# CCSU Atomic Structure Inorganic Chemistry Study Material in English

CCSU Atomic Structure Inorganic Chemistry Study Material in English

## CCSU Atomic Structure Introduction Study Material

Bohr achieved the major breakthrough in the understanding of structure of atom, by presenting his model of atom. He gave the concept of discrete energy levels (or orbits) in which the electrons can revolve. He arrived at these energy levels using Planck’s quantum theory. According to Bohr’s model of atom, as long as the electron moves in a particular orbit, if does not gain or lose energy. Bohr postulated that the angular momentum of an electron revolving around the nucleus has definite values which are whole number multiples of h/2π.

The appearance of various spectral lines in the emission spectra of hydrogen can be explained in terms of Bohr’s theory of structure of atom. The Lyman series in the hydrogen spectrum are produced when the electrons jump from second, third, fourth or a higher energy level to the first energy level. The Balmer series in the hydrogen spectrum are obtained when the electrons jump from third, fourth or a higher energy level to the second energy level. Similarly, Paschen series in the hydrogen spectrum are obtained when the electrons jump from fourth, fifth or higher energy levels to the third energy level.

But, Bohr theory had to be abandoned because of many shortcomings. The Main objection to Bohr’s theory came from new ideas of de Broglie and Heisenberg.

## CCSU Shortcomings of Bohr’s theory

The main shortcomings of Bohr’s theory are:

- Bohr’s atomic model failed to account for the effect of magnetic field on the spectra of atoms pr ions. It was observed that when the source of a spectrum is placed in a strong magnetic fields, each spectral line is further split into a number of lines. This is called
**Zeeman effect.**This observation could not be explained on the basis of Bohr’s model. - Bohr’s theory of an atom was successful in accounting for the spectra of hydrogen and hydrogen-like atoms (e.g., He
^{+}, Li^{2+}). But, it failed to explain the spectra of multi-eletron atoms. The spectroscopic measurements with high precisian revealed the fine structure in the line spectra of atoms. Bohr’s theory could not explain the occurrence of these fine spectral lines. - Another objection of Bohr’s theory came from Heisenberg’s uncertainty principle. According to this principle, it is impossible to determine simultaneously the exact position and momentum of a small moving particle like an electron. The postulate of Bohr that electrons revolve in well defined orbits around the nucleus is, thus not tenable.
- In 1923, de Broglie suggested that electron like light has a dual character. It has particle as well as wave character. Bohr treated the electron only as a particle.

## DE BROGLIE HYPOTHESIS (DUAL NATURE OF MATTER AND RADIATION) Study Material

Earlier, it was thought that light is a stream of particles which area photons (or called corpuscles). However, this concept failed to explain the phenomena of interference and diffractions which could be explained only if light is considered to have wave nature. But at the same time, it was observed that the phenomena of black body radiation and photoelectric effect could be explained only if light considered to have particle character. Therefore, it was concluded that light has a particle nature as well as wave nature i.e., it has a dual nature.

Louis de Broglie, a French physicist, in 1924, advanced the idea that like photons, all material particles such as electron, proton ,atom, molecule a piece of chalk, a piece of stone or an iron ball (i.e. microscopic as well as macroscopic objects) also possessed dual character. The wave associated with a particle is called a** matter wave** or** de Broglie wave.**

### The de Broglie Relation. **The wavelength of the wave associated with any material particle was calculated by analogy with photon as follows :-**

Incase of a photon , if is assumed to have wave character, its energy is given by

E=hv ………(i)

(according to the Planck’s quantum theory)

Where y is the frequency of the wave and h is Planck’s constant.

If the photon is supposed to have particle character, its energy is given by

E=mc^{2 } ………..(ii)

(according to Einstein equation)

Where m is the mass of photon and c is the velocity of light.

