Vector Atom Model (Detailed)

phy-1109 PHY-1109 Physics

---

Postulates of Bohr’s Model of an Atom

• In an atom, electrons (negatively charged) revolve around the positively charged nucleus in a definite circular path called orbits or shells.
• Each orbit or shell has a fixed energy and these circular orbits are known as orbital shells.
• The energy levels are represented by an integer (n=1, 2, 3…) known as the quantum number. This range of quantum number starts from nucleus side with n=1 having the lowest energy level. The orbits n=1, 2, 3, 4… are assigned as K, L, M, N…. shells and when an electron attains the lowest energy level, it is said to be in the ground state.
• The electrons in an atom move from a lower energy level to a higher energy level by gaining the required energy and an electron moves from a higher energy level to lower energy level by losing energy.

Limitations of Bohr’s Theory

Bohr’s planetary model, while foundational, had several significant deficiencies that necessitated further development of atomic theory:

• It introduced only one quantum number ($n$).
• It was insufficient to explain the fine structure of optical lines (which required additional quantum numbers).
• It applied primarily to one-electron atoms and could not be easily extended to describe more complicated or many-electron atoms.
• It could not be used to calculate transition rates or determine selection rules.
• It failed to provide a quantitative explanation of chemical bonding.

Related Previous Year Questions (PYQs)

Question	Year(s) of Appearance
What are the basic postulates of Bohr Atom Model? Calculate the limiting values of wavelengths of different spectral series of Hydrogen atom.	2018, 2020
State the postulates of Bohr atom model. Obtain the expressions for the radius and electron energy levels in the $n^{th}$ orbit for $H_2$ atom.	2024
Find the wavelength of the spectral line corresponding to the transition in the hydrogen atom from $n=4$ to $n=2$ state.	2020
How much energy is required to remove an electron in the $n=2$ state from a Hydrogen atom?	2018

II. Sommerfeld’s Relativistic Model

Sommerfeld partially succeeded in explaining fine structure by extending Bohr’s theory.

Extension Method	Description
Elliptical Orbits	Allowed the possibility of elliptical electron orbits in addition to Bohr’s circular orbit.
Relativistic Mass	Incorporated the relativistic variation of electron mass with velocity, as mass differs at various points of the elliptical orbit.

Permitting elliptical orbits requires dealing with two variable quantities in polar coordinates: the varying distance ($r$) and the varying angular position ($\varphi$).

To deal with these two variables, two quantum numbers are required:

1. The original Principal Quantum Number ($n$), which determines the electron’s energy.
2. The Orbital or Azimuthal Quantum Number ($l$), which is useful in finding the possible elliptical orbits.

(No PYQs explicitly)

III. Quantum Numbers and Angular Momenta

The sources define two main sets of four quantum numbers, depending on whether the spin-orbit interaction is considered.

A. Four Quantum Numbers (Neglecting Spin-Orbit Interaction)

1. Principal quantum number ($n$)
2. Orbital quantum number ($l$)
3. Orbital magnetic quantum number ($m_l$)
4. Magnetic spin quantum number ($m_s$)

B. Four Quantum Numbers (Including Spin-Orbit Interaction)

When spin-orbit interaction is taken into account, the following four quantum numbers are required:

Quantum Number	Range and Definition	Function
1. Principal ($n$)	Range: $n=1,2,3,\dots ,\infty$ .	Determines the energy of the electron and the size of the orbit.
2. Orbital/Azimuthal ($l$)	Range: $l$ varies from $0$ to $(n-1)$ (or $1$ to $(n-1)$ ).	Determines the shape of the electron orbits and the orbital angular momentum.
3. Total Angular ($j$)	Values: $j=l+s$ and $j=l–s$ , with the restriction that $j$ must be positive.	Describes the coupling between the spin and orbital angular momenta.
4. Magnetic ($m_j$)	Range: $+j\ge m_j\ge -j$ , excluding zero. Can have $(2j+1)$ values.	Represents the numerical value of the total angular momentum vector ($j$) on the direction of the applied external field . Results from space quantization of the total angular momentum ($p_j$).

C. Magnetic Moments

• The ratio of the orbital magnetic dipole moment (${\mu }_l$) to the orbital angular momentum ($p_l$), (${\mu }_l/p_l$), is called the gyromagnetic ratio (G).
• The unit of magnetic moment, ${\mu }_B$, is the Bohr Magneton.
• The spin magnetic moment (${\mu }_s$) of a spinning electron is also defined.

Related Previous Year Questions (PYQs)

Question	Year(s) of Appearance
Mention name of quantum numbers associated with vector atom model. Define space quantization.	2024
Discuss magnetic quantum number and total angular quantum number.	2022
State correspondence principle. Mention name of quantum numbers associated with vector atom model. Define space quantization.	2024

IV. Electronic Distribution and Principles

The electronic distribution within an atom is governed by specific rules and principles.

A. Electron Configuration Rules

Shell/Subshell Property	Governing Rule
Total Electrons in a Shell ($n$)	The maximum total number of electrons with the same principal quantum number $n$ is $2n^2$.
Subshells	In the $n$ th shell, there are $n$ subshells having different values of $l$, ranging from $0,1,2,\dots ,(n-1)$.
Maximum Electrons in a Subshell ($l$)	Each subshell can have a maximum of $2(2l+1)$ electrons.

B. Pauli’s Exclusion Principle

This principle dictates that in a single atom, no two electrons can have the same exact set of values for the four quantum numbers ($n,l,m_l,$ and $m_s$). This rule fundamentally restricts the number of electrons an atom can possess.

(No PYQs)