Q.The simple Bohr model is not applicable to $\text{He}^4$ atom because

The key idea is that the Bohr model applies only to single-electron systems (hydrogen-like atoms). It fails for multi-electron atoms because electron-electron repulsion breaks the simple central-force assumption.

Reasoning:

1. The Bohr model assumes a single electron orbiting a nucleus under a pure Coulomb force.
2. $\text{He}^4$ has two electrons (atomic number $Z=2$). …

The Bohr model is a beautiful, simple picture: an electron moves in a circular orbit around a point‑like nucleus, held by the Coulomb attraction. That works perfectly for hydrogen (one electron) and for ions like $\text{He}^+$, $\text{Li}^{2+}$, etc. — any system with exactly one electron.

But the moment you have two or more electrons, the story changes. Each electron feels not only the pull of the nucleus but also the push from every other electron. Those electron‑electron repulsions are not directed toward the nucleus — they are off‑centre, making the net force on any electron non‑central. The Bohr model’s entire foundation (a central, inverse‑square force) collapses.

Let’s go through the options one by one.

1. Option (A): “$\text{He}^4$ is an inert gas.”

   Inertness is a chemical property — it tells you about bonding, not about the validity of a physical model. The Bohr model fails for helium not because it’s inert, but because it has two electrons. In fact, the Bohr model also fails for lithium, beryllium, etc. — none of which are inert. So this is a red herring.

2. Option (B): “$\text{He}^4$ has neutrons in the nucleus.”

   Neutrons add mass but do not affect the Coulomb force (they are neutral). The Bohr model treats the nucleus as a point charge $+Ze$ — the number of neutrons is irrelevant. $\text{He}^+$ (one electron) has the same neutrons and the Bohr model works perfectly for it. So this is not the reason.

3. Option (C): “$\text{He}^4$ has one more electron.”

   This is almost correct — but it’s not the number of electrons alone that matters; it’s the fact that the extra electron introduces electron‑electron repulsion. The phrasing “one more electron” is vague. Compared to hydrogen, yes, helium has one more electron. But the Bohr model also fails for $\text{Li}$ (two more electrons), $\text{Be}$ (three more), etc. The key is that the forces are no longer central, not just the count.

4. Option (D): “electrons are not subject to central forces.” …

Method: Trace the Assumption Chain Back to Its Source

Rather than listing surface facts about helium, work backwards through the Bohr model's derivation chain to find exactly which mathematical step breaks down — this pinpoints the true reason a candidate cause fails, rather than a superficially plausible one.

Steps

Step 1: Recall where "central force" enters the Bohr derivation

The entire Bohr quantization result, $L=n\hbar \Rightarrow E_n = -13.6/n^2$, is derived by balancing the Coulomb force against the required centripetal force for a single electron orbiting a single point charge:

This equation assumes the net force on the electron points straight at the nucleus at every instant — i.e. it's a central force problem.

Step 2: Check what changes with a second electron present

With two electrons (as in He$^4$), each electron feels the nuclear attraction plus a repulsion from the other electron. Since the second electron is generally not sitting exactly opposite or exactly at the nucleus, that repulsion is not directed along the line to the nucleus — so the net force on either electron is no longer central.

Step 3: Rule out superficially-plausible but non-causal options …