# Atomic Structure B.Sc. 1st Semester

### Atomic Structure MCQs

### 2nd Semester(Session: 2023-2027) Batch Start: 20th December 2023 @ 7.30AM

## Atomic Structure

Bohr's Theory, its limitations and atomic spectrum of hydrogen atom. Wave mechanics: de Broglie equation, Heisenberg's Uncertainty Principle and its significance, Schrodinger's wave equation, significance of wave function.Quantum Numbers and their significance. Normalized and orthogonal wave functions. Sign of wave functions. Radial and angular wave functions for hydrogen atom. Radial and angular distribution curves. Shapes of s, p, d and f orbitals. Contour boundary and probability diagrams.

Pauli's Exclusions Principle, Hund's Rule of maximum multiplicity. Aufbau's principle and its limitations, Variations of orbital energy with atomic number.

## Bohr's Theory

The Bohr model of an atom was proposed by Neil Bohr in 1915. It came into existence with the modification of Rutherford's atomic model. Bohr theory modified the atomic structure model by explaining that electrons move in fixed orbitals or shells and not anywhere in between and he also explained that each orbit (shell) has a fixed energy. Bohr's model consists of a positively charged small nucleus surrounded by negative electrons moving around the nucleus in orbits. Bohr found that an electron located away from the nucleus in three dimensional space has more energy, and the electron which is closer to nucleus has less energy.## Postulates of Bohr's Model of an Atom

The main postulates of Bohr's Atomic Model are as follows-1. The atom has a central massive core nucleus where all the protons and neutrons are present. The size of the nucleus is very small.

2. The electron in an atom revolves around the nucleus in certain discrete orbits. Such orbits are known as stable orbits or non – radiating or stationary orbits.

3. The force of attraction between the nucleus and the electron is equal to centrifugal force of the moving electron.

Force of attraction towards nucleus = centrifugal forcebr

4. An electron can move only in those permissive orbits in which the angular momentum (mvr) of the electron is an integral multiple of h/2π.

Thus, mvr = nh/2π

Where, m = mass of the electron, v = velocity of the electron in its orbit, r = radius of the electronic orbit, n = 1,2,3, ... from the nucleus.

5. The emission or absorption of radiation by the atom takes place when an electron jumps from one stationary orbit to another.

6. The radiation is emitted or absorbed as a single quantum (photon) whose energy h𝜈 is equal to the difference in energy ΔE of the electron in the two orbits involved.

Thus, h𝜈 = ΔE

Where h = Planck's constant, 𝜈 = frequency of the radiant energy.

Hence the spectrum of the atom will have certain fixed frequency.

7. The lowest energy state (n = 1) is called the ground state. When an electron absorbs energy, it gets excited and jumps to an outer orbit. It has to fall back to a lower orbit with the release of energy.

8. The maximum number of electrons in an orbit is 2n

^{2}.

Where n = orbit number.

## Limitations of Bohr's Theory

Followings are the limitations of Bohr's Theory-1. It violates the Heisenberg Uncertainty Principle because it considers electrons to have both a known radius and orbit.

2. The Bohr Model provides an incorrect value for the ground state orbital angular momentum.

3. It does not explain the atomic spectra of elements having more than one electron.

4. It does not predict the relative intensities of spectral lines.

5. The Bohr Model does not explain fine structure and hyperfine structure in spectral lines.

6. It does not explain the Zeeman Effect.

## Atomic Spectrum of Hydrogen atom

When an electric discharge is applied on gaseous hydrogen atom at low pressure, a bluish light is emitted. When a ray of this bluish light is passed through a prism, a spectrum of several isolated sharp line is obtained. The wavelength of various lines shows that the spectrum lines lie in the visible, ultraviolet and infrared region. These lines are grouped in different series. These series of lines are named after the scientists who discovered them. The limiting line (i.e. last line) of any spectral series in the hydrogen spectrum is the line when n_{2}in the Rydberg's formula is infinity, i.e. n

_{2}= ∞. The wavelength of all these series can be expressed by a single formula which was given by Rydberg.

