Atomic Structure.pdf

Atomic Structure

Discipline Course-1 (DC-1) Semester-1 Paper -2 Unit-I: Atomic Structure Lesson: Atomic Structure Lesson Developer: Geetika Bhalla College/Department: Hindu College, University of Delhi

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Atomic Structure 1.1 Introduction

Though the atom has been in existence since the beginning of creation, it was only in 465 BC that it was first thought of and mentioned by the Greek philosopher Democritus (460-370 B.C). He believed that matter was indestructible and made up of tiny particles called atoms. The structure of atom as known to our ancient Hindu philosophers (600500 BC) was made up of exceedingly small, indivisible and eternal particles called "Paramanu". For the next 2000 years and more the atom was forgotten until in 1808, Dalton put forward the living hypothesis called Dalton's atomic theory. Dalton in 1808, proposed that atom was the smallest indivisible particle of matter. However, later on, the researches carried out by various eminent scientists like J. J. Thomson, Goldstein, Chadwick, and many others established beyond doubt that the atom was not the smallest particle but had a complex structure of its own and was made up of still smaller particles like electrons, protons and neutrons. However, an atom is the smallest particle, which retains the properties of an element. The arrangement of particles within the atom was put forward by Rutherford in 1911 on the basis of his  ‘scattering experiments’ or nuclear structure of atoms. This model, however, suffered

from a serious drawback that it could not explain the stability of the atom. Niels Bohr in 1913 proposed a model of the hydrogen atom that not only did away with the problem of

Rutherford’s unstable atom but also satisfactorily explained the line spectrum of

hydrogen element. The Bohr model was the first atomic model based on the quantization of energy. Biographic Sketch

Biographic Sketch

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Atomic Structure 1.1 Introduction

Though the atom has been in existence since the beginning of creation, it was only in 465 BC that it was first thought of and mentioned by the Greek philosopher Democritus (460-370 B.C). He believed that matter was indestructible and made up of tiny particles called atoms. The structure of atom as known to our ancient Hindu philosophers (600500 BC) was made up of exceedingly small, indivisible and eternal particles called "Paramanu". For the next 2000 years and more the atom was forgotten until in 1808, Dalton put forward the living hypothesis called Dalton's atomic theory. Dalton in 1808, proposed that atom was the smallest indivisible particle of matter. However, later on, the researches carried out by various eminent scientists like J. J. Thomson, Goldstein, Chadwick, and many others established beyond doubt that the atom was not the smallest particle but had a complex structure of its own and was made up of still smaller particles like electrons, protons and neutrons. However, an atom is the smallest particle, which retains the properties of an element. The arrangement of particles within the atom was put forward by Rutherford in 1911 on the basis of his  ‘scattering experiments’ or nuclear structure of atoms. This model, however, suffered

from a serious drawback that it could not explain the stability of the atom. Niels Bohr in 1913 proposed a model of the hydrogen atom that not only did away with the problem of

Rutherford’s unstable atom but also satisfactorily explained the line spectrum of

hydrogen element. The Bohr model was the first atomic model based on the quantization of energy. Biographic Sketch

Biographic Sketch

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Atomic Structure

did you know

Amongst the various scientists who proposed different models for structure of an atom, the Nobel Prize was conferred to Thomson in 1906, Rutherford in 1908, Bohr in 1922, Heisenberg in 1932 and Schrödinger in 1933. Although the models proposed by some of these scientists were rejected later but all of them were significant contributions in the progress of science and were a major breakthrough in their times.

Historical Context

The history of the study of the atomic nature of matter illustrates the thinking process that went on as science progressed. The models used by various scientists might mi ght not provide an absolute understanding of the atom but only a way of abstracting so that they can make useful predictions about them. Scientist

Year

Contributions and Postulates

Democritus

460 B.C.

Developed the idea of tiny particles called atoms, which could not be subdivided or made any smaller.

John Dalton

1803

Matter is made up of tiny particles called atoms. The atom is the smallest particle of matter that takes part in a chemical reaction. Atoms are indivisible and cannot be created or destroyed. Atoms of the same element are identical in every respect.

J.J. Thomson

1897

Discovered electrons in Cathode Ray experiments. His model of the atom looked like raisins stuck on the surface of a lump of pudding.

E. Goldstein

1900

Discovered protons in Anode Ray experiments.

Max Planck

1900

Showed that when you vibrate atoms strong enough, such as when you heat an object until it glows, you can measure the energy only in discrete units. He called call ed these energy packets, quanta.

Albert Einstein

1905

Explained that light absorption can release electrons from atoms, a phenomenon called the

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Atomic Structure "photoelectric effect." Published the famous equation E =mc 2 R.A. Millikan

1909

Oil drop experiment determined the charge (e=1.602 x 10-19 coulomb) and the mass ( m = 9.11 x 10-28 gram) of an electron.

E. Rutherford

1911

Discovered the nucleus and provided the basis for the modern atomic structure through his alpha particle scattering experiment. He established that the nucleus was: very dense, very small and positively charged. He also assumed that the electrons were located outside the nucleus. Drawback: The theory of electricity and magnetism predicted that opposite charges attract each other and the electrons should gradually lose energy and spiral inward.

Niels Bohr

1912

Electrons do not spiral into the nucleus. Electrons can orbit only at certain allowed distances from the nucleus. Atoms radiate energy when an electron jumps from a higher-energy orbit to a lower-energy orbit. Also, an atom absorbs energy when an electron gets boosted from a low-energy orbit to a high-energy orbit. Drawbacks: The theory only worked roughly in case of heavier elements. Could not explain Zeeman effect and Stark effect. Gave the concept of stationary orbits.

Louis de Broglie

1924

Showed that if light can exist as both particles and waves, then atomic particles also behave like waves; gave the principle of dual behaviour of matter.

Pauli

1924

Gave a rule governing the behavior of electrons within the atom that agreed with experiment: If an electron has a certain set of quantum numbers, then no other electron in that atom can have the same set of quantum numbers. Physicists call this "Pauli's exclusion principle."

Werner Heisenberg

1925

Had a theory of his own called matrix mechanics, which also explained the behavior of atoms. Heisenberg based his theory on mathematical quantities called matrices that fit with the conception of electrons as particles.

Schrödinger

1926

Schrödinger's wave mechanics. He based his theory on waves and used partial differential equations to solve the system of equations.

Max Born

1926

Gave statistical interpretation of wave function 'psi'. Born thought they resembled waves of chance and hence gave the idea of probability of finding the particle in space (a given volume element) around the nucleus.

Heisenberg

1927

Formulated an idea, which agreed with tests, that no experiment can measure the position and momentum of a quantum particle simultaneously. Scientists call this the "Heisenberg uncertainty principle." This implies that as one measures the certainty of the position of a particle, particl e, the uncertainty in the momentum gets correspondingly larger. Or, with an accurate momentum measurement, the knowledge about the particle's position gets correspondingly less.

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Atomic Structure James Chadwick

1932

Found the weight of atom did not check out when working with isotopes and thus found the predicted particles in nucleus with no charge, called the neutron. Neutron weighs about the same as a proton. Neutron has no electrical charge.

The main postulates of Bohr’s theory of atom are: i.

An atom consists of a small, heavy positively charged nucleus in the centre and the electrons revolve around bit in circular paths called orbits.

ii.

Out of an infinite number of circular orbits possible, the electrons revolve only in certain permissible orbits having a fixed value of energy.

These orbits (shown in Figure 1.1) are called energy levels or stationary states. The energies of the different stationary states in case of hydrogen atom are given by expression,

where, m = mass of the electron, h = Planck’s cons tant e = electronic charge and ε 0 = permittivity = 8.8542 × 10 -12 C2N-1m-2 and n is an integer that can take the values 1, 2, 3, …, etc. for first, second, third…, etc. energy levels, respectively Since the electrons revolve only in certain permissible orbits, they can have only certain definite or discrete values of energy and not any arbitrary value of their own. This is expressed by saying that the energy of an electron is quantized. Only those orbits are permissible whose angular momentum, mvr , is an integral multiple of h/2π . Mathematically it can be expressed as:

Hence, the angular momentum of an electron in an atom is also quantized, i.e., it can have certain definite or discrete values and not any arbitrary value of its own.