From equations (i) and (ii), we get

hv= mc^{2 }

But v=c/λ

h.c/ λ = mc^{2 }…………………………(iii)

or λ= h/mc

de Broglie pointed out that the above equation is applicable to any material particle big or small. The mass of the photon is replaced by the mass of material particle and the velocity of the photon is replaced by the velocity v of the moving particle. Thus, for any material particle like electron, we may write

λ=h/mv

or λ=h/p ……………………. (iv)

where mv=p is the momentum of the particle.

The above equation is called **de Broglie equation.**

**Significance of de Broglie equation**. *Although the Broglie equation is applicable to all material objects but is has significance only is case of microscopic particles*. This is because the wavelength produced by a bigger particle (say a ball) using eq. (iv) come out to be too small to be observed. Only particles like electrons, atoms, etc. give an observable value of λ according to eq. (iv).

## Heisenberg’s Uncertainty Principle for CCSU

Werner Heisenberg, a German physicist, in 1927 gave a principle about the uncertainties on simultaneous measurements of position and momentum of small position and momentum of small particles. It is known as Heisenberg’s uncertainty principle and it states as follows :

It is impossible to measure simultaneously the position and momentum of small particle with absolute accuracy or certainly. If an attempt is made to measure any one of these two quantities with greater accuracy, the other becomes less accurate. The product of the uncertainly in the position (Δx) and the uncertainty in the momentum (Δp=m, Δv where m is the mass of the particle and is the uncertainly in velocity) is always constant and is equal to h/4π , where h is Planck’s constant i.e.

Δx. Δp=h/4 π …(i)

Putting Δp=m x Δv equ. (i) becomes

Δx.(m Δv)=h/4π …(ii)

Or Δx. Δv = h/4πm ….(iii)

This implies that the position and velocity of a particle cannot be measured simultaneously with certainty.

**Explanation of Heisenberg’s uncertainly principle.** Suppose we attempt to measure both the position and momentum of an electron. To locate 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 by the photon, the position as well as the velocity of the electron are altered.

**Significance of Heisenberg’s uncertainty principle.** Although Heisenberg’s uncertainty principle holds good for all objects* but it is of significance only for microscopic particles.* Obviously the energy o fthe of bigger bodies when id collides with them. For example, the light from a torch falling on a running ball in a deak room neither changes the speed of the ball nor its direction i.e. position.

This may be further illustrated by comparing Δx. Δv values for a particle of mass 1g and a microscopic particle.

For a particle of mass 1g, we have

Δx. Δv = h/πm=6.626×10^{-34}kgm^{2}s^{-1}/4×3.1416x(10^{-3}kg)=10^{-25}m^{-2}s^{-1}

Thus the product of Δx and Δv is extremely small. Hence these values are negligible. For a microscopic particle like an electron, we have

Δx. Δv = h/πm=6.626×10^{-34}kgm^{2}s^{-1}/4×3.1416x(9.11×10^{-31} kg)=10^{-4}m^{2}s^{-1}

Thus if uncertainty in position is 10^{-8}m, uncertainty in velocity will be 10^{4}ms^{-1} which is quite significant.

### Derivation of Schrodinger Wave Equation Study Material

Schrodinger developed a mathematical wave equation to describe the wave motion of the electron in a hydrogen atom. Here the nucleus is surrounded by a vibrating electron wave which may be mathematically compared with stationary as standing wave formed by a vibrating string fixed between two points as shown in the figure. The electron in a hydrogen atom is under constraint imposed by the attraction of the nucleus. An electron motion can be described by an equation analogous to that used to describe a “standing or stationary” wave system whose solution in one dimension is given by the sine wave equation (i).

Ψ=A sin 2πx/λ …..(i)

Where = amplitude of the wave of wavelength at a distance x from the origin A is a constant.

