Bohr Model of the Atom: Class 11 Chemistry Notes on Principles to Limitations

The Bohr model of the atom proposes that electrons occupy fixed, quantised orbits around the nucleus. They do not lose energy. But these electrons change orbits. They either absorb or emit specific amounts of energy. 

Rutherford initially described a dense, positively charged nucleus with electrons orbiting it.

But the main flaw was that Rutherford based his model on classical electromagnetic theory, as proposed by Maxwell. The old thought was that an accelerating charged particle, like an orbiting electron, should continuously emit energy.

So it was considered that the orbiting electron would cause it to spiral inward and collapse into the nucleus. This, of course, does not happen.

How Does an Electron's Energy Change Like Moving Seats in a Stadium?

Let’s understand Bohr's model intuitively. One easy example is to observe spectators moving between different seating sections in a stadium. 

1. Moving from the lower rows to the upper sections of seats requires more energy. The same is with electrons. They need specific amounts of energy to jump to higher orbits. 
2. Spectators must sit in designated rows (sections A, B, C...).  The same goes for electrons. They can exist only in specific energy levels (n=1, 2, 3...).
3. Higher stadium sections are farther from the field (nucleus). Higher energy electrons orbit farther from the atomic nucleus. 
4. In the same way, we can consider the movement to better seats closer to the field. That will release potential energy. This would be similar to electrons emitting photons when dropping to lower energy levels. 
5. Lower sections offer better views and stability. Just as lower energy levels provide greater electron stability. 
6. Spectators move section by section, not to random spaces. That is also how electrons transition between specific energy levels. 

It’s truly fascinating about the arrangement of electrons in Bohr’s atomic model. It follows a systematic, shell-based structure. Without this, Bohr wouldn’t be able to explain how atoms are stable and that they have spectroscopic properties. 

Shell-Based Electron Architecture

Circular Orbits of Electrons are in Orbits

Electrons are arranged in concentric circular shells around the central nucleus. Each shell represents a distinct energy level, with electrons maintaining fixed distances from the nucleus while in these stable orbits.

Shell Designations and Electrons

• K shell (n=1): This is the closest to the nucleus. It has a maximum capacity of 2 electrons
• L shell (n=2): The second shell, L, has a maximum capacity of 8 electrons
• M shell (n=3): M is the third shell. It can have up to 18 electrons
• N shell (n=4): N, being the fourth shell, has a maximum capacity of 32 electrons

 

The general formula for maximum electrons in any shell is 2n². Remember, n is the principal quantum number.

Alright, so, the principle quantum number is the energy and size of the electron orbital's within an atom. It is always a positive integer. 

Electrons are Distributed Based on their Energy

Ground State of Electrons

In the ground state, we know that electrons have the lowest energy. Having next to no energy means that low-energy electron will occupy the lowest energy level first. 

As the K shell is right next to the nucleus, the electrons with lowest energy will be here. Then electrons higher above this energy, will occupy the next shell, that is the L shell.  

How an Electron Transitions Between Shells

• When there is an energy supply, electrons jump from lower shells (lower n) to higher shells (higher n). That is energy absorption. 
• Suppose electrons fall from higher shells to lower shells, they will release energy. That will be electromagnetic radiation. And it is energy emission. 
• Electrons can only exist in these specific shells. They don’t exist between them. 

How Electrons are Positioned

Here, you will need to use your physics mind.

Fixed Orbital Radii

Each shell has a definite radius that increases with the principal quantum number. 

For the Bohr’s model of the hydrogen atom, we need to first note this calculation that comes from Bohr radius formula. 

• K shell (n=1): radius = 0.529 Å
• L shell (n=2): radius = 4 × 0.529 = 2.116 Å
• M shell (n=3): radius = 9 × 0.529 = 4.761 Å

Angular Momentum Quantisation 

Electrons in each shell have quantised angular momentum values given by mvr = nh/2π. 

This systematic electron arrangement in discrete energy shells explains the periodic properties of elements. This arrangement and forms the basis for understanding chemical bonding patterns in the periodic table.

 

When reading the Structure of Atoms chapter in Class 11 Chemistry, Section 2.4 of the textbook outlines the principles and postulates.

Postulate 1: Fixed Circular Orbits 

Electrons are negatively charged particles that revolve around the positively charged atomic nucleus. The path is well-defined and circular. Bohr says that electrons do not emit electromagnetic radiation while moving in orbits are that stationary. 

Postulate 2: Quantised Energy Levels 

Each orbital shell had a fixed energy value. These circular orbits are termed as quantised energy levels or orbital shells. The energy associated with each orbit remains constant. This happens as long as the electron remains in that particular shell.

Postulate 3: Principal Quantum Number 

The fixed energy levels have a positive integer value (n = 1, 2, 3, 4...). That’s the principal quantum number. This quantum number sequence begins from the innermost orbit. You know by now, that’s the orbit closest to the nucleus. So, n = 1 represents the lowest energy state. These orbits are systematically labeled as K, L, M, N... shells corresponding to n = 1, 2, 3, 4... respectively. When an electron occupies the lowest possible energy level (n = 1), the atom is said to exist in its ground state.

Postulate 4: Energy Transitions and Photon Exchange 

Electrons can transition between different energy levels through energy absorption or emission. When an electron absorbs a specific quantum of energy, it moves from a lower energy orbit to a higher energy orbit (excitation). 

Alternately, when an electron moves from a higher energy level to a lower energy level, it releases energy in the form of electromagnetic radiation (photon emission).

 

Let's go over the maths and physics behind the postulates of the Bohr's model of the atom. 