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Atomic Structure 1. While revolving in a particular orbit, the electron does not lose or gain any energy. Energy is absorbed or emitted only when the electrons jump from one orbit to another. the amount of energy absorbed or emitted during this transition is given by:

v. –= Δ= or Δ = E 2E 1E hv where, E 2= energy of higher orbit, E 1= energy of lower orbit, ΔE = difference in energy between higher and lower orbits, v = frequency of light absorbed or emitted, λ = wavelength of light absorbed or emitted, c = velocity 8 -1 of light (3.0 x 10 m s ) and h = Planck’s constant.

1.2.1 Limitations of The Bohr’s Model of Atom Bohr’s model of atom could not explain the spectra of atoms containing more than 1 one electron. It failed even for the simple helium atom, which has only two electrons. 2 Although Bohr’s model was s uccessful in explaining the spectra of hydrogen atom + 2+ and hydrogen like systems such as He , Li etc., it could not account for the fine structure in the spectrum observed using refined spectroscopic techniques. It is found that each spectral line on high resolution is in fact a doublet, i.e., two closely spaced lines. 3 Bohr’s theory was also unable to account for the splitting of the spectral lines in the presence of a magnetic field (Zeeman Effect) or an electric field (Stark Effect). 4 Bohr proposed a two dimensional or flat model of the atom which now has become obsolete as it has been established that atom is a three dimensional entity. 5 It failed to account for Heisenberg uncertainty principle. 6 It failed to account for the shapes of molecules.

Need for a New Approach to the Atomic Model

In view of the shortcomings of the Bohr model, attempts were made to develop a more suitable and general model for atom. Two important developments that contributed significantly in the formulation of such a model were: (i) Dual behavior of matter (ii) Heisenberg uncertainty principle 1.3 Dual Behaviour of Matter and Radiation In case of light, some phenomenon, like interference [shown in Fig. 1.2(a)], diffraction, etc., can be explained if the light is supposed to possess wave character. However, certain other phenomenon, such as black body radiation and photoelectric effect [shown in Fig. 1.2(b)] can be explained only if it is believed to be a stream of photons, i.e., it has particle character (or is corpuscular in nature). Thus, light is said to have a dual character. Comment

The quanta of light (radiant energy) are called ‘ photons’ (from the Greek ‘phos’ = light;

introduced by G. N. Lewis in 1926). They are defned by the equation: E  =

hυ

where h is Planck’s constant (named for the German physi cist Max Planck, 1858 –1947) and υ is the frequency of radiation.

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Atomic Structure

l

Figure1.2: (a) Interference pattern shown by a beam of light (b) photoelectric effect Louis de-Broglie, the French physicist in 1924 proposed that matter, like radiation, should also exhibit dual behaviour, i.e., both particle and wave-like properties (the distinction between the properties of a particle and a wave is given in Table 1.1). This means that just as a photon has momentum as well as wavelength, electrons should also have momentum as well as wavelength. de-Broglie, from this analogy, gave the following relation between wavelength ( λ) and momentum ( p) of a material particle, i.e.,

where ‘m’ is the mass of the particle, ‘ v ’ its velocity, and  ‘ p’ its momentum.

The wave character of matter should not be confused with the wave character of electromagnetic waves. They differ in various aspects as given in Table 1.2. Although, de-Broglie equation is applicable to all material objects but it has significance only in case of microscopic particles. -1

For example, consider a ball of mass 0.1 kg moving with a speed of 60 m s . From -34 de-Broglie equation, the wavelength of the associated wave is h /mv = (6.62 x 10 -34 )/(0.1 x 60) or 10 m. It is apparent that this wavelength is too small for ordinary -31 observation. On the other hand, an electron with a rest mass equal to 9.11 x10 kg, -30 i.e., approximately 10 kg moving at the same speed would have a wavelength = (6.62 -34 -30 -5 x 10 )/(10 x 60) =10 m, which can be easily measured experimentally. Davisson and Germer, in 1927, confirmed de- Broglie’s prediction experimentally. They showed that a crystal diffracts that electron beam in the same way as light radiation is diffracted, a phenomenon characteristic of waves. This provided a proof of wave nature of electrons. It needs to be noted that according to de-Broglie, every object in motion has a wave character. The wavelengths associated with ordinary objects are so short Institute of Lifelong, University of Delhi

Atomic Structure (because of their large mass) that their wave properties cannot be detected. The wavelengths associated with electrons and other subatomic particles (with very small mass) can however be detected experimentally. Table 1.1 Distinction between a particle and a wave S.No.

Particle

Wave

1.

A particle occupies a well-defined position in space, i.e., a particle is localized in space. For example, a cricket ball.

A wave is spread out in space. For example, on throwing a stone in a pond of water, the waves start moving out in the form of concentric circles. Thus, wave is delocalized in space.

2.

When a particular space is occupied by one particle, the same space cannot be occupied simultaneously by any other particle. Particles do not undergo interference.

Two or more waves can co-exist in the same region of space and hence undergo interference.

3.

When a number of particles are present in a given region of space, their total value is equal to their sum.

When a number of waves are present in a given region of space, due to interference, the resultant wave can be larger or smaller than the individual waves.

The wave character of matter should not be confused with the wave character of electromagnetic waves. They differ in various aspects as given in Table 1.2. Table 1.2 Characteristics of matter and electromagnetic waves S.No.

Electromagnetic Waves

Matter Waves

1.

The electromagnetic waves (photons and radiations) are associated with electric and magnetic fields perpendicular to each other and to the direction of propagation.

Matter waves (associated with material particles, like electron, proton etc.) are not associated with electric and magnetic fields.

2.

They do not require any medium for propagation, i.e., they can pass through vacuum.

They require medium for their propagation, i.e., they cannot pass through vacuum.

3.

They travel with the same speed as that of The speed of these waves is not the light. same as that of light. Moreover, it is not constant for all material particles.

4.

Their wavelength is given by

Their wavelength is given by

Example A moving particle has 4.55 x 10 -25 J of kinetic energy. Calculate its wavelength. (mass = 9.1 x 10-31 kg and h = 6.6 x 10 -34 kg m2 s-1) Solution: Since, kinetic energy = 1/2 mv 2 = 4.55 x 10-25 J (given) m = 9.1 x 10 -31 kg h = 6.6 x 10 -34 kg m2 s-1 Thus, ½ (9.1 x 10 -31) v 2 = 4.55 x 10 -25 or v 2 = 106 or v  = 103 m s-1 λ = h /mv  = 6.6 x 10 -34 / (9.1x10-31) x 103 = 7.25 x 10 -7 m. Two particles A and B are in motion. If the wavelength associated with particle A is 5 x 10-8 m, calculate the wavelength associated with particle B if its momentum is half of A. Solution: Institute of Lifelong, University of Delhi

Atomic Structure Using de-Broglie equation,  λA = h/pA and λB= h/pB so λA / λB = pB / pA But pB = ½ pA (given)

or λB = 2 λA = 2 (5 x 10 -8) m =10-7 m Interesting fact A very interesting fact about the life of two scientists lies in the concept of the wave-particle duality of matter.

It was J.J. Thomson who first showed in 1895 that electrons are subatomic particles and his son G.P. Thomson was one of the first who showed in 1926 that electrons could also behave as waves. J.J. Thomson, the father, won the Nobel Prize in 1906 for showing particle nature of electrons and his son won Nobel Prize in 1937 for showing that electrons behave as waves.

1.4 Heisenberg's Uncertainty Principle Werner Heisenberg, a German physicist, in 1927, stated the uncertainty principle as a consequence of dual behaviour of matter and radiation. It states that it is impossible to determine simultaneously the exact position and velocity of an electron . The uncertainty principle can best be understood with the help of an example. Suppose you are asked to measure the thickness of a sheet of paper with an unmarked meter-stick, obviously, the result obtained would be extremely inaccurate and meaningless. In order to obtain any accuracy you would have to use an instrument graduated in units smaller than the thickness of a sheet of paper. Analogously, in order to determine the position of an electron, we must use a 'metrestick' calibrated in units smaller than the dimensions of an electron. To observe an electron, we can illuminate it with light or electromagnetic radiation. The light used must have a wavelength smaller than the dimensions of an electron, but photon of such light would possess very high energy. The high momentum photons of such light ( p = h /λ ) would change the velocity of electrons on collision. In this process, although we are able to calculate the position of the electron accurately, we would know very little about the velocity of the electron after the collision. On the other hand, if we reduce the energy ( h ) of the photons, i.e., we increase wavelength ( λ) of light, the effects of impacts could be considerably diminished. But at the same time the electron would be less accurately defined owing to the decreased resolving power of the microscope. Therefore, with light of low frequency, it is possible to know the velocity of an electron, but not its position. The uncertainty principle is inherent in the nature of things. For example, let us try to measure the momentum of an electron moving with a velocity 8 -1 -30 8 -22 -1 of say 10 m s . Momentum ( mv ) of the electron = 10 x 10 = 10 kg m s . The γ -15 photon needed for looking at the electron has wavelength of the order of 6 x 10 m.