Series | Discovered by | Region | n_{2}→n_{1} | Number of Lines |
---|---|---|---|---|

Lyman | Lyman | U.V. Region | n_{2}=2,3,4... & n_{1}=1 | n_{2}−1 |

Balmer | Balmer | Visible Region | n_{2}=3,4,5... & n_{1}=2 | n_{2}−2 |

Paschen | Paschen | IR Region | n_{2}=4,5,6... & n_{1}=3 | n_{2}−3 |

Brackett | Brackett | IR Region | n_{2}=5,6,7... & n_{1}=4 | n_{2}−4 |

Pfund | Pfund | IR Region | n_{2}=6,7,8... & n_{1}=5 | n_{2}−5 |

Humphery | Humphery | Far IR Region | n_{2}=7,8,9... & n_{1}=6 | n_{2}−6 |

### Rydberg Formula

Rydberg gave a theoretical equation for the calculation of wavelength of various lines of hydrogenic spectrum in 1890.Where, R is Rydberg Constant and its value is equal to 109678cm

^{−1}.

Derivation of Rydberg Formula

## de-Broglie Wave Equation

The de Broglie equation is used to describe the wave properties of matter, specifically, the wave nature of the electron.de Broglie equation states that a matter can act as waves and particles like light and radiation.

We know that Einstein's equation-

E = mc

^{2}---------(eq.1)

where, E = energy, m = mass the particles and c is the velocity of light

According to Plank's rdiation theory-

E = hν = h.c/λ ---------(eq.2)

where, h = Plank's constant and λ = wave length of radiation

From equation 1 and 2 we have-

mc

^{2}= h.c/λ

or, mc = h/λ

or, p = h/λ [as mass(m) X velocity(c) = momentum(p)]

or,

**λ = h/p**---------(eq.3)

Equation '3' is called de-Broglie Wave Equation.

### Find the de Broglie wavelength for an electron moving at the speed of 5.0×10^{6}m/s (mass of an electron is 9.1×10^{−31}kg )

[Hints: λ = h/mvh = 6.63×10

^{−34}J⋅s]

Answer: 1.46×10

^{−10}m

## Heisenberg Uncertainty Principle

Werner Heisenberg in 1927, stated the uncertainty principle which is the consequence of dual behaviour of matter and radiation. It states that__it is not possible to determine precisely and simultaneously the momentum and position of small moving particles.__

If position of the particle is known then momentum is unknown and vice-versa.

Mathematically, it can be given as-

Δx. Δp ≥ ℏ/2 = h/4π

where, Δx = uncertain position, Δp = uncertain momentum,

ℏ = h/2π

where, h = Plank's constant

Δx. mΔv ≥ ℏ/2 (as, P = MV)

Δx. Δv ≥ ℏ/2m

Δx. Δv ≥ h/4πm

This equation states that the product of ∆x and ∆p can either be greater than or equal to (≥) but never smaller than h/4π. If ∆x is measured more precisely (i.e. ∆x is small) then there is large uncertainty or error in the measurement of momentum (∆p is large) and vice versa.

## Significance of the Heisenberg Uncertainty Principle

Uncertainty principle holds good for all the objects but this principle is significant for only microscopic particles and is negligible for that of macroscopic particles. The energy of a photon is insufficient to make change in velocity or momentum of bigger particles when collision occurs between them.### A particle is moving with constant momentum. The uncertainty in the momentum of the particle is 3.3 x 10^{-2} kg ms^{-1}. Calculate the uncertainty position.

[Hints: Δx = ℏ/2.ΔpΔx = 5.27 ☓ 10

^{-35}/3.3 ☓ 10

^{-2}

1.59 ☓ 10

^{-33}m]

### Heisenberg uncertainty principles has no significance in our everyday life. Explain.

## Quantum Number

An atom contains number of orbits and orbitals. These are distinguished from one another on the basis of their size, shape and orientation 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. Quantum number tells us the complete address of the electron (i.e. location, energy, the type of orbital occupied and orientation of that orbital) in an atom.continue...

