The plots of radial distribution functions for various orbitals of hydrogen atom against 'r' are giv

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

A radial node is a spherical surface at a specific distance $r$ from the nucleus where the electron probability is zero. An angular node is a flat plane or a conical surface passing through the nucleus where the probability is zero, determined by the $l$ quantum number.

Q: Why does the radial probability curve start at zero for all orbitals?

The radial probability is given by $4\pi r^2 R^2(r)$. Even though the wave function $R(r)$ for an $s$ orbital is maximum at the nucleus ($r=0$), the volume of the spherical shell $4\pi r^2$ is zero at $r=0$. Multiplying by zero makes the total radial probability zero at the nucleus.

Q: Why do the peaks get taller as distance $r$ increases for the 3s orbital?

As $r$ increases, the volume of the spherical shell ($4\pi r^2 dr$) grows significantly. Even though the electron density $|\psi|^2$ generally decreases as we move away from the nucleus, the rapidly increasing volume factor causes the outer peaks to represent a higher total probability than the inner peaks.

Q: How is radial probability different from probability density $|\psi|^2$?

Probability density $|\psi|^2$ tells you the probability of finding an electron in a tiny, specific point in space. Radial probability $4\pi r^2 |\psi|^2$ tells you the probability of finding the electron *anywhere* on the surface of a sphere of radius $r$.

Q: Can the number of peaks be zero?

No, the number of peaks is given by $n - l$. Since the maximum value of $l$ for a given $n$ is $n - 1$, the minimum number of peaks is $n - (n - 1) = 1$. Every orbital must have at least one region of maximum probability.

Question

The plots of radial distribution functions for various orbitals of hydrogen atom against 'r' are given below.
The correct plot for 3s-orbital is

Options:

(A) (A)

(B) (B)

(C) (C)

(D) (D)

Sigma Percentile · Accepted Answer

The behavior of an electron in an atom is one of the most fascinating concepts in quantum mechanics. We can't pinpoint exactly where an electron is, but we can determine the *probability* of finding it at a certain distance from the nucleus. This is exactly what a **radial probability distribution curve** shows us.

In this problem, we are given four different graphs and asked to identify which one represents the **$3s$ orbital** of a hydrogen atom. Let's break down the quantum mechanics behind these curves to find the answer.

### Decoding the Quantum Numbers

Every orbital is defined by a set of quantum numbers. For the $3s$ orbital, the name itself gives us the first two crucial pieces of information:
*   **Principal Quantum Number ($n$):** This tells us the main energy level or shell. For $3s$, $n = 3$.
*   **Azimuthal Quantum Number ($l$):** This defines the subshell and the shape of the orbital. For any $s$ orbital, $l = 0$.

These two numbers are the master keys to unlocking the shape of the radial distribution curve.

### Finding the Radial Nodes

A **radial node** is a specific spherical boundary around the nucleus where the probability of finding an electron drops exactly to zero. On our graph, this corresponds to the points where the curve touches the horizontal $r$-axis.

The number of radial nodes can be calculated using a simple formula:
$$ 	ext{Radial Nodes} = n - l - 1 $$

Let's substitute the values for our $3s$ orbital:
$$ 	ext{Radial Nodes} = 3 - 0 - 1 = 2 $$

This tells us that the correct graph must touch the $x$-axis exactly **two times** before it finally tapers off towards infinity.

### Counting the Peaks

The regions of high probability between the nodes appear as peaks on the graph. The total number of peaks in a radial probability distribution curve is given by another straightforward formula:
$$ 	ext{Number of Peaks} = n - l $$

For the $3s$ orbital:
$$ 	ext{Number of Peaks} = 3 - 0 = 3 $$

So, we are looking for a graph that features exactly **three distinct peaks**.

### The Final Verdict

Let's examine the given options with our findings:
*   Graph (A) has 1 peak.
*   Graph (B) has 2 peaks.
*   Graph (C) has 1 peak.
*   **Graph (D) has exactly 3 peaks and 2 radial nodes.**

Furthermore, for an $s$ orbital, the peaks get progressively taller as we move further from the nucleus because the volume of the spherical shell ($4\pi r^2$) increases rapidly, dominating the exponential decay of the wave function at moderate distances. Graph (D) perfectly illustrates this behavior.

Therefore, Graph (D) is the correct radial distribution curve for the $3s$ orbital.

Animated Solution for Chemistry - Atomic Structure: The plots of radial distribution functions for various orbitals of hydrogen atom against 'r' are given below. The correct plot for 3s-orbital is

Select Answer:

Visualized Solution

$3 s$ Orbital

Radial Nodes Formula

Substituting Values

Calculating Nodes

Peaks Formula

Calculating Peaks

Matching the Graph

The Way Forward

The Sigma Insight: Quantum Mechanical Model

Decoding the Quantum Numbers

Finding the Radial Nodes

Counting the Peaks

The Final Verdict

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