But So the momentum of photon,

It is clear that the γ-photon required to detect the electron will hit with a momentum 1000 times greater than its own. So, the very act of observing the electron disturbs it. In other words, it can be stated that if the position of the electron is measured accurately, there is an uncertainty in its velocity or momentum and vice-versa.

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Atomic Structure

where Δ x = uncertainty with regard to position, Δv = uncertainty with regard to velocity and m = mass of the particle. Example

Calculate the uncertainty in the velocity of an electron if the uncertainty in its position is1 -34 2 -1 Å or 100 pm. ( h = 6.6 × 10 kg M s ) Solution:

∆ -10

= 10

m (given) m = 9.1 x 10 -34

-31

= (6.6 x 10 )/(4 x 3.14 x 9.1 x 10

kg Applying uncertainty principle :

-31

-10

5

x 10 ) = 5.77 x 10 m s

-1

Did you know The nonexistence of electron in the nucleus can be explained on the basis of Heisenberg’s uncertainty principle. The app roximate dimension of a nucleus of an atom is –14 of the order of 10 . Now, if the electron were supposed to be present inside the –14 nucleus, the maximum uncertainty in its position would be 10 . Using the uncertainty expression, the maximum uncertainty in its velocity would be:

This uncertainty in the velocity is much higher even than the velocity of light, which is not possible. Hence, such an electron cannot exist.

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Atomic Structure FAQs Is the uncertainty in Heisenberg is principle due to limitations in our technology for making precise measurements?

No, the uncertainty is not due to limitations in our technology for making precise measurements. The uncertainty is inherent in the nature of subatomic particles and in the very process of making measurements. For making a measurement, the instrument must interact with the particle on which the measurement is made. During this interaction it changes the properties of the particle 'uncontrollably'. If the interaction is through photons (which have a dual nature) then shorter wavelength photons can determine the position of the particle more accurately. But such photons will have very high energy and at the instant of measurement of the position it will grossly change the momentum of the particle. If, in order to accurately determine the momentum, we use low energy photons, then the wavelength would become very large and the uncertainty in position will increase.

1.4.1 Concept of Probability

According to the uncertainty principle, the position and the momentum of an electron cannot be known accurately at the same time. So, the Bohr's concept of well-defined paths or orbits for a moving electron becomes meaningless. There is always some R

uncertainty associated with the location the electron. In other words, we can only talk about probability of finding an electron at a point in space rather than describing it accurately. Bohr model of the hydrogen atom not only ignores dual behavior of matter but also contradicts Heisenberg's uncertainty principle. In view of this inherent weakness in the Bohr's model, there was no point in extending his model to other atoms. What was needed was an insight into the structure of the atom that takes into account wave-particle duality of matter and was consistent with Heisenberg uncertainty principle. This came with the advent of quantum mechanics.

1.5 Quantum Mechanical Model of Atom

In classical mechanics, the ‘state’ of a syst em is defined by specifying all the forces

acting on the system and all the positions and velocities of the particles. Classical mechanics is based on Newton’s laws of motion that successfully describes the motion of

all macroscopic objects such as a falling stone, orbiting planets etc., which have essentially a particle-like behaviour. However, it fails when applied to microscopic objects like electrons, atoms, molecules etc. This is mainly because of the fact that classical mechanics ignores the concept of dual behavior of matter and the uncertainty principle. The branch of science that takes into account this dual behavior of matter is called quantum mechanics. Quantum mechanics is a theoretical science that deals with the study of the motion of the microscopic objects that have both observable wave like and particle like properties. It specifies the laws of motion that these objects obey. When quantum mechanics is applied to macroscopic objects (for which wave like properties are insignificant) the results are the same as those from the classical mechanics. Wave mechanical model of atom is based on the following postulates: Heisenberg uncertainty principle governs atomic particles. Thus, the idea of uncertainty in the position and the velocity overruled Bohr's picture of fixed circular orbits. The term probability distribution is used to describe position of an electron. A stream of electrons is associated with a wave, whose wavelength ( λ) is given • • Since electron is under the influence of nuclear forces, it can be compared to a stationary or standing wave formed by a vibrating string fixed between two points. Energy values of the electron are quantized as calculated by Bohr. • •

Quantum mechanics was developed independently in 1927 by Werner Heisenberg and Erwin Schrödinger. Here, however, we shall be discussing the quantum mechanics of Schrödinger, which is based on the ideas of wave motion. With the wave nature of a Institute of Lifelong, University of Delhi

Atomic Structure moving electron confirmed, it was expected that its behaviour could be described by a suitable wave equation just as for waves of light, sound, etc.

The fundamental equation of quantum mechanics is the Schrödinger equation:

where ψ = amplitude of the wave, m = mass of the electron, h = Planck’s constant, E = total energy of the electron, V = potential energy of the electron and  x , y and z are the coordinates of the electron with nucleus at the origin. This equation was proposed by Schrödinger, which won him the Nobel Prize in Physics in 1933. In quantum mechanics, the state of the system is defined by a mathematical function (ψ) called the state function or wave function. A wave equation gives a complete description of a system. Just as we can write a volume function for a gas, V = nRT   /P , we can write a wave function for a wave in the form of ψ. where x = displacement, A = constant and ψ = amplitude of the wave. where x = distance from the nucleus Differentiating with respect to  x For waves of a vibrating string, we have the simplified relation,

where x  = displacement, A = constant and ψ = amplitude of the wave. For electron waves, we can write, where x  = distance from the nucleus Differentiating with respect to x  Differentiating again,

From de Broglie’s relation we have,

Replacing λ in equation (1.1), we have,

The total energy of a particle is sum of potential energy and kinetic energy, i.e., E = P.E. + K.E. = V  + ½ mv 2 E – V  = ½ mv 2 v 2 = 2(E  - V )/ m

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Atomic Structure

The Schrödinger equation can be rearranged to the form

and shortened to Ĥψ =

Eψ where Ĥ  is called a Hamiltonian operator and it defines the operation or sequence of operations to be performed (i.e., taking second differentials, multiplying their sum by – 2 2 h  /8π m and adding the potential energy, V ) on the function ψ. The result of carrying out these operations on function ψ is the same as multiplying ψ by the electronic energy, E . This form of the Schrödinger wave equation (i.e. Ĥψ= E ψ) is called an eigenvalue equation. FAQs What is an operator?

An operator is basically a mathematical command, which operates on a function that follows it. For every observable/ measurable property in classical mechanics, there is a corresponding operator in quantum mechanics. Like d/d x (first derivative of function), 2 2 d  /d x  (second derivative of function), √   (square root of function) etc, are some examples of operators.

Does the Hamiltonian operator, Ĥ, have units? The Hamiltonian operator defines the sequence of mathematical operations to be performed on the wave function, Ψ  , which follows it in the Schrödinger Equation. ĤΨ = EΨ i.e.,

The wave function, y, is an eigenfunction and the energy, E is an eigenvalue of the Hamiltonian operator. This indicates a correspondence between the Hamiltonian operator and the energy. In fact, the Hamiltonian operator in the above equation is constructed in a manner such that it contains both the ki netic energy term and the potential energy term and therefore it is evident that the Hamiltonian operator has the dimensions of energy (energy units) in the Schrödinger Equation.

The solution of the Schrödinger wave equation led to the concept of the most probable regions where the chance to find the electrons is maximum. This is in contrast to the well-defined circular paths proposed by Bohr. In accordanc e with Heisenberg’s uncertainty principle, the electron cannot exist at a definite point but at certain regions in space around the nucleus where the probability of finding electron is about 90-95%. Institute of Lifelong, University of Delhi

Atomic Structure These regions in space are called orbitals.  An orbital may be defined as a region in space, around the nucleus, where the probability of finding electron is maximum .

