Animated Solution for Chemistry - Atomic Structure: Given below are two statements. Statement I Bohr's theory accounts for the stability and line spectrum of ${\text{Li}}^{+}$ ion. Statement II Bohr's theory was unable to explain the splitting of spectral lines in the presence of a magnetic field. In the light of the above statements, choose the most appropriate answer from the options given below :

The journey of understanding the atom is one of the most fascinating stories in modern physics and chemistry. When Niels Bohr proposed his atomic model in 1913, it was nothing short of a scientific revolution. By introducing the radical concept of quantized angular momentum, Bohr successfully explained the stability of the hydrogen atom and perfectly predicted its emission spectrum, solving mysteries that classical physics could not. However, like all early scientific theories, Bohr's model had its limitations. This problem tests our deep understanding of exactly where Bohr's brilliant theory hits a wall and fails to describe physical reality.

Let's break down the two statements provided in the question to uncover the boundaries of the Bohr model and understand the evolution of atomic theory.

The Single-Electron Constraint

Statement I claims: Bohr's theory accounts for the stability and line spectrum of ${\text{Li}}^{+}$ ion.

To evaluate the truth of this statement, we must first recall the fundamental mathematical and physical assumptions of Bohr's model. Bohr derived his elegant equations for the radius, velocity, and energy of an electron by setting up a simple two-body problem: a single positively charged nucleus at the center and a single negatively charged electron orbiting it. The only forces at play in Bohr's derivation are the electrostatic Coulombic attraction between the nucleus and the electron, and the centripetal force keeping the electron in its circular orbit.

The moment you add a second electron into the mix, the mathematical simplicity completely breaks down. Why? Because now you have to account for the electron-electron repulsion. This creates a complex three-body problem (one nucleus, two electrons) that cannot be solved with Bohr's simple algebraic equations. The repulsive forces between electrons alter the energy levels in ways that Bohr's formula, $E_n=-13.6\frac{Z^2}{n^2}\text{ eV}$, simply cannot predict.

Therefore, Bohr's theory is strictly valid only for single-electron species, often referred to as hydrogenic atoms. Examples of species that obey Bohr's model include Hydrogen ($\text{H}$), singly ionized Helium (${\text{He}}^{+}$), doubly ionized Lithium (${\text{Li}}^{2+}$), and triply ionized Beryllium (${\text{Be}}^{3+}$). Notice that all of these have exactly one electron.

Now, let's look at the lithium ion (${\text{Li}}^{+}$) mentioned in the statement. The atomic number of Lithium ($Z$) is 3. This means a neutral lithium atom has 3 protons in its nucleus and 3 electrons orbiting it. When it forms a ${\text{Li}}^{+}$ ion, it loses exactly one electron:

Since ${\text{Li}}^{+}$ has 2 electrons, it is a multi-electron system. The electron-electron repulsions make it impossible for Bohr's theory to accurately predict its stability or the wavelengths of its line spectrum.

Thus, Statement I is absolutely false.

The Magnetic Mystery

Statement II claims: Bohr's theory was unable to explain the splitting of spectral lines in the presence of a magnetic field.

When scientists began examining the emission spectra of atoms with more powerful and precise spectrometers, they discovered something peculiar. If a glowing gas (which is emitting a characteristic line spectrum) is placed in a strong external magnetic field, the single spectral lines suddenly split into multiple, closely spaced lines. This fascinating phenomenon is known as the Zeeman effect, named after the Dutch physicist Pieter Zeeman.

Could Bohr's model explain this splitting? To answer that, we have to look at the geometry of Bohr's atom. Bohr envisioned electrons moving in flat, two-dimensional circular orbits. His model only utilized one quantum number: the principal quantum number ($n$), which determined the size of the orbit and the energy of the electron.

However, the Zeeman effect is a fundamentally three-dimensional phenomenon. It occurs because an electron orbiting a nucleus acts like a tiny current loop, which in turn creates a tiny magnetic dipole moment. When you apply an external magnetic field, this atomic magnet interacts with it. Depending on how the electron's orbit is oriented in 3D space relative to the external magnetic field, the energy of the electron shifts slightly up or down. This slight shift in energy levels causes the emitted photons to have slightly different frequencies, resulting in the splitting of the spectral line.

Because Bohr's model lacked the concept of spatial quantization—the idea that orbits can only have specific, quantized orientations in 3D space—it was completely blind to this effect. Bohr's flat orbits had no way to account for different spatial alignments.

It wasn't until the development of the more advanced Quantum Mechanical Model, which introduced the magnetic quantum number ($m_l$), that the Zeeman effect was finally understood. Similarly, Bohr's model also failed to explain the Stark effect, which is the splitting of spectral lines in the presence of an external electric field.

Therefore, Bohr's theory indeed failed to explain this magnetic splitting. Statement II is perfectly true.

By carefully analyzing the limitations of Bohr's atomic model, we have determined that: 1. Statement I is false because ${\text{Li}}^{+}$ is a multi-electron species, and Bohr's model cannot handle electron-electron repulsions. 2. Statement II is true because Bohr's 2D orbits could not account for the 3D spatial quantization required to explain the Zeeman effect.

Matching this logical deduction with our given options, the correct choice is (b) Statement I is false but statement II is true.

This problem serves as a beautiful reminder of how science progresses. Bohr's model was a crucial stepping stone that bridged classical and quantum physics, but its ultimate failures paved the way for the more profound, mathematically rigorous, and accurate Quantum Mechanical Model that we use today.