Which of the following combination of statements is true regarding the interpretation of the atomic

Q: Why does statement I use Bohr's model when we usually use wave mechanics for orbitals?

While wave mechanics is more accurate, the general principle that higher angular momentum (associated with higher principal quantum number $n$) corresponds to a larger average distance from the nucleus holds true conceptually, which is why the simpler Bohr analogy is often used to justify it.

Q: Isn't the size of an orbital dependent on $l$ as well?

While $l$ affects the shape and the exact radial distribution (like penetration), the primary determiner of the overall 'size' or extent of the electron cloud is the principal quantum number $n$.

Q: Why is the ground state angular momentum zero in wave mechanics?

In the $1s$ state, the electron cloud is perfectly spherical and has no preferred axis of rotation. It doesn't 'orbit' the nucleus in a classical sense, hence its orbital angular momentum is exactly zero.

Q: Why do the peaks shift to the right for higher $l$ in the $\psi$ vs $r$ graph?

The radial wavefunction contains a term proportional to $r^l$. For higher $l$, this term suppresses the wavefunction near the nucleus (a phenomenon related to the centrifugal barrier), pushing the peak probability further out.

Q: What is the difference between $\psi$ and $\psi^2$?

$\psi$ is the wavefunction amplitude, which can be positive or negative and has no direct physical meaning. However, $\psi^2$ (or more accurately, $4\pi r^2 \psi^2$) represents the actual probability density of finding the electron at a distance $r$.

Question

Which of the following combination of statements is true regarding the interpretation of the atomic orbitals?
I. An electron in an orbital of high angular momentum stays away from the nucleus than an electron in the orbital of lower angular momentum.
II. For a given value of the principal quantum number, the size of the orbit is inversely proportional to the azimuthal quantum number.
III. According to wave mechanics, the ground state angular momentum is equal to $\frac{h}{2\pi}$.
IV. The plot of $\psi$ vs $r$ for various azimuthal quantum numbers, shows peak shifting towards higher $r$ value.

Options:

(A) I, III

(B) II, III

(C) I, II

(D) I, IV

Sigma Percentile · Accepted Answer

The behavior of electrons within an atom is one of the most fascinating subjects in quantum mechanics. This problem tests our conceptual understanding of atomic orbitals, angular momentum, and the visual representation of wavefunctions. Let's break down each statement to uncover the truth.

### Analyzing Statement I: The Classical vs Quantum Dance

Statement I claims that an electron in an orbital of high angular momentum stays further away from the nucleus.

To understand this, we can look back at the Bohr model. According to Bohr, the angular momentum of an electron is quantized and given by the formula:
$$mvr = \frac{nh}{2\pi}$$

Here, we can clearly see that the angular momentum is directly proportional to the principal quantum number, $n$.

We also know that the radius of an orbit increases as $n$ increases. Therefore, a higher angular momentum naturally correlates with a larger distance from the nucleus. Statement I is **correct**.

### Analyzing Statement II: What Really Determines Size?

Statement II suggests that for a given principal quantum number $n$, the size of the orbit is inversely proportional to the azimuthal quantum number $l$.

Is this true? Not quite. The overall "size" or the spatial extent of an electron cloud is primarily dictated by the principal quantum number $n$.

While the azimuthal quantum number $l$ determines the **shape** of the orbital (spherical for $s$, dumbbell for $p$, etc.) and affects how close the electron can penetrate towards the nucleus, it does not make the overall size inversely proportional to $l$. Thus, Statement II is **incorrect**.

### Analyzing Statement III: The Surprising Ground State

Statement III states that according to wave mechanics, the ground state angular momentum is $\frac{h}{2\pi}$.

Let's consult the quantum mechanical formula for orbital angular momentum:
$$L = \sqrt{l(l+1)}\frac{h}{2\pi}$$

For the ground state of an atom (the $1s$ orbital), the principal quantum number is $n=1$, and the azimuthal quantum number is $l=0$.

Substituting $l=0$ into our formula yields an angular momentum of exactly **zero**. The electron cloud is perfectly spherical and has no net rotation. Therefore, Statement III is **incorrect**.

### Analyzing Statement IV: Visualizing the Wavefunction

Statement IV describes the plot of the wavefunction $\psi$ against the radial distance $r$ for various azimuthal quantum numbers, claiming that the peaks shift towards higher $r$ values.

When we plot the radial wavefunctions for states where $n = l+1$ (such as $1s, 2p, 3d, 4f$), we observe a distinct pattern.

For $l=0$ ($1s$), the maximum probability density is right at the nucleus ($r=0$).

As we increase $l$ to $1$ ($2p$), the peak shifts to the right. Increasing $l$ further to $2$ ($3d$) and $3$ ($4f$) pushes the peaks even further away from the nucleus. This is due to the centrifugal barrier term $r^l$ in the radial wavefunction, which suppresses the probability of finding the electron near the nucleus for higher $l$ states.

This visual behavior perfectly matches the description. Statement IV is **correct**.

### Final Conclusion

After a thorough analysis, we have determined that Statements I and IV are the only correct interpretations.

This leads us to the final answer, which corresponds to option **(d)**. Understanding these nuances is key to mastering the quantum mechanical model of the atom!

Select Answer:

Visualized Solution

$Analyzing the Statements$

$Statement I: Angular Momentum$

$Statement II: Size of Orbit$

$Statement III: Ground State Angular Momentum$

$Statement IV: Radial Wavefunction Plot$

$Peak Shifting in ψ vs r$

$Conclusion$

The Sigma Insight: Quantum Mechanical Model

Analyzing Statement I

Analyzing Statement II

Analyzing Statement III

Analyzing Statement IV

Final Conclusion

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