Table 1.3. The difference between an orbital and orbit is given in Table 1.3. Difference between an Orbit and an Orbtal

Table 1.3 Difference between an orbit and an orbital S.No

Orbit

Orbital

1.

An orbit is a well-defined circular path around the nucleus in which an electron revolves.

It represents the region in space around the nucleus in which the probability of finding the electron is the maximum.

2.

It represents the planar motion of It represents the three-dimensional motion of an electron. an electron around the nucleus.

3.

The concept of a well-defined orbit This concept is in accordance to is against the Heisenberg's the Heisenberg Uncertainty principle and the de-Broglie relation. uncertainty principle and the de-Broglie concept.

4.

All orbits are circular.

Orbitals have different shapes. For example, s- orbital is spherically symmetrical and  p-orbital is dumbbell shaped.

5.

Orbits do not have directional characteristics.

Except the s-orbital, all others have directional characteristics and, thus, they clearly explain the shape of a molecule during bonding.

6.

The maximum number of electrons An orbital can accommodate only two electrons and that too with opposite spins. in an orbit is 2n2, where n represents the number of orbit.

1.5 Quantum Mechanical Model of Atom

Schrödinger equation cannot be derived, i.e., it cannot be proved from fundamental principles of quantum mechanics. It was the genius of Schrödinger to arrive at equation (1.4) by intuition and to justify it by solving this equation to give values for ‘ E ’ in agreement with the experiments. It may be emphasized again that the above treatment is in no way a ‘proof’ of the Schrödinger equation, it merely shows that if the deBroglie’s

relationship is assumed and if the motion of the electron is analogous to a system of standing waves, equation (1.4) is the type of the wave equation to be expected. The Schrödinger equation shows a logical coherence to a vast amount of experimental observations. It has been found to hold true in both microscopic and macroscopic systems. The mathematical task in applying the Schrödinger equation to a particular problem is to obtain a suitable expression for ψ showing how the wave amplitude varies with distance along the x , y , z -axes and then to derive solutions of the differential equation. There will be, in every case, many expressions for ψ satisfying the Schrödinger equation, all of which are, however, not acceptable. Only those wave functions which satisfy certain conditions are acceptable and are called eigenfunctions of the system, while the energy E, corresponding to the these eigenfunctions are called eigenvalues. For hydrogen atom, the eigenvalues correspond to the discrete sets of energy values postulated by Bohr theory, i.e., the occurrence of definite energy levels in an atom follows directly from the wave mechanical concept. Institute of Lifelong, University of Delhi

Atomic Structure The conditions which a wave function, ψ, must satisfy before it is accepted are: (i) ψ must be continous (all real values are continous). 2 (ii) ψ must be finite (ψ must be finite since it represents probability). (iii) ψ must be single valued, i.e., it must have only one definite value at a particular point in space. (iv) The probability of finding the electron over the entire space from –∞ to + ∞ must

be equal to one, that is,

1.5.1 Significance of

Ψ and Ψ2

The physical significance of ψ is nebulous. The wave function ψ has no physical significance except that it represents the wave amplitude. ψ depends on the coordinates of the particle. In some cases, it may turn out to be a complex function of the form: ψ = a + ib where a and b are real functions of the coordinates and i is iota This complex nature of ψ is necessary if one is to use superposition of wave functions to describe interference effects in matter waves. The complex conjugate of ψ is: 2

2

ψ* = a – ib therefore, ψψ* = (a + ib) (a – ib) = a + b If ψ is real, ψψ* is identical to 2 ψ. 2

Neither ψ nor ψ* has any physical significance but ψψ* (or ψ ) has. The product of ψ ψ* 2 will always be real, whereas, ψ (ψψ) can possibly be imaginary. In dealing with all forms of wave motions such as light waves, sound waves or matter waves, the square of the wave amplitude at any point is interpreted 2 ) is ΨΨ Ψ as the intensity of the effect at that point. So, the value of * or at equal to any point around the nucleus gives measure of the electronic charge unity. density at that point, the density of Mathemati which varies from point to point around the nucleus. According to the cally, it statistical interpretation, the electron 2 is still considered as a particle and the can be value of Ψ at any point is taken to represent the probability of finding the electron at that point at a given instant. According to uncertainty principle, the position of the electron cannot be determined with 2 certainty. Thus Ψ is interpreted as giving a direct measure of probability. Consequently, the greater the intensity of wave function at a particular point, greater is the probability 2 of locating the electron at that point. When Ψ is high, electron density is high, i.e., the probability of finding an electron is high. Did you know ψ is known as the “storehouse” of information because the information regarding any property of the system like energy, radius etc., can be obtained from ψ.

1.5.2 Normal and Orthogonal Wave Functions The wave function, Ψ, is said to be normalized if the probability of finding an electron over the entire space (from – to + ) is equal to unity. Mathematically, it can be expressed as: This is quite reasonable, as the probability of finding the electron in the entire universe has to be 100%. Sometimes Ψ may not be a normalized wave function. In that case, we multiply the function Ψ with a constant, N such that, the function N Ψ is also a solution to the wave equation. The value of N is so chosen that the new function N Ψ be normalized. Mathematically, the expression can be written as: Institute of Lifelong, University of Delhi

Atomic Structure

where N is called normalization constant. The wave function Ψ is said to be orthogonal if it is different from all the other wave functions over the entire space (from – to + ). Mathematically, it can be expressed as: a

This is reasonable, as each wave function (that is solution to the wave equation) is unique and different. Thus, any two wave functions (say, Ψ 1 and Ψ2) should be independent of each other, i.e., they should be orthogonal to one another. The quantum mechanical study of any system consists of (i) Writing Schrödinger wave equation for the system. (ii) Solving Schrödinger wave equation for the meaningful solutions of the wave functions and the corresponding energies. (iii) Calculation of all the observable properties of the system from ψ. 1.5.3 Quantum Mechanical Treatment of The Hydrogen Atom

Hydrogen atom is the simplest chemical system consisting of one proton and one electron. Schrödinger equation can be solved exactly for hydrogen-atom but not for atoms having more than one electron. Assuming that the electron moves at a distance around the stationary nucleus, then the nucleus can be taken as the origin in a coordinate system. Schrödinger equation for the hydrogen atom can be written in terms of the Cartesian coordinates ( x, y, z ) as given below:

The solutions of the Schrödinger equation for the hydrogen atom lead to important conclusions. It gives the values of energy, which agrees well with those obtained experimentally, and also with those given by Bohr for his model of atom. It also gives the allowed energy levels for electrons, which are same as that derived by Bohr. The problem here is that of calculating the amplitude of electron waves at various points in a hydrogen atom. These points can be defined by drawing a set of Cartesian ( x, y, z ) axes through an origin at the nucleus of the atom and locating points by  x, y, z coordinates. However, it turns out that the mathematics involved is much simpler if we use an alternative way of specifying position namely the polar coordinates ( r, θ , φ). Since the atom has spherical symmetry, it is apparent that in the polar coordinates, the separation of the variables for the hydrogen system is easier than in Cartesian coordinates. Hence, Schrödinger equation is transformed into polar coordinates. The coordinate r , measures radial distance from the origin, θ is a lattitude and φ is a longitude. Since the electron is moving in three dimensions, three coordinates are sufficient to describe its position at any time. Transformation of Cartesian Coordinates to Polar Coordinates

The relation between the two coordinate systems is shown in the fig. 1.3. It can be seen from the fig. 1.3 that the coordinates  x, y, and  z of the electron with respect to nucleus Institute of Lifelong, University of Delhi

Atomic Structure in terms of polar coordinates can be written as:

Transformation of Cartesian Coordinates to Polar Coordinates By replacing  x, y, z coordinates with the polar coordinates in the Schrödinger

we get,

equation,

The above equation is the Schrödinger wave equation for H-atom in terms of polar coordinates. A series of wave functions exist that are solutions to the above equation. A standard method of solving the classical wave equation is by the method of separation of variables. This can be done if it is assumed that the actual wave function ψ (r, θ  , φ ) is the product of three separate wave functions each containing one of the three variables. This can be represented as:

l

Figure 1.3: Relationship For convenience, the solutions are given in terms of radial part R(r ) and angular part Θ(θ).ф(φ) or A(θ ,φ). When Schrödinger equation in polar coordinates is solved for the hydrogen atom, it gives the possible energy states and the corresponding wave functions [ ψ(r , θ, φ)] (called atomic orbitals or hydrogenic orbitals). The quantum mechanical solution of the hydrogen atom successfully predicts all aspects of the hydrogen atom spectrum and other phenomena that could not be explained by the Bohr model.

