This article is on the **Structure Of Atom Class 11 Notes** of **Chemistry**. The notes on the** structure of atom** of **class 11 chemistry** have been prepared with great care for the students so that they can give a quick glance of the chapter. The **Notes** on the **Structure of atom** of Class 11 has been divided into two articles. This article (Part 2) includes **Bohr’s model**, **Heisenberg’s Uncertainty principle**, **Quantum mechanical model**, **Quantum numbers**, **Pauli’s principle**, **Aufbau principle** and **Hund’s Rule**. The second article (Part 1) is on **Dalton’s Atomic Theory**, **Discovery of electrons**, **protons** and **neutrons**, **Atomic models**, **electromagnetic radiations** and **Atomic spectrum**.

**Structure of Atom (Atomic Structure) Part 2**

**BOHR’S ATOMIC MODEL**

Bohr proposed a quantum mechanical model of the atom, to overcome the objections of Rutherford’s model and to explain the hydrogen spectrum. This model was based on the quantum theory of radiation and the classical laws of physics. The important postulates on which Bohr’s model is based are the follows:

**(a) **The atom has a nucleus where all the protons and neutrons are present. The size of the nucleus is very small. It is present at the centre of the atom.

**(b) **Negatively charged electrons are revolving around the nucleus in a similar way as the Planets are revolving around the Sun.

**(c) **Out of infinite number of possible circular orbits around the nucleus, the electron can revolve only on those orbits whose angular momentum is an integral multiple of h/2π, that is, mvr = n(h/2π) where m is the mass of the electron, v is the velocity of electron, r is the radius of the orbit and n = 1, 2, 3,…. number of the orbit. The angular momentum can have values such as h/2π, 2h/2π, 3h/2π, etc., but it cannot have a fractional value. Thus, the angular momentum is quantized. The specified or circular orbits (quantized) are called stationary orbits.

**(d) **When the electron remains in any one of the stationary orbits, it does not lose energy. Such a state is called ground of normals. In the ground state potential energy of electron will be minimum, hence it will be the most stable state.

**(e) **Each stationary orbit is associated with a definite amount of energy. The greater is the distance of the orbit from the nucleus, more shall be the energy associated with it. These orbits are also called energy levels and are numbered as 1, 2, 3, 4, ….or K, L, M, N,… from nucleus outward, i.e. E_{1 }< E_{2} < E_{3} < E_{4} …. (E_{2} – E_{1}) > (E_{3} – E_{2}) > (E_{4 }– E_{3}).

**(f) **The emission or absorption of energy in the form of radiation can occur only when an electron jumps from one stationary orbit to another. ∆E = E_{high} – E_{low }= hν; energy is absorbed when the electron jumps from an inner to an outer orbit and it is emitted when the electron moves from an outer to an inner orbit. When the electron moves from an inner to an outer orbit by absorbing a definite amount of energy, the new state of the electron is said to be an excited state.

**Interpretation of Hydrogen Spectrum**

The maximum number of lines produced when an electron jumps from nth level to ground level is equal to n(n – 1)/2. For example, in the case of n = 4, the number of lines produced is six (4 → 3, 4 → 2, 4 → 1, 3 → 2, 3 → 1, 2 → 1). When an electron returns from n_{2} to n_{1 }state, the number of lines in the spectrum will be equal to (n_{2} – n_{1} )( n_{2} – n_{1} + 1)/2.

If the electron returns from energy level having energy E_{2} to energy level having energy E_{1}, then the difference may be expressed in terms of energy of the photon as E_{2} – E_{1} = ∆E = hν. Or the frequency of the emitted radiation is given by ν = ∆E/h.

Since ∆E can be only definite values depending on the definite energies of E_{2} and E_{1}, ν will have only fixed values in an atom, or ν = c/λ = ∆E/h or λ = hc/∆E.

**Limitations of Bohr’s Model**

**(a) **It does not explain the spectra of multi-electron atoms.

**(b) **It is observed by using a high resolving power spectroscope that a spectral line in the hydrogen spectrum is not a simple line but a collection of several lines which are very close to one another. This is known as fine spectrum. Bohr’s theory does not explain the fine spectra of even the hydrogen atom.

**(c) **Spectral lines split into a group of inner lines under the inﬂuence of the magnetic field (Zeeman effect) and electric field (Stark effect); but, Bohr’s theory does not explain this.