## Significance of Quantum Numbers

Followings are the Significance of Quantum Numbers-1. Principal quantum number defines the size and energy of the orbit.

2. Azimuthal quantum number defines the shape of the orbitals.

3. Magnetic quantum number defines the spatial orientation of the orbital.

4. Spin quantum number defines the spin of the electron.

### Which of the following is a possible set of quantum numbers that describes an electron?

A. n = 3, ℓ = 2, m = −3, s = −1⁄2

B. n = 0, ℓ = 0, m = 0, s = +1⁄2

C. n = 4, ℓ = 2, m = −1, s = 0

D. n = 3, ℓ = 1, m = −1, s = +1⁄2

E. n = 4, ℓ = −3, m = −1, s = +1

[Hints: A. For 'd' orbtal, the value of 'm' ranges from +2 to −2.

B. Principal quantum number can never be zero.

C. Spin quantum number can never be zero.

D. For 'p' orbital, the value of 'm' ranges from +1 to −1.

E. Agimuthal quantum number can never be negative.

The correct answer is option D.]

## Schrödinger Wave Equation

Schrödinger Wave Equation is a mathematical expression which describe the energy and position of the electron in space.The Bohr's model violates the uncertainty principle. So, the correct theory is expected to deal with the probability of finding an electron in a given space.

Schrödinger think over the de-Broglie wave particle duality principle and worked out a mathematical equation using the principle of quantum mechanics.continue...

## Physical Significance of Wave Function

Wave function ψ represents the amplitude of electron wave i.e. probability amplitude and has no physical significance indivisually because it may be positive, negative or imaginary. ψ^{2}is known as probability density and determines the probability of finding an electron at a point within small volume (d𝜏) (atom). This means that if:continue...

## Normalization of the wave function

Normalization of the wave function means probability of finding an electron in limited area (i.e. somewhere in space) is one. So, the normalization is possible when there are some boundary conditions placed on the particle.continue...## Orthogonalization of the wave function

continue...## Radial and Angular Wave Functions for Hydrogen Atom

### Radial wave function, R (r)

The value of ψ appearing in Schrodinger's wave equation in polar coordinates, (r,θ,∅), can be determined only when ψ is written in the following form-Ψ (r, θ, ɸ) = R (r) Θ(θ) Ф(ɸ) Equ.-----1

Where ψ (r, θ, ɸ), is known as total wave function, R (r) is the radial wave function and other two are angular wave functions. The radial wave function, R (r) is dependent on r only where r is the distance of electron from the nucleus and is independent of θ and ɸ. Therefore, R (r) deals with the distribution of the electron charge density as a function of distance (r) from the nucleus. R (r) depends on two quantum numbers n and l and can be denoted as R

_{n,l}(r) or simply R

_{n,l}. Both Rn,l and R

^{2}n,l are significant only for drawing the probability curves for various orbitals. The radial wave functions for all s-orbitals are spherically symmetrical.

### Angular wave function, ψ (θ, ɸ)

The angular wave functions depend on the angles θ and ɸ and are independent of the distance (r). As given above in equation-1, these are represented as Θ (θ) and Θ (ɸ). Their values depend on the quantum numbers l and m and can be written as Θ_{l,m}and Θ

_{m}, respectively. Therefore, the equation-1 can also be written as-

Ψ

_{n,l,m}= R

_{n,l}Θ

_{l,m}Ф

_{m}Equ.-----2

This equation shows that the total wave function besides depending on r, θ, ɸ, also depends on the quantum numbers viz., n, l and m. Each permitted combination of n, l and m gives a distinct wave function and hence a distinct orbital. The angular wave functions together are used to predict the shapes of the orbitals.