1.6 Quantum Numbers

The application of Schrödinger equation to hydrogen atom and other atoms yields three quantum numbers n, l and m, characterized as principal quantum number, azimuthal quantum number and magnetic quantum number, respectively. These three quantum numbers arise as a natural consequence in the solution of the Schrödinger equation . The restrictions on the values of these three quantum numbers also come naturally from this solution. The fourth quantum number, s, characterized as the spin quantum number, arises from the spectral evidence. Using refined spectroscopic techniques, it was found that in high-resolution spectrum of H-atom, each spectral line is in fact a doublet, i.e., two Institute of Lifelong, University of Delhi

Atomic Structure closely spaced lines. This can be accounted for only if we assume that electron in its motion around the nucleus, not only rotates but also spins about its own axis. So, four quantum numbers are actually required in order to define an electron completely in an atom. These quantum numbers describe the whole atomic state. Principal or Radial quantum number ( n) The principal quantum number, n, is a positive integer with values of 1,2,3…, etc. The

principal quantum number determines the size and to a large extent the energy of the orbital. ‘n’ specifies location and energy,

+

2+

For hydrogen and hydrogen-like species (e.g., He , Li ), it alone determines the energy and size of the orbital. The lower the value of n, the more stable will be the orbital. The principal quantum number also identifies the shell to which the electron belongs. Each value of n corresponds to a shell, which can be represented by the following letters. n = 1 2 3 4 shell = K L M N Azimuthal or Orbital Angular Momentum Quantum Number( l )

A shell consists of one or more subshells or sublevels. Each subshell in a shell is assigned an azuimuthal quantum number, l . The number of subshells in a principal shell is equal to the value of n, i.e. for a given value of n, l can have values ranging from 0 to ( n-1). For example, for n =1, the only value of l is 0 and there is only one sub-shell. For n=2, there are two subshells that have values of l as 0 and 1. For n =3, the values of l are 0,1 and 2 for three subshells. Subshells corresponding to different values of l are represented by the following symbols: l

notation

= =

0, s,

1, p,

2, d,

3, f,

4, g,

5 h

The symbols s, p, d and f stands for the initial letters as in the words sharp, principal, diffuse and fundamental respectively which had been used to define the spectral lines. The other symbols, i.e., g, h,…. and so on proceed alphabetically. Table 1.4 shows the permissible values of l for a given principal quantum number and the corresponding subshell notations. Table 1.4 Subshell notations n

1

2

3

4

l 

0

0

1

0

1

2

0

1

2

3

Subshell

1s

2s

2 p

3s

3 p

3d 

4s

4 p

4d 

4f 

The azimuthal quantum number gives an idea about the shape of the orbital. For example, orbital in a ‘s’ subshell is spherical, orbitals in a  ‘ p’ subshell are dumbbell shaped, orbitals in a ‘d ’ subshell are double dumbbell shaped and so on. It also describes

the motion of the electron in terms of orbital angular momentum. The orbital angular momentum ( L) of an electron is given by the relation:

For hydrogen atom, energies of the subshells belonging to the same shell are equal, i.e., ns = np = nd = nf . However, for atoms with more than one electron, the energy increases slightly with an increase in the value of l , i.e., ns < np < nd < nf. Magnetic Quantum Number ( ml)

Each subshell consists of one or more orbitals and each orbital is identified by a magnetic orbital quantum number ml, which gives information about the orientation of the orbital. The number of orbitals in a subshell is given by (2 l + 1). For instance, when l =0(s-subshell) there is only one orbital, when l =1( p-subshell) there are three orbitals, when l =2(d -subshell) there are five orbitals and so on. This is also equal to the number of values that ml may assume for a given value of l. Since ml determines the orientation of orbitals, the number of orbitals must be equal to the number of ways in which they are oriented. For a given value of l , the values of ml are -l ,…..0,….,+l . Table 1.5 Values of ml  for different values of  l Institute of Lifelong, University of Delhi

Atomic Structure l 

Subshell

ml 

0

s

0

1

 p

-1, 0, +1

2

d 

-2, -1, 0, +1, +2

It may be noted that the values of ml are derived from l and that the values of l are derived from n. For any shell represented by the quantum number ‘ n’, The number of subshells (or values for the quantum number l )= n 2 The number of orbitals (or values for the quantum number ml )= n Orbitals within a given subshell differ in their orientation in space, but not in their energies, i.e., they are degenerate (having same energies). However, on applying an external magnetic or electric field, the degeneracy is lifted (orbitals with different orientations acquire different energies in presence of the magnetic or electric field). Spin Quantum Number (s) and Magnetic Spin Quantum Number ( ms)

The spinning of an electron is untenable from quantum mechanics. It arises from the spectral evidence. The electron not only revolves around the nucleus but also spins about its own axis. This spin of the electron is associated with spin angular momentum, which is characterized by spin quantum number, s (=½ specifically for electrons). In an analogous way of orbital angular momentum, L, the spin angular momentum can be obtained by the expression:

where s = spin quantum number and h = Plank’s constant

The component of the spin angular momentum along the  z -axis is given by the expression:

A spinning electron acts like a tiny magnet. So, two electrons with opposite spins will behave as two magnets with opposite poles towards each other and hence there is attraction. For every value of the quantum number ml (orbital), there are only two permitted values of ms, i.e., + ½ and –½ . That is to say that every orbital can have not more than two electrons and that too having opposite spins.

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Atomic Structure Interesting fact The Mystery of the fourth quantum number ( s).

Quantum Mechanics is not just a theoretical science but is a result of an interplay of theory and experiment. The advent of and the progress made in the field of quantum mechanics in the first quarter of the 20th century had provided a much better solution to the problem of hydrogen atom and addressed the anomalies in the Bohr’s model,

however, still many experimental observations were still raising the eyebrows of many scientist of the time and these observations were the following: • -On passing a beam of silver atoms through an inhomogeneous magnetic field they

unexpectedly observed a splitting of the beam into two parts. Stern-Gerlach observation

• A doublet in the spectra of alkali metals.

In order to address these questions, Wolfgang Pauli postulated that an electron can exist in two distinct states and introduced a fourth quantum number in a rather ad-hoc manner. This fourth quantum number is now called the spin quantum number, s and has a value ½ for electrons. It is interesting to know that Pauli did not give any interpretation to this fourth quantum number and therefore this fourth quantum number was just a mystery at that time though it provided explanation to various experimental observation listed above. It was finally the attempt of George Uhlenbeck and Samuel Goudsmit in 1925 who showed that there are two intrinsic states of an electron, which are due to the motion of the electron about its own axis and identified these two intrinsic states with two intrinsic angular momentum or spin states. That is how the concept of spin and thereby the spin quantum number came in to existence. It did not come from the solution of the Schrodinger wave equation but due to the explanations for the various experimental observations at that time.

Discovery of electron spin and spin quantum number

As stated earlier, the spin quantum number, s, does not follow directly from the solution of the Schrödinger wave equation for the hydrogen atom. It arose out of necessity to explain two types of experimental evidences, which suggested an additional property of the electron. One was the closely spaced splitting of the hydrogen spectral lines, called fine structure. The other was the Stern-Gerlach experiment, which showed in 1922 that a beam of silver atoms directed through an inhomogeneous magnetic field would be forced into two beams. Both of these experimental situations were consistent with the possession of an intrinsic angular momentum and a magnetic moment by individual electrons. Classically, this could occur if the electron was a spinning ball of charge, and this property was called electron spin. George Uhlenbeck and Samuel Goudsmit, in 1925, proposed the idea that each electron spins with an angular momentum of one half Planck -24 constant and carries a magnetic moment of one Bohr magneton ( µB= 9.27 x 10 J/T ). Figure 1.4: An electron spin angular momentum vector of length units (one unit = h /2π) can take only two orientations with axis respect to a specifed

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Atomic Structure The experimental evidences suggest just two possible states for this angular momentum described by spinning of electron in either clockwise or in anti-clockwise direction. In the presence of an external magnetic field an electron can have one of the two orientations corresponding to magnetic spin quantum number, ms =±½ as shown in Figure 1.4. The magnetic spin quantum number, ms, is used to define the  z -component of the spin angular momentum.