On double differentiation w.r.t. x, we get

dψ/dx=A.2π/λ.cos2πx/λ …..(ii)

d^{2}ψ/dx^{2 }= A.2π/λ.2π/λ[-sin2πx/λ]

=-4π^{2}/λ^{2}-Asin2πx/λ …….(iii)

Substituting ψ=A sin2πx/λ from eq. (i) into eq. (iii), we get

d^{2}ψ/dx^{2}=-4π^{2}/λ^{2}ψ ……(iv)

or d^{2}ψ/dx^{2}+4π^{2}/λ^{2}ψ =0 ……(v)

This is the equation for one dimensional standing wave which can be extended to describe motion of an electron in three dimensions by a second order partial differential equation (vi)

d^{2}ψ/dx^{2}+ d^{2}ψ/dy^{2}+ d^{2}ψ/dz^{2}+4π^{2}/λ^{2}ψ=0

where ψ is a function of Cartesian co ordinates x, y, z. This may be written more precisely as

Δ^{2}ψ+4π^{2}/λ^{2}ψ=0 …….(vii)

Where Δ^{2 }stands for d^{2}ψ/dx^{2}+ d^{2}ψ/dy^{2}+ d^{2}ψ/dz^{2}

Combining de Broglie equation λ=h/mv and eq. (vii), we have

Δ^{2}ψ+4π^{2}m^{2}v^{2}/h^{2}ψ=0 …..(viii)

Considering the electron as a particle, the total energy (E) of the system is sum of kinetic (1/2mv^{2}) and potential (V) energier, i.e.,

E=1/2 mv^{2}+V

Or v^{2}=2(E-V)/m ……(ix)

In order to calculate values for the energy states value of v2 (Eq.(ix)) is substituted into equation (viii) to obtain a fundamental form of Schrodinger wave equation for a single particle in three dimensions.

Δ^{2}ψ+8π^{2}/h^{2}(E-V)ψ=0 ….(x)

Or d^{2}ψ /dx^{2}+ d^{2}ψ/dy^{2}+ d^{2}ψ/dz^{2}+8π^{2}/h^{2}(E-V)ψ=0 ….(xi)

Where, ψ=amplitude of the wave function associated with the electron

x,y,z = Cartesian coordinates

m=mass of the electron

h=Planck’s constant

E=total energy of the system

V=potential energy of the system

The potential energy term V in Schrodinger wave equation (x) is given by the following equation

V=-Ze^{2}/r ….(xii)

Where +Ze = charge of the nucleus

Z= atomic number

-e= charge on the electron

Consequently, the equation (xi) for one electron atomic system in three dimensions may be written as

d^{2}ψ /dx^{2}+ d^{2}ψ/dy^{2}+ d^{2}ψ/dz^{2}+8π^{2}m/h^{2}(E+ Ze^{2}/r)ψ=0 ……(xiii)

### Application of Schrodinger wave equation to hydrogen atom Study Material

Schrodinger wave equation for one electron atomic system is

d^{2}ψ /dx^{2}+ d^{2}ψ/dy^{2}+ d^{2}ψ/dz^{2}+8π^{2}m/h^{2}(E+ Ze^{2}/r)ψ=0

For hydrogen atom, Z=1; therefore V=-e^{2}/r and Schrodinger wave equation for H atom may be written as

d^{2}ψ /dx^{2}+ d^{2}ψ/dy^{2}+ d^{2}ψ/dz^{2}+8π^{2}m/h^{2}(E+ e^{2}/r)ψ=0 ……(xiv)

Equation (xiv) results from the assumputions:

(i) That the behavior of an electron in an atom is analogous to a system of standing wave; and

(ii) De Broglie relationship λ=h/mv describes the wavelength of an electron.

Justification for the above equation for hydrogen atom is found in the fact that the solution of equation (xiv) gives values of energy which agree well with those obtained experimentally from atomic spectra or calculated from the equation (xv) given by Bohr for his model of the atom.

