The graph between |\psi|^2 and r (radial distance) is shown below. This represents
(a) 1s-orbital
(b

Q: Why does the graph start from a maximum value at $r=0$?

For s-orbitals ($l=0$), the electron probability density is maximum at the nucleus. For p, d, and f orbitals, the wave function has a node at the nucleus, so their graphs start from zero.

Q: What is the difference between a radial node and an angular node?

Radial nodes are spherical surfaces where probability is zero, depending purely on the distance $r$ from the nucleus. Angular nodes are planes or cones where probability is zero, depending on the direction (angles).

Q: Why isn't the answer 2p?

A 2p orbital has $n=2$ and $l=1$, giving $2-1-1 = 0$ radial nodes. Also, its $|\psi|^2$ graph would start from zero at the origin, not a maximum.

Q: Does the curve ever touch the axis again after the node?

No, it approaches the axis asymptotically as $r \to \infty$. This means the probability gets infinitesimally small at large distances but never exactly reaches zero again.

Q: How do I quickly identify s-orbital graphs?

Any $|\psi|^2$ or $\psi$ graph that starts from a non-zero maximum on the y-axis belongs to an s-orbital. All other orbitals (p, d, f) start exactly at the origin.

Question

The graph between $|\psi|^2$ and $r$ (radial distance) is shown below. This represents

Options:

(A) 1s-orbital

(B) 2p-orbital

(C) 3s-orbital

(D) 2s-orbital

Sigma Percentile · Accepted Answer

### Decoding the Quantum Graph

Imagine you are an explorer trying to map out the exact location of an electron around a nucleus. The graph provided in the question is your map. It plots the **probability density**, denoted mathematically as $|\psi|^2$, against the radial distance $r$ from the nucleus.

Notice how the curve starts at a high value on the y-axis. This immediately tells us a profound secret: the electron has a high probability of being found very close to the nucleus. This behavior is an exclusive signature of **s-orbitals** ($l=0$). For any other orbital (like p, d, or f), the probability density at the nucleus is strictly zero.

### The Secret of Radial Nodes

As we trace the curve moving away from the nucleus, the probability density drops sharply, hits absolute zero, and then rises again to form a smaller secondary peak before finally fading away into the distance.

That exact point where the curve touches the horizontal axis ($|\psi|^2 = 0$) is called a **radial node**. It is a spherical region in space where the probability of finding the electron is absolutely zero.

In quantum mechanics, the number of radial nodes for any given orbital is governed by a beautifully simple formula:

$$ 	ext{Radial Nodes} = n - l - 1 $$

Where $n$ is the principal quantum number and $l$ is the azimuthal quantum number.

### Testing the Candidates

By simply looking at our graph, we can count exactly **one** radial node. Now, let's put our four candidates to the test using our formula:

For the **1s-orbital** ($n=1, l=0$):
$$ 1 - 0 - 1 = 0 	ext{ nodes} $$

For the **2p-orbital** ($n=2, l=1$):
$$ 2 - 1 - 1 = 0 	ext{ nodes} $$

For the **3s-orbital** ($n=3, l=0$):
$$ 3 - 0 - 1 = 2 	ext{ nodes} $$

For the **2s-orbital** ($n=2, l=0$):
$$ 2 - 0 - 1 = 1 	ext{ node} $$

### The Final Verdict

The math aligns perfectly with our visual evidence. The 2s-orbital is the only candidate that possesses exactly one radial node. Furthermore, because it is an s-orbital, it correctly starts from a maximum probability density at the nucleus. Therefore, the graph undeniably represents the **2s-orbital**.

Animated Solution for Chemistry - Atomic Structure: The graph between $∣ ψ ∣^{2}$ and $r$ (radial distance) is shown below. This represents

Select Answer:

Visualized Solution

$∣ ψ ∣^{2} vs r Graph$

$Concept of Radial Nodes$

$Nodes in the Given Graph$

$Evaluating the Options$

$Conclusion$

$Food for Thought$

The Sigma Insight: Quantum Mechanical Model

Decoding the Quantum Graph

The Secret of Radial Nodes

Testing the Candidates

The Final Verdict

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Animated Solution for Chemistry - Atomic Structure: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The graph between ∣ψ∣2 and r (radial distance) is shown below. This represents