Soon, a new version of the wave equation was developed by relativistically invariant, (called equation the additional quantum the electron arose naturally during its solution.

r ,

of

Schrödinger Paul Dirac, which was Dirac's equation). In this number ‘s’ intrinsic to

1.7 Atomic Orbitals and Their Pictorial Representations

An atomic orbital is a one-electron wave function ψ(r, θ , φ) obtained from the solution of the Schrödinger wave equation. It is a mathematical function of the three coordinates of the electron (r, θ , φ) and can be factorized into three separate parts each of which is a function of only one coordinate: where R (r ) is the radial wave function which gives the dependence of the distance, the electron from the nucleus and are the angular wave functions giving the direction of the electron in terms of the angles θ and φ respectively. On solving the three wave equations (each involving only one variable) separately, it was found that the radial wave function depends upon the quantum numbers wave function depends upon quantum numbers l and m depends on quantum number ml only. The total wave function ψ may, therefore, be more explicitly written as:

2

It would be interesting to know how ψ and |ψ| vary as a function of the three coordinates r, Θ and φ for different orbitals. Such representation of the variations of ψ or 2 ψ in space would however need a four dimensional graph -three dimensions for the 2 coordinates and the fourth for ψ or |ψ| . It is not possible to show such variation in a single diagram since we can draw only two-dimensional diagrams on paper. We can get over this difficulty by drawing separate diagrams for: (i) variation of radial function and (ii) angular function. 1.7.1 The Radial Wave Function, R(r)

The solutions of radial wave functions are in the form of polynomials in ‘ r ’ and are known

as the associated Laguerae polynomials. The normalized solutions of these wave equations are quite complex. However, they reduce to relatively simple forms on introduction of particular values of the parameters, i.e., n parameter and l parameter. (n = 1, 2, 3……

and l = 0 to n-1). Mathematical expressions for radial functions of 1 s,2s, and 2 p are given in the Table 1.6. Redial Functions of 1 s, 2 s and 2 p orbitals S. No.

n

l

Orbital

1.

1

0

1s

2.

2

0

2s

3.

2

1

2 p

Radial Wave function, R(r )*

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Atomic Structure

*where Z = nuclear charge, e = base of natural logarithm, = 0.529 Å for 1 s in H atom and in wave mechanics it is the most probable radii. The important aspects of radial functions may be made apparent by grouping constants (Table 1.7). For a given atom, Z is constant. *where k 1s, k 2s and k 2 p are the constants. All these radial wave functions represent an exponential decay and that for n = 2, the decay is slower than for n = 1. 1.7.2 Plots of the Radial Wave Function, R(r)

The radial part of the wave function R(r ) gives the distribution of the electron with respect to its distance, r , from the nucleus. The importance of these plots lies in the fact that they give information about how the radial wave function changes with distance, r , and about the presence of nodes. As stated earlier, the radial wave function depends only on the values of quantum numbers n and l . It is mainly goverened by exponential - zr/naº term, e . In all cases wave function, R(r ), approaches zero as r approaches infinity. However, the actual shape of the curve is determined by actual wave function that depends on the value of the quantum numbers n and l . For 1s orbital (n =1; l =0), the wave function depends only on the exponential function. So, a plot of R(r ) versus r is an exponential curve as shown in fig. 1.5(a). In this plot, at r = 0, R(r ) has a maximum value and as r increases, R(r ) decreases exponentially. For a 2 s orbital (n =2; l =0), the wave function R(r ),

in addition to the exponential term also depends on the term

wave

function, R(r ), initially decreases as r increases and becomes zero at it passes through a minimum and then increases to an infinitesimally small negative value. This is shown in fig 1.5(b). R(r ) for the 2 s orbital becomes zero at a particular value of r  between r  = 0 and r = , at this point, the so called nodal point, R(r ) changes sign from positive to negative.In general, the number of nodes in ns-orbital is given by ( n – 1). There is only one node in 2 s radial function. The radial plot for a 3 s-orbital (n =3; l  =0), is shown in fig.1.5(c). The radial wave function, R(r ), for the 3 s-orbital becomes zero at two points between 0 and therefore, it has two nodes.

.

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Atomic Structure Figure 1.5: The plots of the radial wave function, R(r ), as a function of distance, r , of the electron from the nucleus for (a) 1s-orbital (b) 2s-orbital and (c) 3s- orbital

The radial function for a 2 p-orbital (n =2; l =1), depends on the term

in addition to

the exponential term . So, in the plot of R(r ) versus r , the function R(r ) becomes zero at r  = 0. It increases initially with r , reaches a maximum value and then decreases to an infinitesimally small value. The function R(r ) remains positive throughout as shown in fig. 1.6(a), hence there are no nodes in the 2 p radial function. The radial wave function for 3 p orbital [fig. 1.6(b)] increases to a maximum in the beginning and then decreases passes through zero and decreases further and finally increasing again to an infinitesimally small value. There is one node in the radial function plot of 3 p-orbital. The radial wave function plot for 3 dorbital is shown in fig. 1.6(c). The general formula for calculating the number of nodes in any orbital is (n – l – 1).

(a) Figure 1.6: T of and

(b) he plots of the radial wave function,

(c) R(r), as a function of distance, r ,

the electron from the nucleus for (a) 2 p-orbital (b) 3 p-orbital (c) 3d -orbital

Animation Plots of the Radial Wave Function, R(r)

http://www.illldu.edu.in/mod/resource/view.php?id=5634

1.7.3 Plots of Radial Probability Density R2(r) or R2 2

The square of the radial wave function R for an orbital gives the radial density. The radial density gives the probability density of finding the electron at a point along a 2 particular radius line. To get such a variation, the simplest procedure is to plot R against r (fig. 1.7 and 1.8). These plots give useful information about probability density or relative electron densitys at a point as a function of radius. It may be noted that while for s-orbitals the maximum electron density is at the nucleus; all other orbitals have zero electron density at the nucleus. Institute of Lifelong, University of Delhi

Atomic Structure (a) (b) (c)

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Atomic Structure 2

Figure 1.7: The plots of the radial probability density, R , as a function of distance, r, of the electron from the nucleus for (a) 1 s orbital (b) 2s orbital and (c) 2 p orbital

(a) (b) (c) 2

Figure 1.8: The plots of the radial probability density, R , as a function of distance, r , of the electron from the nucleus for (a) 3 s orbital (b) 3 p orbital and (c) 3d (a) Figure 1.6: T of

(b) he plots of the radial wave function,

(c) R(r), as a function of distance, r ,

the electron from the nucleus for (a) 2 p-orbital (b) 3 p-orbital (c) 3d -orbital

and

Polts of Radial probability Density R(r) 2 or R 2

http://www.illldu.edu.in/mod/resource/view.php?id=5634 FAQs What is the role of quantum mechanics in chemistry, if it is a science dealing with "laws of motion" of microscopic objects? Quantum mechanics caters to three

fundamental problems in chemistry namely, structure, bonding and reactivity. Solutions to these problems depend on the detailed understanding of the behavious of these microscopic particles in matter. Microscopic particles can never be at rest. An in depth understanding of

orbital. 1.7.4 Plots of Radial Probability Distribution Function, 4 πr2R2(r) 2

The radial density, R ,for an orbital, as discussed earlier, gives the probability density of finding the electron at a point at a distance r from the nucleus. Since the atoms have spherical symmetry, it is more useful to discuss the probability of finding the electron in a spherical shell between the spheres of radius ( r + dr ) and r (fig. 1.9). The probability of finding the electron between the shell with radius r and the shell with radius ( r +dr ) is called the radial probability. Volume of sphere with radius r is given as:

Multiplying by R

2

Figure 1.9: Spherical shell of thickness d r The probability of finding the electron in the spherical shell of thickness d r   is equal to Institute of Lifelong, University of Delhi

Atomic Structure (volume of the shell x probability function). This represents radial probability distribution function ( ), which gives the probability of finding the electron in a shell at a distance, r , from the nucleus regardless of direction. FAQs 2

2

2

What is the difference between ψ and 4πr  R terms if both represent the probability of finding an electron around the nucleus? 2

2

*

ψ or ψ ψ represents the probability of finding the electron at a point. The value of ψ at a point represents the probability of finding the electron at that point. ψ depends on variables ( x , y , z ) in Cartesian system and (r , θ, f ) in polar system. When we perform the separation of variables of the wave function ψ(r , θ, f ) into radial part R(r ) and 2 angular part Y (θ, f ), the R term gives us the radial probability distribution of electron around the nucleus because R, the radial part of wave function, is only dependent on r (distance of electron from the nucleus) and not on the other two polar coordinates ( θ and f ). The radial probability represents the total probability of finding the electron in a spherical shell situated at a distance r from the nucleus and having a thickness d r .