## Radial and angular distribution curves or Functions

In the atomic orbital, there is probability of finding an electron in a particular volume element at a given distance and direction from the nucleus. This gives two types of probability of finding electrons, the radial probability distribution function i.e. probability of finding an electron at a given radial distance from the nucleus without considering the direction from the nucleus (i.e. how far away from the nucleus the orbital extends and the number of nodes the orbital has). The radial distribution functions depend on both n and l. This means that the number of nodes an orbital has and how far that orbital extends from the nucleus depends on the principle quantum number or energy level of the orbital (the 1 in 1s, the 2 in 2s, the 3 in 3s, etc.) and the type of orbital (s vs. p vs. d).The number of Radial nodes can be clculated by the given formula-

Number of Radial nodes = n - l - 1 = n - (l + 1)

Where n = principal quantum number, l = Azimuthal quantum number

The other is angular distribution curve i.e. the probability of finding an electron in any given direction from the nucleus without considering its distance from the nucleus. The Angular distribution function or curve describes the basic shape of the orbital or the number of lobes in an orbital. The angular distribution functions depend only on the quantum numbers l and m. That is the angular distribution functions of all electrons with the same l and m values are the same. Simply, we can say that all s orbitals have the same basic shape. For example The 2s orbital (n = 2, l = 0, m = 0), the 3s (n = 3, l = 0, m = 0) and the 4s (n = , l = 0, m = 0) have the same basic shape spherical.

The planes or planar areas around the nucleus where the probability of finding an electron is zero are called angular nodes. The value of the angular nodes does not depend upon the value of the principal quantum number. It only depends on the value of the azimuthal quantum number.

Example-

In 3d orbital, the value of Azimuthal quantum number (l)= 2

So, the number of Angular nodes = l = 2

Atom is spherical. If we consider it as composed of layers much like onion. The layer extends from r to r + dr and then, the volume of the thin shell as dV.

The volume of the sphere is given by-

V = 4/3 πr

^{3}

dV/dr = 4/3 π(3r

^{2})

or, dV = 4πr

^{2}dr

or, R

^{2}dV = 4πr

^{2}R

^{2}dr

When the radial portion (R) of the wave function is squared an multiplied by 4πr

^{2}, we get the probability 4πr

^{2}R

^{2}(r) verses the distance from the nucleus(r).

If, r = 0,

Then, 4πr

^{2}R

^{2}= 0

Hence, the volume at the nucleus is approach zero. In between them, both r and R have finite values, so there is a maximumin the curve at r = a

_{o}(Bohr radius).

The above plot shows that the maximum probability occurs at distance of 0.52 Å from the nucleus. This is equal to the Bohr radius. It indicates that the maximum probability of finding an electron around the nucleus is at this distance. However, there is a probability to find the electron at other distances also.

The radial distribution function of 2s, 2p, 3s, 3p and 3d orbitals of the hydrogen atom are represented as follows-

## Shapes of s, p, d and f Orbitals

## Radial Nodes

Radial nodes are the spherical surface region where the probability of finding an electron is zero. This sphere has a fixed radius. Therefore, radial nodes are determined radially. Radial nodes occur as the principal quantum number increases.When finding radial nodes, the radial probability density function can be used. The radial probability density function gives the probability density for an electron to be at a point located the distance 'r' from the proton. The following equation is used for this purpose.

Ψ(r,θ,Φ) = R(r) Y(θ,Φ)

Where Ψ is the wave function, R(r) is the radial component (depends upon only the distance from the nucleus) and Y(θ,φ) is the angular component. A radial node occurs only when R(r) component becomes zero.