E=-2π^{2}me^{4}Z^{2}/h^{2}n^{2} ….(xv)

That is, Schrodinger wave equation leads to exactly the same allowed energies as those deduced from the Bohr model. Besides that the “most probable” radius for 1s electron in H atom from wave mechanics comes out to be the same as determined by Bohr (0.529 Å) from equation (xvi) from Bohr model

R=h^{2}n^{2}/4πme^{2}Z^{2} …….(xvi)

### Physical significance of ψ and ψ^{2}

In all wave equations, the square of the wave function is the property which has physical significance. For example, in case of light waves, square of the amplitude of the wave at a point is proportional to the intensity of the light wave. Similarly in electron wave motion, square of ψ (ψ2) should represent intensity. According to uncertainty principle, the position of electron cannot be determined with certainty. All we can say is that greater the intensity of wave function at a particular point, greater is the probability of locating the electron at that point. Thus may be interpreted as being proportional to electron density. When is high, electron density is high, that is, the probability of finding an electron is high. If ψ^{2 }is low, probability is low. In order to avoid imaginary values, ψ. Ψ^{*} is used for ψ^{2}. Ψ^{*} is a complex conjugate of Ψ. We expect that if Ψ extends over a finite volume in space dx.dy. dz (=dτ), then the electron would be found in that volume. So ΨΨ^{* }dτ gives the probability of finding the electron in the volume dτ. The wave function Ψ may be positive, negative or imaginary but the probability density ΨΨ^{* }will always be positive and real. The probability of fmding the electron is finite, which means ΨΨ^{* }dτ must give real values. If Ψ has real value, then Ψ = Ψ^{* }and we can Ψ^{2}dτ use in place of ΨΨ^{*} dτ.

### CCSU Main conditions which Ψ must satisfy to give meaningful solution (Eigen function)

Ψ as a wave function must obey the following conditions if it is to give meaningful results.

- Ψ must be single valued, i.e., it must have only one definite value at a particular point in space.
- Ψ must be continuous at all points in space, i.e., there must not be sudden changes in the values of Ψ when its variables are changed.
- Ψ must have a finite value over the space dτ that the electron can occupy.
- Ψ must be normalized, i.e., the probability of finding the electron in the space under consideration must be unity.
- Ψ must become zero at infinity.

This means that of all the values of Ψ obtained by solving Schrodinger wave equation, only those values of Ψ are accepted which obey the above conditions. Thus only acceptable values of Ψ have significance.

The significant values of Ψ are called Eigen functions or wave functions. Eigen Functions give significant values of the total energy of electron (E) called Eigen values.

### CCSU Physical significance of Ψ^{*}

Solution of Schrodinger wave equation gives Ψ values which can be real as well as imaginary but Ψ^{2 }is always real and has a positive value. This is because the square of even negative values is positive. If Ψ has imaginary values, then it can be expressed as

Ψ = a + ib

Where a and b are real numbers and i (known as iota) = (-1)^{1/2}. In such a case, Ψ is made real by multiplying it by its own complex conjugate (Ψ^{*}) which is expressed as

Ψ^{*}= a-ib

The product obtained by multiplying Ψ and Ψ^{* }for real values of Ψ is as :

ΨΨ^{* }= (a + ib)(a – ib) = a^{2} – (i^{2}b^{2})

= a^{2} –(-1Xb^{2}) [i = (-1)^{1/2}]

=a^{2 }+ b^{2 }(which is real)

Thus Ψ Ψ^{* }becomes a real quantity.

However, if Ψ is real, then Ψ = Ψ^{* }, then ΨΨ^{*} = Ψ^{2 }

### Origin of quantum numbers n, l and m Study Material

The three quantum numbers n, l and m and their allowed values arise as a consequence of the solution of the wave equation for hydrogen atom. These quantum numbers and their allowed values are discussed below:

#### (i) Principal or Radial Quantum Number, n

It arises from the solution of the radial part of Ψ and identifies the shell or level to which the electron belongs. These shells are regions in space indicating the “most probable” position of the electron from the nucleus. The quantum number n is restricted to have any positive integral values from 1 to infinity. The larger the value of n, the further is the shell from the nucleus.

n=1, 2, 3, 4, 5…………∞.

The oretically there is no limit to the number of shells or levels but seventh shell (n = 7) js the highest shell occupied by the known elements.