The radial probability curves, (obtained by plotting radial probability distribution function, , versus distance, r ) for 1s. 2s and 2 p orbitals are shown in fig. 1.10. An important contrasting feature of the radial probability distribution curves with regard to radial wave function and radial probability density can be seen in the plots of s-orbitals. It may be noted that at r = 0, the latter two functions have a maxima while the former has a zero value. The maxima in the former is very close to r = 0 but not at r = 0. This is justified as the radial probability distribution function, , also depends on ‘r ’ (in addition to

), so its value is zero at r = 0.

The importance of these curves lies in the fact that they give the true picture of the probability of finding the electron with respect to the nucleus as it has been established that an electron can never exist at the nucleus (i.e., at r = 0). 1 s orbital

The radial probability distribution function for the 1 s orbital, [fig. 1.10.(a)], initially increases with increase in distance from the nucleus. It reaches a maximum at a distance very close to the nucleus and then decreases. The maximum in the curve corresponds to the distance at which the probability of finding the electron is maximum. This distance is called the radius of maximum probability or most probable distance . For 1s orbital, it is equal to 52.9 pm, same as Bohr's radius for hydrogen atom. Did you know Bohr was the first to introduce the concept of quantization of angular momentum through the relation mvr

= nh /2π

but this concept had no proof at that time and hence became a limitation of Bohr model of the atom. (a) (b) (c) Figure 1.10: The plots of the radial probability distribution function, 4πr 2drR2, as a function of distance, r , of the electron from the nucleus for (a) 1s-orbital (b) 2s-orbital and (c) 2 p-orbital.

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2 s and 2 p-orbitals

The radial probability function curve for 2 s orbital, [fig. 1.10.(b)], shows two maxima, a smaller one near the nucleus and a bigger one at a larger distance. In between these two maxima it passes through a zero value indicating that there is zero probability of finding the electron at that distance. The region at which the probability of finding the electron is zero is called a node. The most probable distance for a 2s electron corresponds to the value of r where we have a bigger maxima. The distance of maximum probability for a 2 p electron, [fig. 1.10.(c)], is slightly less than that for a 2 s electron. However, in contrast to 2 p curve, there is a small additional maxima in the 2 s curve, which lies at or around the maxima for a 1 s orbital. This indicates that the electron in 2 s orbital spends some of its time near the nucleus. In other words, the 2 s electron penetrates into the inner 1 s shell and therefore, is held more tightly than the 2 p electron. That is the reason why 2 s electron is more stable and has lower energy than a 2 p electron. 3 s,3 p and 3d -orbitals

Radial probability distribution curves for 3s,3 p and 3d -orbitals are shown in fig. 1.11.

(a) (b)

2R 2

Figure 1.11: The plots of the radial probability distribution function, 4π r , as a function of distance, r , of the electron from the nucleus for (a) 3 s-orbital (b) 3 s,3 p and 3d -orbitals.

For 3s orbital, there are three regions of high probability separated by two nodes. The first two peaks indicate penetration of the electrons. In general, the number of high probability regions = ( n – l ) And the number of radial nodes = (n – l – 1) For example, in a 3 p-orbital, number of peaks = n – l =3–1=2 and number of nodes = n – l –1=3–1 –1=1 It can be seen in Fig. 1.11(b) that the most probable distance for 3 s,3 p and 3d orbitals decreases in the order 3s >3 p >3d . That is to say that the 3 s orbital is more extended in space than the 3 p orbital which is in turn more extended than 3 d orbital. However, the radial distribution of 3s orbital spreads into the curve for 2 s and 1s orbital. Similarly, the radial distribution of 3 p-orbital spreads into the curve for 2 p orbital and so on. This is called penetration of orbitals to inner cores. Hence, an electron in an outer orbital is not fully screened or shielded by the inner electrons from the nuclear charge. It is only partially screened from the nuclear charge. The extent of penetration decreases from s to f orbitals, i.e. s > p > d > f . This variation in the extent of penetration greatly influences the effective nuclear charge and relative energies of orbitals in multi –electron Institute of Lifelong, University of Delhi

Atomic Structure atoms (discussed later in this chapter). 2

2

Plots of Radial Probability Distribution Function , 4nr R(r)

http://www.illldu.edu.in/mod/resource/view.php?id=5634 1.7.5 Plots of Angular Wave Function

Θ(θ)Φ(φ)

The angular part of the wave function determines the shape of the electron cloud and varies depending upon the type of orbitals involved i.e., s, p, d , f and their orientation in space.

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Atomic Structure As already mentioned, the angular wave function ' Θ(θ)Φ(φ)’ depends only on the quantum numbers l and ml and is independent of the principal quantum number n. It therefore means that all s orbitals will have same angular wave function irrespective of the shell they belong to. Similarly, all  px orbitals will have same angular wave function and so on. The plots of angular wave function with θ  and φ) are 3-dimensional plots, which give an idea about the shapes of the orbitals. Mathematical expressions for angular wave functions of s-and p-orbitals are given in the table 1.8: (a) Figure 1.6: T of

(b) he plots of the radial wave function,

(c) R(r), as a function of distan

the electron from the nucleus for (a) 2 p-orbital (b) 3 p-orbital (c) 3d -orbital

and

Polts of Radial probability Density R(r) 2 or R 2

http://www.illldu.edu.in/mod/resource/view.php?id=5634 FAQs What is the role of quantum mechanics in chemistry, if it is a science dealin "laws of motion" of microscopic objects? Quantum mechanics caters to three

fundamental problems in chemistry namely, structure, bonding and reactivity. Solu these problems depend on the detailed understanding of the behavious of these mi particles in matter. Microscopic particles can never be at rest. An in depth understa the behaviour of these particles is achieved by quantum mechanical studies. Table 1.8 Angular wavefunctions for s-and p-orbitals l

ml

Orbital

0

0

s

Angular wave functions

The angular wave function for s-orbital is independent of angle θ and φ and therefore, its plot is spherically symmetrical. This is shown in fig. 1.12. There are no angular nodes in s-orbital. The angular wave function plots for  p-orbitals give two tangent spheres. The plots for px, py and  pz orbitals are identical in shape but are oriented along the  x , y and  z axes respectively.

Figure 1.12: Angular wave function plot for 1 s orbital Each  p-orbital has an angular node (represented by a plane) as shown in fig. 1.13. The angular wave function plots for d and f orbitals are four-lobed and six-lobed respectively. There are two nodal planes in each d -orbital except for d z2 , which has a nodal surface as shown in the fig. 1.14. It is necessary to keep in mind that in the angular wave function plots, the distance from the center is proportional to the numerical values of θ and φ in that direction and is not the actual distance from the center of the nucleus.

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Figure 1.13: Angular wave function plots for 2 p-orbitals

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Figure 1.14: Angular wave function plots for 3 dxy and 3dz 2-orbitals showing the angular nodes.

1.7.6 Plots of Angular Probability Density: Shapes of Atomic Orbitals

The actual shapes of the orbitals are obtained by plotting square of the angular wave function as it represents the probability distribution. On squaring, different orbitals change in different ways. For an s orbital, the squaring causes no change in shape since the function everywhere is the same; thus another sphere is obtained. For both  p and d orbitals, however, on squaring, the plot tends to become more elongated as shown for  pz in fig. 1.15. Instead of two tangent spheres, we get two dumbbells symmetrical about  z -axis. Similarly, for  px and  py orbitals, we get dumbbell-shaped lobes symmetrical about  x -axis and y -axis respectively. For d -orbitals, four dumbbell shaped lobes are obtained except for d z2 orbital, which consists of two lobes along  z -axis with a ring of electron density around the nucleus in  xy -plane as shown in fig. 1.20(a).

