposite phases (shown as + and - or different colors). This is important for bonding!

For JEE
The three 2p orbitals have identical shapes but different orientations. Together they fill space uniformly (spherical distribution when all three are equally occupied).

d Orbitals

Shape

More complex shapes with four lobes or a dumbbell with a ring.

Five d Orbitals

OrbitalShapeNodal planes
dₓᵧ4 lobes between x and y axesxz, yz
dₓ_z4 lobes between x and z axesxy, yz
dᵧ_z4 lobes between y and z axesxy, xz
d_{x²-y²}4 lobes along x and y axesx=y, x=-y
d_{z²}Dumbbell along z + ring in xy plane2 conical surfaces

Characteristics

  1. Start from n = 3 (no 1d or 2d orbitals)
  2. Two angular nodes (l = 2)
  3. All five are degenerate in free atom
  4. d_{z²} has unique shape but same energy as others
d Orbital Geometry
  • dₓᵧ, dₓ_z, dᵧ_z: Lobes between axes (at 45°)
  • d_{x²-y²}: Lobes along x and y axes
  • d_{z²}: Lobes along z-axis with a “donut” in xy plane

f Orbitals

For completeness (rarely asked in JEE Main):

  • 7 orbitals (l = 3, mₗ = -3 to +3)
  • Complex multilobed shapes
  • Start from n = 4
  • 3 angular nodes

Nodes

Types of Nodes

A node is a region where the probability of finding an electron is zero (ψ = 0).

TypeDescriptionFormula
Radial nodesSpherical surfacesn - l - 1
Angular nodesPlanes or conesl
Total nodesAll nodesn - 1

Node Counting Examples

OrbitalnlRadial nodesAngular nodesTotal
1s10000
2s20101
2p21011
3s30202
3p31112
3d32022
4s40303
Common Mistake

Don’t confuse radial and angular nodes!

  • Radial nodes: Depend on n and l, are spherical surfaces
  • Angular nodes: Depend only on l, are planes or cones

Radial Probability Distribution

What is it?

The probability of finding an electron at a distance r from the nucleus, considering all directions.

$$P(r) = 4\pi r^2 |\psi|^2$$

Key Observations

  1. Maximum probability occurs at specific distances
  2. For 1s orbital: maximum at r = a₀ (Bohr radius)
  3. For 2s orbital: two maxima, with node between them

Distance of Maximum Probability

For hydrogen atom:

  • 1s: r = a₀ = 0.529 Å
  • 2s: r ≈ 5.2a₀ (second maximum)
  • 2p: r = 4a₀

Boundary Surface Diagrams

These are the common representations of orbitals showing surfaces that enclose ~90-95% probability.

What They Show

  1. Shape of the orbital
  2. Size (relative)
  3. Phase (+ and - regions)
  4. Nodal planes (where lobes meet)

What They Don’t Show

  1. Actual electron path
  2. Radial nodes (internal structure)
  3. Exact probability values

Orbital Energies

In Hydrogen (Single electron)

Energy depends only on n:

$$E_n \propto -\frac{1}{n^2}$$

So 3s = 3p = 3d in energy.

In Multi-electron Atoms

Energy depends on both n and l:

$$1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < ...$$

This is due to electron-electron repulsion and shielding effects.

Summary Table

OrbitallShapeAngular nodesmₗ values
s0Sphere00
p1Dumbbell1 plane-1, 0, +1
d24-lobed/ring2 planes-2 to +2
f3Complex3 planes-3 to +3

Practice Problems

  1. How many radial nodes are in a 4p orbital?

  2. Sketch the shape of dₓᵧ and d_{x²-y²} orbitals. How do they differ?

  3. At what distance from the nucleus is the probability of finding a 1s electron maximum?

  4. Which has more nodes: 3d or 4s? Calculate each type.

Quick Check
Why is there no 2d orbital? Explain using quantum numbers.