**Number of Radial nodes = n – l – 1 = n – (l + 1)**

Where n = principal quantum number, l = Azimuthal quantum number

Examples-

**Number of radial nodes of 1s orbital-**

In 1s orbital-

Principal quantum number (n) = 1 and

Azimuthal quantum number (l) = 0

Number of Radial nodes = n – l – 1 = 1 – 0 – 1 = 0

**Number of radial nodes of 2s orbital-**

In 2s orbital-

Principal quantum number (n) = 2 and

Azimuthal quantum number (l) = 0

Number of Radial nodes = n – 0 – 1 = 2 – 0 – 1= 1

**Number of radial nodes of 3s orbital-**

In 3s orbital-

Principal quantum number (n) = 3 and

Azimuthal quantum number (l) = 0

Number of Radial nodes = n – 0 – 1 = 3 – 0 – 1= 2

## Angular Nodes or Nodal Planes

The planes or planar areas around the nucleus where the probability of finding an electron is zero are called angular nodes. This means we cannot ever find an electron in an angular (or any other) node. While radial nodes are located at fixed radii, angular nodes are located at fixed angles. The number of angular node present in an atom is determined by the angular momentum (Azimuthal) quantum number only. Angular nodes occur as the angular momentum quantum number increases.**Number of Angular nodes = l**

Where l = Azimuthal quantum number

Examples-

**Angular nodes or nodal planes of 1s orbital-**

In 1s orbital-

Azimuthal quantum number (l)= 0

Number of Angular nodes = l = 0

**Angular nodes or nodal planes of 2s orbital-**

In 2s orbital-

Azimuthal quantum number (l)= 0

Number of Angular nodes = l = 0

**Angular nodes or nodal planes of 2p orbital-**

In 2p orbital-

Azimuthal quantum number (l)= 1

Number of Angular nodes = l = 1

**Angular nodes or nodal planes of 3d orbital-**

In 3d orbital-

Azimuthal quantum number (l)= 2

Number of Angular nodes = l = 2

## Total Number of Nodes

The total number of nodes is the sum of the number of radial nodes and angular nodes.Total number of nodes = Number of radial nodes + Number of Angular nodes

Total number of nodes = (n – 1) + l = (n – 1)

**Total number of nodes = (n – 1)**

**Note:**If the node at r = ∞ is also considered then the number of nodes will be 'n' (not n – 1)

**Total number of nodes of 2s orbital-**

In 2s orbital-

Principal quantum number (n) = 2 and

Azimuthal quantum number (l) = 0

Total number of nodes = n – 1 = 2 – 1 = 1

## Pauli's Exclusions Principle

Austrian physicist and Noble laureate Wolfgang Pauli formulated this principle in 1925. He stated that__No two electrons in an atom can have the same set of all the four quantum numbers__.

There are two salient rules that the Pauli exclusion principle follows-

1. Only two electrons can occupy the same orbital.

2. The two electrons that are present in the same orbital must have opposite spins, or they should be antiparallel.

↑↓ ✔ | ↑↑ ✗ | ↓↓ ✗ |
---|

Example-

Quantum numbers of 5th and 6th electrons of nitrogen-

5th electron-

Quantum Number | 5th Electron | 6th Electron |
---|---|---|

n | 2 | 2 |

l | 1 | 1 |

m | +1 | 0 |

s | +1/2 | +1/2 |

## Applications of Pauli's Exclusion Principle

Pauli exclusion principle applies to a set of fundamental particles called Fermions that have half-integer spin (S = 1/2, 3/2, 5/2 …). It does not apply to Bosons that have integer-spin (S = 1,2,3…). Pauli principle also applies to atoms with half-integer spin. For example, helium-3 has spin of +1/2.Pauli exclusion principle can explain the electron-shell structure of atoms and the electron configuration of elements. It can explain how atoms form chemical bonds and how chemical properties vary among the elements.

Hund’s rule uses the Pauli principle to fill up electrons in the various energy sublevels, thereby building up the periodic table elements.

In quantum mechanics, the Pauli exclusion principle states that Fermions must have an antisymmetric wave function, unlike Bosons, which have symmetric wavefunctions.