Figure 1.15: Angular probability density function for  pz , px and py orbital. Check your Progress

How do electrons cross nodes when they can never be present at the nodes? A mathematical expression, which describes the probability that an electron of an atom will be at a certain point in space, is called an atomic orbital. A complete picture of an orbital can be obtained by summing up both the radial and angular distribution functions together. These pictures of orbitals are called the plots of total probability density. Ideally, probability distribution extents till infinity but most of it exists very near to the 2 nucleus. Chemists tend to think of electron clouds and hence, ψ probably gives the best intuitive picture of an orbital. Although, electron density may be shown either by shading or by contours of equal electron density, only the latter method is quantitatively accurate. Figure 1.16 shows the pictorial representation of the electron density in a hydrogen like 3 p-orbital .

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Figure 1.16: Countour diagram for a 3 pz -orbital.

1.7.7 Plots of Total Probability Density: Shapes of Atomic Orbitals

We can also construct boundary surfaces or boundary surface diagrams such that they contain a volume within which there is a 99% probability of finding the electron. Such a boundary surface for an s-orbital (l = 0) has the shape of a spherical shell centered on the nucleus [fig. 1.17(b)]. For each value of n, there is one s orbital. As n increases, there are (n-1) concentric spherical shells like the successive layers in an onion [fig. 1.17(a)]. An alternative approach is to draw charge cloud diagrams. In this approach, the 2 probability density |ψ| is shown as a collection of dots such that the density of dots in any region represents the electron probability density in that region. The greater the density of dots at a region, the greater is the probability of an electron to be found there. These diagrams for s-orbitals are shown in [fig. 1.17(c)]. Charge cloud diagrams give a better picture of orbitals as compared to boundary surface diagrams because in the former plots, density of dots is proportional to electron probability density. Hence, just by looking at these plots, we can predict where the probability of finding an electron is maximum. Moreover, the radial nodes can be seen in these plots, which are not visible in boundary surface diagrams.

Figure 1.17: Total probability density plots for s-orbitals (a) crossection of boundary surface plots showing radial nodes (b) boundary surface diagrams for 1 s,2s and 3s orbitals (c) charge cloud diagrams for 1s,2s and 3s-orbitals.

Boundary surface diagrams for the three 2p-orbitals (l = 1) are shown in fig. 1.18(a). In these diagrams, the nucleus is at the origin. Each  p-orbital consists of two lobes on either side of the plane that passes through the nucleus. The size, shape and energy of the three orbitals are identical. They differ, however, in the way the lobes are oriented. Since the lobes may be considered to lie along the  x , y or  z axis, they are given the designations 2 px ,2 py and 2 pz . Like s orbitals,  p-orbitals increase in size with increase in the principal quantum number and hence 4 p >3 p >2 p. Figure 1.18(b) shows the charge cloud diagram for 2 py orbital.

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Figure 1.18: Plots for 2p-orbitals (a) boundary surface diagrams for 2 px ,2 py and 2 pz orbitals (c) charge cloud diagram for 2 py orbital. Boundary surface diagrams for the three 3 p-orbitals (l = 1) are shown in fig. 1.19(a). There are three 3 p-orbitals, as there were three 2 p-orbitals. The shapes of the 3 porbitals are similar to those of the 2 p-orbitals, except that the 3 p-orbitals have a spherical node that cuts each lobe into two distinct sections. Figure 1.19(b) shows the charge cloud diagram for 3 pz orbital.

Figure: 1.19: Total probability density plots for 3 p orbitals (a) boundary surface diagrams for 3 px ,3 py and 3 pz orbitals (c) charge cloud Diagram for 3 pz orbital. Boundary surface diagrams of the d-orbitals (l =2) are shown in fig. 1.20(a). There are five d orbitals which are designated as dxy, dyz, dxz, d x2 -y2 and dz 2. The shape of 3dz 2 orbital is different from that of others but all five 3 d orbital are equivalent in energy. The d orbitals for which n is greater than 3 (4d ,5d ……) have similar shapes. Figure 1.20(b) shows the charge cloud diagram for 3 dxz ,3d x2-y2 and 3dz2 orbitals.

Figure 1.20: Plots for five d -orbitals (a) boundary surface diagrams for 3 dxz ,3d yz , 3d xy 3d x2-y2 and 3d z2 orbitals (b) charge cloud diagram for 3d xz , 3d x2-y2 and 3d z2 orbitals. An important feature to be noted here is that all angular nodes except for dz 2 orbital are represented as two-dimensional planes or surfaces while all radial nodes are represented as spheres. The general formula for calculating the number of nodes is: Total number of nodes (radial + angular) in an orbital = ( n – 1) Number of radial nodes = (n – l – 1) Number of angular nodes = l For instance, in a 4 p-orbital, there are in all three nodes out of which two are radial nodes and one is an angular node. Institute of Lifelong, University of Delhi

Atomic Structure 1.7.8 Sign of Wave Function

The sign of the wave function can be obtained from its angular part. The wave function, ψ , can be either positive or negative depending on the values of θ and Φ. The s-orbital being independent of angle θ and Φ is positive all through. The  p-orbitals, on the other hand, have one positive and one negative lobe as shown in the fig. 1.18(a). Angular dependence of d -orbitals gives opposite lobes with identical signs as shown in fig. 1.20(a). In d  z 2 orbitals, two opposite lobes have the positive sign and the ring in the  xy -plane caries the negative sign. These positive and negative sign does not indicate the electron density in any way; they only represents the sign of the wave function. The true significance of the sign is reflected only when orbitals combine to form bonds.

1.8 Relative Energies of Orbitals

For hydrogen like systems (i.e., systems with only one electron), the energies of orbitals depend only on the quantum number, n and therefore as n increases the energy of the orbital increases. Thus the relative energies of the orbitals of hydrogen like system have the following order:

On the other hand, for multi-electron systems (i.e. atoms/molecules with more than one electron), the relative energies of orbitals not only depends on the principal quantum number, n, but also on the azimuthal quantum number, l . The following factors are responsible for the relative energies of orbitals for multi-electron systems: (i) Effective nuclear charge (depends on ‘ n’ – distance from the nucleus) (ii) Penetration effect (depends on ‘ l ’ – shape of the orbitals)

Effective nuclear charge may be defined as the actual charge felt by the electron. The electron present in the innermost orbital (i.e., 1 s) experiences the full charge of the nucleus. However, the charge experienced by the electron in the outer orbitals is influenced by the repulsions from the inner core electrons. As a result of these repulsions, the net charge experienced by an electron is always less than the actual nuclear charge. The reduced effect of the nuclear charge by the inner electrons is called shielding. Consider for example the lithium atom, which has the ground state electronic 2 1 configuration as 1s 2s . Lithium has three protons, which gives the nucleus a charge of +3, but the full attractive force of this charge on the 2 s electron is partially offset by 2 electron – electron repulsion from the inner 1 s electrons. Consequently, the 1 s electrons shield the outer 2 s electron from the nucleus. The effective nuclear charge (Zeff), experienced by an electron can be expressed as:  ) where Z is the actual nuclear charge (atomic number of the element) and σ = (Z -σ  (sigma) is called shielding constant. The shielding constant is the sum of shielding contributions from all the inner electrons and the electrons in the same shell. About 520 kJ of energy is required to remove the first electron from 1 mol of Li atoms and 7298 kJ + of energy to remove the second electron from 1 mol of Li ions. Thus, much more amount of energy is required to remove second electron as the electron feels the greater effective nuclear charge. Effective nuclear charge depends on distance of the electron from the nucleus. The greater the distance, the lesser is Z eff. Z eff

1.8.2 Penetration Effect

As already mentioned, the radial probability distribution curves show that the 2 s-orbital also has a maximum at or around the maximum for 1 s orbital. Thus the electrons in the 2s-orbital penetrates the lower 1s orbital as shown in fig. 1.21(a). This means that although 2s electron is shielded by the 1 s core yet the 2 s electron penetrates it to some extent and experiences somewhat higher nuclear charge. This penetration exposes the electrons to more influence of the nucleus and causes them to be more tightly bound and Institute of Lifelong, University of Delhi

Atomic Structure thus lowering their associated energy states. For example, in a sodium atom, 3 s electron penetrates the inner core more than the 3 p and is significantly lower in energy. This is shown in fig. 1.21(b).

Figure 1.21: (a) Penetration of a 2s-orbital to the inner 1s core (b) Penetration of 3 s and 3 p-orbitals to the inner 1 s core in a sodium atom. Therefore, the order of energies depends on the degree of penetration, lesser is the energy associated with that orbital. As discussed earlier, the degree of penetration decreases in the order : s

>

 p

>

d

>

f

Hence, the order of energy is 2 s