Principal quantum number n alone does not give complete energy but designates the main energy level to which an electron belongs. The lowest energy state of an atom corresponds to n = 1 and the energy increases with increasing n.

#### (ii) Azimuthal or Subsidiary Quantum Number, l

It refers to the subshell or sublevel to which an electron belongs and describes the motion of the electron. The magnitude of the angular momentum of an electron (angular velocity X moment of inertia) is related to l by the expression:

Angular momentum = [l(l+1)]^{1/2} h/2π

Where l= 0,1,2,3,…….(n-1)

This means that the values of l for the subshells of a shell are governed by n, when n=1,there is only one subshell, with the value l=0, when n=2, there are two subshells with the values l=0,1. When n=3, there are three subshells that have the values l=0,1,2. Similarly when n=4, number of shells is 4 with values, l= 0,1,2,3. For every value of l, there is a separate notation to indicate subshell. For example,

Quantum No. l=0 1 2 3 4 5 ……….

Subshell Notation = s p d f g h…….

The first four notations : s, p, d and f are adjectives used earlier to identify spectral lines;** sharp, principal, diffuse and fundamental** but subsequent notations used for l>3 proceed alphabetically e.g. , g, h, i, j……. so on.

Knowing the value of n and finding out l, we designate a subshell. For example, subshell with n=1,l=0 is called 1s subshell and others are given as under:

n |
l |
Subshell notation |

2 | 0 | 2s |

1 | 2p | |

3 | 0 | 3s |

1 | 3p | |

2 | 3d | |

4 | 0 | 4s |

1 | 4p | |

2 | 4d | |

3 | 4f |

#### (iii) Magnetic Quantum Number, m

Each subshell is comprised of one or more orbitals (regions with maximum probability of finding the electron) whose number is equal to the number of ways the electrons in a subshell can orient themselves in space. The number of orientations is given by 2l + 1. Thus, the number of orientations or orbitals is one when l=0 (s-subshell), three for l=1 (p-subshell), five for l=2 (d-subshell) and seven for l=3 (f-subshell). In other words, s-subshell has one s-orbital, p-subshell has three p-orbitals d-subshell are degenerate, that is, they have identical energy. For example, for l=1, the three orbitals P_{x}, P_{y} and P_{z} are degenerate but this degeneracy is split in the presence of a magnetic field.

In this way three p-orbitals of different energies with the lowest energy orbital aligned with the magnetic field (m = +1) and five d-orbitals of different energies with the lowest aligned with magnetic field (m= +2) and are obtained. Each orbital within a subshell is identified in term of the component of the angular momentum m h/2π where m = magnetic quantum number. In other words, m is associated with different orientations of the orbital with reference to some defined directions.

For a given subshell, m can have values

m = -l, -(l-1),-(l-2)…..0…..(l-2),(l-1),l

Thus for l=0, m=0, we get an s-orbital.

For l=1, m can have three values m=-1, 0, +1 corresponding to three p-orbitals; P_{x}, P_{y}, P_{z},

For l=2, m can have five values m=-2, -1, 0, +1, +2 corresponding to five d-orbitals; d_{x2-y2}, d_{xy}, d_{yz}, d_{zx}, d_{z2}

**Some salient features of these quantum numbers are given below:**

(i) Radial quantum number n and its allowed values arise from the solution of radial part of ψ, whereas l and m quantum numbers arise from solution of angular part of ψ.

(ii) The lowest energy state of an atom corresponds to n=1 and energy increases as n increases.

(iii) The number of subshells in a shell is equal to the value of n where n=2, there are 2 subshells (l=0, 1); for n=3, there are 3 subshells (l=0, 1, 2) and so on.

(iv) The number of orbitals of a type in a given subshell is given by 2l+1. That is, we have one s-type orbital of l=0; three p-type orbitals for l=1; five d-type orbitals for l=2 and seven f-type orbitals for l=3.

