JEE Mains Chemistry · Structure of Atom

Orbitals: Nodes, Shapes and Probability Plots

An orbital is a region where the electron is likely to be found. Its nodes, its shape and its probability plots all follow from n and l.

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Why this matters

Fifteen PYQs, twelve of them multiple choice. Seven count radial and angular nodes or find where a node sits from the wave function; eight read probability-density and radial-distribution plots, boundary surfaces and orbital shapes. Two ideas cover the page.

Concept 1 of 2: Radial and angular nodes

A node is where the wave function is zero, so the electron is never found there. An orbital has

n-1

nodes in all.

l

of them are angular: flat planes or cones through the nucleus. The rest,

n-l-1

, are radial: spherical shells at a fixed distance. A radial node sits where the radial part of

\psi

changes sign.

Definition

Radial nodes $=n-l-1$ .
Angular nodes (nodal planes) $=l$ .
Total nodes $=n-1$ .
s orbitals have no angular node; p have one plane; d have two.
To locate a radial node, set the factor of $\psi$ that can change sign to zero. For hydrogen 2s, $\psi\propto\left(2-\dfrac{r}{a_0}\right)e^{-r/2a_0}$ gives $r=2a_0$ .

Node counts

\text{radial}=n-l-1,\qquad \text{angular}=l

Worked example

Find the radial and angular nodes of a 5f and a 6p orbital.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 6 Apr 2026 Shift 1 · Q28Moderate

Example 1 · Structure of Atom · Orbitals: Nodes, Shapes and Probability Plots

Match List-I with List-II :

	List-I (Orbital)		List-II (Radial nodes and nodal plane)
(A)	2s	(I)	1 Radial node + two nodal planes
(B)	3s	(II)	1 Radial node + one nodal plane
(C)	3p	(III)	2 Radial nodes + No nodal plane
(D)	4d	(IV)	1 Radial node + No nodal plane.

Choose the correct answer from the options given below:

Radial nodes are n − l − 1

Not

n-l

and not

n-2

. A 3s orbital has

3-0-1=2

radial nodes; a 3p has 1.

A nodal plane is an angular node

Nodal planes count

l

, not radial nodes. A 2s orbital has one radial node and no nodal plane.

Drill 6 more on radial and angular nodes

Concept 2 of 2: Probability plots, boundary surfaces and shapes

Two different plots describe an orbital.

\psi^2

is the probability density at a point. The radial distribution

4\pi r^2\psi^2

is the probability of finding the electron in a thin shell at distance

r

. For 1s, the density is highest at the nucleus, but the shell is most likely at

a_0

, because a shell far out has more volume.

Definition

$\psi^2$ is never negative; it can only touch zero at a node.
s orbitals have $\psi^2$ greatest at the nucleus; p and d orbitals have $\psi^2=0$ there.
For hydrogen 1s, $4\pi r^2\psi^2$ peaks at $a_0$ . The electron can still be found at any distance.
The radial distribution of an orbital has $n-l$ peaks, one more than its radial nodes.
A boundary surface encloses about 90% of the probability, not 100%.
Lobes along the axes: $p_x, p_y, p_z, d_{x^2-y^2}, d_{z^2}$ . Between the axes: $d_{xy}, d_{yz}, d_{xz}$ .
The $2p_x$ nodal plane is the yz plane. The + and − signs on lobes are phases of $\psi$ , not charges.
An orbital is fixed by $n$ , $l$ and $m_l$ ; $m_s$ labels the electron, not the orbital.

Orbital	ψ² at the nucleus	Radial nodes	Peaks in 4πr²ψ²	Shape
1s	Maximum	0	One, at a₀ for H	SphereQ The density ψ² peaks at the nucleus; only the radial probability peaks at a₀.
2s	Maximum	1, at 2a₀ for H	Two, the outer one larger	SphereQ
2p	Zero	0	One, at 4a₀ for H	Two lobes, one nodal plane
3s	Maximum	2	Three	Sphere
3p	Zero	1	Two	Two lobes, one nodal plane
3d	Zero	0	One, at 9a₀ for H	Four lobes (d z² has two lobes and a ring), two nodal surfaces

For hydrogen, an orbital with no radial node (l = n − 1) has its radial peak at

n^2a_0

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 4 April 2025 · Q45Moderate

Example 2 · Structure of Atom · Orbitals: Nodes, Shapes and Probability Plots

Which one of the following about an electron occupying the 1 s orbital in a hydrogen atom is incorrect ? (Bohr's radius is represented by

a_{0}

)

Density is not radial probability

For 1s,

\psi^2

is largest at the nucleus, while

4\pi r^2\psi^2

is largest at

a_0

. Read the axis label before you read the curve.

Signs on lobes are not charges

The + and − on a p or d orbital are the sign of the wave function. They matter for bonding, not for charge.

Drill 7 more on probability plots, boundary surfaces and shapes

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Radial and angular nodes
Node counts
$\text{radial}=n-l-1,\qquad \text{angular}=l$

Reference tables (1)

Probability plots, boundary surfaces and shapes6 rows

Orbital	ψ² at the nucleus	Radial nodes	Peaks in 4πr²ψ²	Shape
1s	Maximum	0	One, at a₀ for H	SphereQ The density ψ² peaks at the nucleus; only the radial probability peaks at a₀.
2s	Maximum	1, at 2a₀ for H	Two, the outer one larger	SphereQ
2p	Zero	0	One, at 4a₀ for H	Two lobes, one nodal plane
3s	Maximum	2	Three	Sphere
3p	Zero	1	Two	Two lobes, one nodal plane
3d	Zero	0	One, at 9a₀ for H	Four lobes (d z² has two lobes and a ring), two nodal surfaces

For hydrogen, an orbital with no radial node (l = n − 1) has its radial peak at

n^2a_0

Watch out for (4)

Radial nodes are n − l − 1
→ Radial and angular nodes
A nodal plane is an angular node
→ Radial and angular nodes
Density is not radial probability
→ Probability plots, boundary surfaces and shapes
Signs on lobes are not charges
→ Probability plots, boundary surfaces and shapes

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