• Angular momentum is quantised: $mvr=\frac{nh}{2\pi },\mathrm{}n=\mathrm{1,2},3,\dots$ where $n$ is the principal quantum number, $m$ is electron mass, $v$ is velocity, $r$ is radius, and $h$ is Planck's constant.
• Energy transitions between orbits either create photon emission or absorption: $\mathrm{\Delta }E=E_{n_2}-E_{n_1}$
• Higher $n$ corresponds to higher energy orbits.

What's the Electron Energy in Bohr's Orbit?

Have you considered why the energy of an electron in a Bohr orbit is quantised?

According to Bohr, the energy of an electron is determined by its quantum number and the atom's nuclear charge. This shows a balance of kinetic and potential energies.

For your exams, you have to learn the mathematical representation of this concept.

The energy of an electron in the $n$ th orbit for a hydrogen-like atom is: $E_n=-\frac{2{\pi }^{2}m{Z}^2{e}^4{k}^2}{n^2{h}^2}=-13.6\frac{Z^2}{n^2}\mathrm{e}\mathrm{V}$

Here $Z$ is the atomic number, $e$ is electron charge, $k$  is the Coulomb's constant.

For hydrogen $(Z=1)$ , the ground state $(n=1)$  energy is -13.6 eV .

Key Features of the Electron's Energy

• Energy in alternative units: $E_n=-2.18\times {10}^{-18}\frac{Z^2}{n^2}\mathrm{J}/\text{atom},\mathrm{}-1312\frac{Z^2}{n^2}\mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$
• Negative energy indicates a bound electron; at $n=\mathrm{\infty },E=0$ (ionized state).
• Energy becomes less negative as $n$ increases.

So, we know the electron orbits. Let's talk about the radius and velocity when considering Bohr's model.

Radius and Velocity of the Atom in Bohr's Model

NCERT mentions,

 
"The radii of the stationary states are expressed as: rn = n2 a0 (2.12), where a0 = 52.9 pm. Thus, the radius of the first stationary state, called the Bohr orbit, is 52.9 pm."

The radius and velocity of an electron's orbit in Bohr's model are quantised.

It varies with the principal quantum number and nuclear charge. That is what governs the electron's spatial and dynamic properties.

You will need to know how the radius of the $n$ th orbit is: $r_n=\frac{n^2{h}^2}{4{\pi }^{2}m{e}^{2}kZ}=\frac{n^2}{Z}\times 0.529\text{Å}$

The velocity is: $v_n=\frac{2\pi e^{2}Zk}{nh}=\frac{2.188\times {10}^{6}Z}{n}\mathrm{m}/\mathrm{s}$

For hydrogen $(Z=1)$ , the first orbit radius is $0.529\text{Å}$ , and velocity is $2.188\times {10}^6\mathrm{m}/\mathrm{s}$ .

Key Features of Orbital Velocity and Radius

• Radius scales with $n^2$ and inversely with $Z$ .
• Velocity decreases with increasing $n$ , proportional to $Z$ .

Exam questions may require calculating these parameters for hydrogen or ions like ${\mathrm{L}\mathrm{i}}^{2+}$ .

 

 

In a hydrogen atom, when an electron drops to a lower energy level, it releases a flash of light. This is the hydrogen spectrum, characterised by a distinct pattern of specific lines. 

Mathematically, we can say that the electron transitions from higher $\left(n_2\right)$ to lower $\left(n_1\right)$ orbits emit photons, with wave number given by: $\bar{v}=\frac{1}{\lambda }=R{Z}^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$ where $R=109678{\mathrm{c}\mathrm{m}}^{-1}$ (Rydberg constant).

Key spectral series include:

• $\mathrm{L}\mathrm{y}\mathrm{m}\mathrm{a}\mathrm{n}\left(n_1=1,\mathrm{U}\mathrm{V}\right):{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{4}{3R},{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}=\frac{1}{R}$ .
• Balmer $\left(n_1=2\right.$ , visible $):{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{36}{5R},{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}=\frac{4}{R}$ .
• Paschen $\left(n_1=3\right.$ , IR $):{\lambda }_{\text{max}}=\frac{144}{7R},{\lambda }_{\text{min}}=\frac{9}{R}$ .

Maximum spectral lines from $n$ -th level: $\frac{n(n-1)}{2}$

JEE questions often involve wavelength calculations or series identification.

Bohr's model of the atom does have some limitations.

• The postulates of the Bohr model apply to atom species that have a single electron. It cannot help determine with multiple electrons.
• The position and angular momentum of the electron are already considered in the Bohr model. That goes against what the Heisenberg Uncertainty Principle tells us, that we can never be certain of the position and momentum of a particle simultaneously.
• Bohr’s model cannot explain the fine structure that spectral lines have. Factors, such as electron spin leading to more energy splitting are not explainable.
• The model still relies on classical concepts like definite orbits, which contradict quantum mechanical principles and relativistic concepts.

The Bohr theory is applicable to only hydrogen and hydrogen-like ions. 

• Hydrogen Atom: The Bohr model of the atom can mathematically explain the emission and absorption spectra of hydrogen. It can predict its spectral lines by using the quantised energy levels.
• Hydrogen-like Ions: The model can also be applied to other hydrogen-like (single-electron) ions. The main condition is that they are ions with one electron only. Top examples include He⁺ (helium ion), Li²⁺ (lithium ion), and Be³⁺ (beryllium ion). In these systems, only the attraction between a single electron and the nucleus needs to be considered. So, the Bohr quantisation works well here.