(v) The total number of orbitals in a given shell = n^{2} (n=principal quantum number). For example for n=2, the total number of orbitals = 2^{2}=4.

### CCSU Spin Quantum Number m_{s}

In 1896, Zeeman observed that spectral lines in the atomic spectrum get split under the influence of a strong magnetic field. This phenomenon, called the** Zeeman effect**, is attributed to the orientation of the components of the angular momentum with respect to the external magnetic field and is given by m h/2π, Where m = magnetic quantum number.

Later on it was observed in the atomic spectra of alkali metals that the spectral lines which were earlier considered to be single lines are actually narrow doublets (two lines quite close together). An explanation to these doublet lines was offered by Uhlenbeck and Goudsmit in 1925. They proposed that the electron is also associated with rotation about its own axis (spinning). Axial spin of the electron is associated with spin angular momentum whose magnitude is given by m_{s} h/2π,where m_{s} is called the spin quantum number. The spin quantum number can have values +1/2 or -1/2 Which arise from the direction of spin (clockwise or anticlockwise). It is also designated as up or down. Thus, the description of an electron in an atom in terms of three quantum numbers (described earlier) is not complete and, in fact, four quantum numbers, n, l, m and m_{s} : are needed to describe the electron in an atom completely.

An electron is a spinning negative charge and a spinning charge is magnetic, therefore, a spinning electron can be considered to behave like a tiny magnet. Two electrons with opposite spins in an orbital will behave as two tiny magnets with opposite poles towards each other and thus attract each other.

**Example 1. Which sets of quantum numbers are not permissible and why?**

**(i) n= 5, l= 4, m= 0, m _{s }=1/2**

**(ii) n= 3, l= 0, m= -1, m _{s }=-1/2**

**(iii) n=3, l= 1, m= 2, m _{s = }½**

**(iv) n=2, l=2, m= 0, m _{s = }1/2**

**Solution.** **(i)** This is allowed.

**(ii)** It is not allowed because value of m (+1, …..0…..-1) are governed by l and when l=0, m cannot have values other than zero (m=0).

**(iii)** It is not allowed because when i=1, m has allowed values = +1, 0, -1 only.

**(iv)** It is not allowed because values of l are governed by n. For n=2, 1 can have values equal = 0,1 only. Therefore for n=2, l=2 is not permissible.

**Example 2. Explain why orbitals like 1p, 2d and 3f are not possible.**

**Solution.** This is because the sets of quantum numbers required for these orbitals are not allowed. 1p refers to n=1, l=1 which is not allowed because values of l are governed by n, that is l=0, 1, 2….(n-1). Thus, for n=1; l cannot have a value equal to one.

Similarly, 2d refers to n=2, l=2 which again is not possible. For n=2; l can have values 0, 1 only.

Also 3f is not possible because for 3f, the combination n=3, l=3 is again not allowed. For n=3, l can’t have a value higher than 2.

**Example 3. Calculate and identify all the individual orbitals for (i) n=3 (ii) n=4.**

**Solution. (i) ** n =3; l=0, 1, 2; Total no. of orbitals =9

These are: 3s; 3p_{x}; 3p_{y}; 3p_{z}; 3d_{xy}; 3d_{yz}, 3d_{z2}, 3d_{x2-y2 }

**(ii)** For n=4; l=0,1, 2, 3. Individual orbitals are:

l |
m |
Orbitals |

0 | 0 | 4s |

1 | +-1,0 | 4p_{x}, 4p_{y}, 4p_{z} |

2 | +-2, +-1,0 | 4d_{xy}, 4d_{yz}, 4d_{xz}, 4d_{z2}, 4d_{x2-y2} |

3 | +-3, +-2, +-1, 0 | 4f_{x3}, 4f_{y3}, 4f_{z3}, 4f_{xyz}, 4f_{(z2-y2)}, 4f_{(z2-x2)},4f_{(x2-y2)} |

For n=4; total number of orbitals = 1+3+5+7 = 16

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