**(d) **Bohr’s theory is not in agreement with Heisenberg’s uncertainty principle.

** PARTICLE AND WAVE NATURE OF ELECTRON**

In 1924, **Louis de Broglie **proposed that an **electron**, like light behaves both as a material particle and as a wave. This proposal gave birth to a new theory as **wave mechanical theory of matter**. According to this theory, the electrons, protons and even atoms, when in motion, possess wave properties. de Broglie derived an expression for calculating the wavelengths of the wave associated with the electron. Using Planck’s equation,

E = hν = h. c/λ …(i)

On the basis of Einstein’s mass-energy relationship, the energy of a photon is

E = mc^{2 } …(ii)

where c is the velocity of the electron

Equating both the equations, we get = h. c/λ = mc^{2 }; λ = h/mc = h/p

The momentum of the moving electron is inversely proportional to its wavelengths.

Let the kinetic energy of the particle of mass ‘m’ is E. E = ½ mv^{2}; 2Em = m^{2}v^{2}

√2Em = mv = p(momentum); λ = h/p = h/√2Em

**HEISENBERG’S UNCERTAINTY PRINCIPLE**

Heisenberg, in 1927 gave a principle about the uncertainties in the simultaneous measurement of position and momentum (mass × velocity) of small particles. This principle is due to the consequence of dual nature of matter.

**This Principle States: **‘It is impossible to measure simultaneously the position and momentum of a small microscopic moving particle with absolute accuracy or certainty’, i.e. if an attempt is made to measure any one of these two quantities with higher accuracy, the other becomes less accurate. The product of the uncertainty in position (∆x) and the uncertainty in the momentum (∆p = m.∆v where m is the mass of the particle and ∆v is the uncertainty in velocity) is equal to or greater than h/4π where h is the Planck’s constant. Thus, the mathematical expression for the Heisenberg’s uncertainty principle is readily written as ∆x.∆p ≥ h/4 π

**Explanation of Heisenberg’s uncertainty principle: **Let us attempt to measure both the position and momentum of an electron; to pinpoint the position of the electron we have to use light so that the photon of light strikes the electron and the reﬂected photon is seen in the microscope. As a result of the hitting, both the position and the velocity of the electron are disturbed. The accuracy with which the position of the particle can be measured depends upon the wavelength of the light used. The uncertainty in position is ± λ. The shorter the wavelength, the greater is the accuracy. But shorter wavelength means higher frequency and hence higher energy. This high energy photon on striking the electron changes its speed as well as direction. But this is not true for a moving macroscopic particle. Hence Heisenberg’s uncertainty principle does not apply to macroscopic particles.

**QUANTUM MECHANICAL MODEL OF ATOMS**

Erwin Schrodinger in 1920 put forward this model by taking into account the de Broglie concept of dual nature of matter and Heisenberg’s uncertainty principle. In this model, the discrete energy levels or orbits proposed by Bohr’s model are replaced by mathematical function ψ (psi) which is related to the probability of finding electrons around the nucleus.

The wave equation for an electron wave propagating in 3-D space is: ∂^{2}ψ/∂x^{2} + ∂^{2}ψ/∂y^{2 }+ ∂^{2}ψ/∂z^{2} + 8π^{2}m/h^{2} (E –V)ψ = 0, where y is the amplitude of the electron wave at point with coordinates x, y, z, E = total energy and V = potential energy of the electron; ψ is also called wave function and ψ^{2} gives the probability of finding the electron at (x, y, z). The acceptable solutions of the above equation for the energy E are called Eigen values and the corresponding wavefunctions are called **Eigen functions**. Every function is not an Eigen function. An acceptable solution for Schrodinger wave equation must satisfy the following conditions:

1. The function should be finite.

2. It should always bear a single value at a particular point in space.

3. It should be a continuous function. Schrodinger wave equation can be written as

(∂^{2}/∂x^{2} + ∂^{2}/∂y^{2 }+ ∂^{2}/∂z^{2}) ψ + 8π^{2}m/h^{2} (E –V)ψ = 0 or ∇^{2}ψ + 8π^{2}m/h^{2} (E –V)ψ = 0

where ∇^{2 }= ∂^{2}/∂x^{2} + ∂^{2}/∂y^{2 }+ ∂^{2}/∂z^{2} is called Laplacian operator This equation can be rewritten as ∇^{2}ψ = – 8π^{2}m/h^{2} (E –V)ψ h; or (- h^{2}∇^{2}/8π2m + V)ψ = Eψ; or Hψ = Eψ, where H = – h^{2}∇^{2}/8π2m + V, is called Hamiltonian operator

In this operator, the first term represents kinetic energy operator (T) and the second term represents potential energy operator (V).

**Significance of ψ: **It represents the amplitude of an electron wave. It can be positive or negative. It has no physical value.

**Significance of ψ ^{2}: **It is a probability function. It determines the probability of finding an electron within a smaller region of space around the nucleus. The space in which there is a maximum probability of finding an electron is termed as orbital.