## Hund's Rule of Maximum Multiplicity

Friedrich Hund in 1925, states that the greatest value of spin multiplicity has the lowest energy term. So, electron pairing in orbitals belonging to the same subshell (p, d, or f) does not occur until each orbital of that subshell has one electron, i.e. it is singly occupied. After singly occupied, pairation takes place with opposite spin.hund’s rule of maximum multiplicity is used to determine the electronic configuration of elements.

Let's consider the filling of 2p orbitals. There are three degenerate(same energy) orbitals in every p subshell. The first electron enters either of three orbitals since they are degenerate. Even though the first-occupied orbital is not completely filled and it can take one more electron, the second electron will not occupy it. Instead, it will enter another orbital with the same spin as the first electron. Similarly, the third will occupy the remaining orbital. Thus, the first three electrons occupy all three orbitals (p

_{x}, p

_{y}, p

_{z}) with either spin up or spin down. After singly filling, the 4th, 5th and 6th electrons(↑ in figure) will fill the remaining empty orbitals, which results in pairing.

Similarly, we can understand the filling of d and f orbitals.

### Multiplicity

In quantum chemistry, the multiplicity or spin multiplicity is the total number of spin orientations and is given by (2S + 1). Here, 'S' is the total spin quantum number, and its value is the sum of all unpaired half spins.According to the Hund's rule, the lowest energy configuration is attained when the multiplicity (2S + 1), is maximum.

The maximum value of 'S' is obtained only when all the spins are either up or down.

## Aufbau's Principle

Aufbau is a German word which means building up. Elements are built up by gradual addition of electrons in orbitals.In a ground state of the atoms, the orbitals are filled in order of their increasing energies." i.e. an electron will initially occupy an orbital of lower energy level and when the lower energy level orbitals are occupied, then only they shall start occupying the higher energy level orbitals.

## Limitations of Aufbau's Principle

There are a few limitations to the Aufbau principle. These mainly come from atoms in the d- (transition metals) and f- (lanthanides and actinides) blocks of the periodic table. The exceptions also usually come from elements with an atomic number greater than 40. These exceptions happen because there is increased stability from half and fully filled orbitals, which lowers the overall energy of the atom. In these instances, stability is increased because electrons can be further away from each other. Additionally, the energy difference between orbitals is smaller at higher atomic numbers.Cr and Cu are the most common example

Cr: [Ar]3d

^{5}4s

^{1}instead of the expected [Ar]3d

^{4}4s

^{2}

This is due to increase in stability from having half-filled orbitals over partially filled orbitals.

Cu: [Ar]3d

^{10}4s

^{1}instead of the expected [Ar]3d

^{9}4s

^{2}

This is due to increase in stability from having full-filled orbitals over less than full filled orbitals.

## Variations of Orbital Energy with Atomic Number

Energies depend upon the distance of the orbital from the nucleus and also the number of electrons present in it. As the atomic number increases the atomic radius increases leading to increase in the distance from the nucleus and thus decrease in energy. Also the inter-electronic repulsion between electrons of the same orbital becomes prominent. Therefore, The energy of the orbitals in the same subshell decreases with an increase in the atomic number.## Factors affecting the energy of orbitals

1. For a given value of the principal quantum number, the s orbital electron will be more tightly bound to the nucleus than the p orbital electron, which is more tightly bound in comparison to a d orbital electron.2. S orbital electrons will have a lesser amount of energy (more negative) than that of p orbital electrons which will have lesser energy than that of d orbital electrons.

3. As the extent of shielding from the nucleus is different for electrons in different orbitals, it leads to the splitting of energy levels having the same principal quantum number. Thus, the energy of orbitals depends upon the values of both the principal quantum number (n) and the azimuthal quantum number (l). Hence, the lower the value of (n + l) for an orbital, the lower its energy.

When the two orbitals having the same value of (n + l), the orbital with a lower value of n (principal quantum number) will have the lower energy.

4. The energy of the orbitals in the same subshell decreases with an increase in the atomic number (Zeff).