**QUANTUM NUMBERS**

An atom contains a large number of shells and sub-shells. These are distinguished from one another on the basis of their size, shape and orientation (direction) in space. The parameters are given in terms of different numbers called **quantum numbers**.

Quantum numbers may be defined as a set of four numbers with the help of which we can get complete information about all the electrons in an atom. It tells us the address of the electron i.e. location, energy, the type of orbital occupied and orientation of that orbital.

**Principal Quantum Number**

** **This is denoted by n, an integer. The values of n are from 1 to n. n = 1 (K shell), n = 2 (L shell), n = 3 (M shell), n = 4 (N shell). ‘n’ represents the major energy shell to which an electron belongs. The values of ‘n’ signify the size and energy level of major energy shells. As the value of ‘n’ increases, the energy of the electron increases and thus, the electron is less tightly held with the nucleus. Angular momentum can be calculated using principal quantum number: mvr = nh/2π.

**Azimuthal Quantum Number**

This is denoted by l. The values of l are from 0 to (n – 1). l = 0, s-subshell, spherical, l = 1, p-subshell, dumbbell, l = 2, d-subshell, double dumbbell or like leaf. The letters s, p, d, f designate old spectral terms – Sharp (s), principal (p), diffuse (d), fundamental (f). For a given value of n, total values of ‘l’ are n. The values of l signify the shape and energy level of sub-shells in a major energy shell. The angular momentum of an electron in an orbital is given by nh/2π. The energy level for sub-shells of a shell shows the order: s < p < d < f.

**Magnetic Quantum Number**

It is denoted by m_{l}, an integer. Zeeman effect: Zeeman studied the fine spectrum of H using a spectroscope of high resolving power as well as putting the source under the inﬂuence of the magnetic field. He noticed that the spectral line splits up to more than one component. Each frequency of radiation emitted by the atom in the presence of magnetic field splits up into components if the angular momentum of the electron along the magnetic field is restricted to the value, m_{l} =h/2π. The values of ml lie from ± l through zero.

The positive values of magnetic quantum number m_{l} represent the angular momentum component of the orbital in the direction of the applied magnetic field whereas the negative values of m_{l} account for the angular momentum component of orbital in the opposite direction of the applied magnetic field. Total values of m_{l} for a given value of n = n^{2}. Total values of ml for a given value of l = (2l + 1). The values of ml signify the possible numbers of orientations of a sub-shell. In the absence of magnetic field, the three p-orbitals are equivalent in energy and are said to be threefold degenerate, i.e. sub subshell (orbitals) having same energy level is known as **degenerate orbitals**.

**Spin Quantum Number**

Wave mechanical treatment required no more than three quantum number n, *l *and m. The existence of multiple, i.e. doublet structure led to the introduction of a spin quantum number m_{s}. The values of m_{s} are + ½ and – ½. This is due to the fact of the doublet structures of spectral lines which can be explained by proposing only two directions of spin of the electron along its own axis. The values of m_{s} signify the direction of rotation or spin of an electron in its axis during its motion. Spin angular momentum is given by h/2π √m_{s} (m_{s }+ 1) = h/2π √½ (½ + 1) = √3 h/4π = √3 h/2. The spin may be clockwise (+ ½ or – ½) or anticlockwise (- ½ or + ½). Spin multiplicity of an atom = √s(s + 1).

** ****Shapes of Orbitals**

The electron cloud represents the shape of the orbital. It is not uniform but it is dense where the probability for finding the electron is maximum. s-orbitals do not vary with angles, i.e. they do not have directional dependence. Thus, all s-orbitals are called spherically symmetrical. Their size increases with increases in the value of n. 1s-orbital has no nodal plane (the plane at which zero electron density is noticed). 2s-orbital has one nodal plane; 3s-orbital has two nodal planes. Thus it is evident that the number of nodal planes increases with increasing value of the principal quantum number.

All orbitals with l ≠ 0 have angular dependence. Therefore, p and d and other higher angular momentum orbitals are not spherically symmetrical. p-orbitals consist of two lobes to form dumbbell-shaped structure. The three p-orbitals along x, y, z-axes named as p_{x} , p_{y }, p_{z }orbitals are perpendicular to each other. All the three p-orbitals of a sub-shell have the same size and shape but differ from each other in orientation. The subscripts x, y and z indicate the axis along which the orbitals are oriented and possess maximum electron density. Also, the orbitals of a sub-shell having same energy are referred to as **degenerate orbitals**.

The point where the wave function changes its sign is called node. The number of radial nodes can be determined by the formula: (n – l – 1). In order to visualize the electron cloud, consider the space around nucleus divided into a large number of small concentric spherical shells of radius dr. The volume of such a shell can be –

dv = 4π/3 (r+dr)^{3} 4π/3r^{3} So, dv = 4πr^{2}dr

This volume is called radial volume and the probability of finding an electron within this shell is called radial probability distribution function.

R.P.F. = (Volume of the spherical shell) × probability density

= (4πr^{2} dr) × R

Radial probability distribution = 4πr^{2}dr R^{2}

For 1s orbital, radial probability increases with increase in distance from the nucleus reach a maximum and then decreases. The maxima are the maximum probability of finding an electron which is also called ‘radius of maximum probability’ and is also same as Bohr’s radius. For 2s, the graph has two maxima. In between these two maxima, the curve passes through a zero value indicating that there is zero probability of finding the electron at that distance. This point is a nodal point which can be a radial/spherical node.

**Angular Node/Nodal Plane **The probability of finding an electron in the nucleus is zero; so it is called a nodal point. Any plane passing through that point where the probability of finding an electron is zero is called a nodal plane.

**PAULI’S EXCLUSION PRINCIPLE**

The principle states that no two electrons in an atom can have the same set of all the numbers. In other words, no orbital can have more than two electrons. The maximum capacity of the main energy shell is equal to 2n^{2} electrons. The maximum capacity of a sub-shell is equal to 2(2*l *+ 1) electrons. The number of sub-shells in the main energy shell is equal to the value of n. The number of orbitals in the main energy shell is equal to n^{2}. One orbital cannot have more than two electrons. If two electrons are present, their spins should be in opposite directions.

**AUFBAU PRINCIPLE**

The word ‘aufbau’ originates from the German word ‘Aufbauen’ which means ‘to build’. This gives us a sequence in which various sub-shells are filled up depending on the relative order of the energy of the sub-shells. The sub-shell of the lowest energy is filled up first, then the next sub-shell of higher energy starts filling. The sequence in which various sub-shells are filled is the following:

1s, 2s, 2p, 3s, 4s, 3d, 5s, 4d, 5s, 4d, 5p, 6s, 4d, 5d, 6p, 7s, 5f, 6d, 7p

**Using (n + l) Value:**

The sequence in which various sub-shells are filled up can also be determined with the help of (n + *l*) value. When two or more sub-shells have same (n + *l*) value, the sub-shell with the lowest value of ‘n’ is filled up first.

The principal quantum number solely determines the energy of the electron in a hydrogen atom and other single electron species like He^{+}, Li^{2+} and Be^{3+}. The energy of orbitals in hydrogen and hydrogen like species increases as follows:

1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f < …..

**Exceptions to Aufbau Principle: **

In some instances, it is noted that actual electronic arrangement is slightly different from the arrangement expected by Aufbau principle. A simple reason behind this is that half-filled and

full-filled sub-shells have got extra stability.

**HUND’S RULE OF MAXIMUM MULTIPLICITY (ORBITAL DIAGRAMS)**

It states that electrons are distributed among the orbitals of sub-shell in such a way as to give the maximum number of unpaired electrons with parallel spins. This means that the orbitals available in a sub-shell are first filled singly before they begin to pair i.e. the pairing of electrons occurs with the introduction of the second electron in the s-orbital, the fourth electron in the p-orbitals, the sixth electron in the d-orbitals and the eighth electron in the f-orbitals.

The rule is based on the fact that electrons have the same charge and repel each other and hence try to keep further apart from each other as much as possible. The electrons thus occupy different orbitals of the subshell as to minimize the inter-electronic repulsion and increase the stability of the atom. Orbitals tend to become half-filled or completely filled since such an arrangement will be more stable on account of symmetry.

**The orbital diagram for nitrogen, oxygen, ﬂuorine and neon are as follows:**

** **The orbital diagrams of elements from atomic number 21 to 30 can be represented on similar lines as below:

All those atoms which consist of at least one of the orbitals singly occupied behave as paramagnetic materials because these are weakly attracted to a magnetic field, while all those atoms in which all the orbitals are doubly occupied behave as diamagnetic materials because they have no attraction for the magnetic field. However, these are slightly repelled by the magnetic field due to induction.

The magnetic moment may be calculated as, 1 BM (Bohr magneton) = where n = No. of unpaired electrons.

This article has tried to highlight the basic concepts of the **structure of atom** in the form of** notes** for **class 11** students in order to understand the basic concepts of the chapter. The **notes** on the** structure of atom** have not only been prepared for class 11 but also for the different competitive exams such as iit** jee**, **neet**, etc.

**Also, Check Part 1 of the chapter: ****Structure of Atom (Atomic Structure) Part 1**

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