Structure of Atom - Wave Mechanical Approach PDF

2 C H A P T E R Structure of Atom —Wave Mechanical Approach C O N T E N T S WAVE MECHANICAL CONCEPT OF ATOM de BROGLIE’S EQUATION HEISENBERG’S UNCERTAINTY PRINCIPLE SCHRÖDINGER’S WAVE EQUATION CHARGE CLOUD CONCEPT AND ORBITALS Q...

2 C H A P T E R Structure of Atom —Wave Mechanical Approach C O N T E N T S WAVE MECHANICAL CONCEPT OF ATOM de BROGLIE’S EQUATION HEISENBERG’S UNCERTAINTY PRINCIPLE SCHRÖDINGER’S WAVE EQUATION CHARGE CLOUD CONCEPT AND ORBITALS QUANTUM NUMBERS PAULI’S EXCLUSION PRINCIPLE ENERGY DISTRIBUTION AND WAVE MECHANICAL CONCEPT OF ATOM ORBITALS Bohr, undoubtedly, gave the first quantitative successful DISTRIBUTION OF ELECTRONS model of the atom. But now it has been superseded completely IN ORBITALS by the modern Wave Mechanical Theory. The new theory rejects the view that electrons move in closed orbits, as was visualised REPRESENTATION OF by Bohr. The Wave mechanical theory gave a major breakthrough ELECTRON CONFIGURATION by suggesting that the electron motion is of a complex nature GROUND-STATE ELECTRON best described by its wave properties and probabilities. CONFIGURATION OF While the classical ‘mechanical theory’ of matter considered ELEMENTS matter to be made of discrete particles (atoms, electrons, protons IONISATION ENERGY etc.), another theory called the ‘Wave theory’ was necessary to interpret the nature of radiations like X-rays and light. According MEASUREMENT OF to the wave theory, radiations as X-rays and light, consisted of IONISATION ENERGIES continuous collection of waves travelling in space. ELECTRON AFFINITY The wave nature of light, however, failed completely to ELECTRONEGATIVITY explain the photoelectric effect i.e. the emission of electron from metal surfaces by the action of light. In their attempt to find a plausible explanation of radiations from heated bodies as also the photoelectric effect, Planck and Einstein (1905) proposed that energy radiations, including those of heat and light, are emitted 43  44 2 PHYSICAL CHEMISTRY discontinuously as little ‘bursts’, quanta, or photons. This view is directly opposed to the wave theory of light and it gives particle-like properties to waves. According to it, light exhibits both a wave and a particle nature, under suitable conditions. This theory which applies to all radiations, is often referred to as the ‘Wave Mechanical Theory’. With Planck’s contention of light having wave and particle nature, the distinction between particles and waves became very hazy. In 1924 Louis de Broglie advanced a complimentary hypothesis for material particles. According to it, the dual character–the wave and particle–may not be confined to radiations alone but should be extended to matter as well. In other words, matter also possessed particle as well as wave character. This gave birth to the ‘Wave mechanical theory of matter’. This theory postulates that electrons, protons and even atoms, when in motion, possessed wave properties and could also be associated with other characteristics of waves such as wavelength, wave-amplitude and frequency. The new quantum mechanics, which takes into account the particulate and wave nature of matter, is termed the Wave mechanics. de BROGLIE’S EQUATION de Broglie had arrived at his hypothesis with the help of Planck’s Quantum Theory and Einstein’s Theory of Relativity. He derived a relationship between the magnitude of the wavelength associated with the mass ‘m’ of a moving body and its velocity. According to Planck, the photon energy ‘E’ is given by the equation E = hν...(i) where h is Planck’s constant and v the frequency of radiation. By applying Einstein’s mass-energy relationship, the energy associated with photon of mass ‘m’ is given as E = mc2...(ii) where c is the velocity of radiation Comparing equations (i) and (ii) c ⎛ c⎞ mc2 = hν = h ⎜∵ ν = ⎟ λ ⎝ λ ⎠ h or mc =...(iii) λ h or mass × velocity = wavelength h or momentum (p) = wavelength 1 or momentum ∝ wavelength The equation (iii) is called de Broglie’s equation and may be put in words as : The momentum of a particle in motion is inversely proportional to wavelength, Planck’s constant ‘h’ being the constant of proportionality. The wavelength of waves associated with a moving material particle (matter waves) is called de Broglie’s wavelength. The de Broglie’s equation is true for all particles, but it is only with very small particles, such as electrons, that the wave-like aspect is of any significance. Large particles in motion though possess wavelength, but it is not measurable or observable. Let us, for instance consider de Broglie’s wavelengths associated with two bodies and compare their values. (a) For a large mass Let us consider a stone of mass 100 g moving with a velocity of 1000 cm/sec. The de Broglie’s wavelength λ will be given as follows :  STRUCTURE OF ATOM—WAVE MECHANICAL APPROACH 45 6.6256 × 10−27 ⎛ h ⎞ λ = ⎜λ = ⎟ 100 × 1000 ⎝ momentum ⎠ = 6.6256 × 10– 32 cm This is too small to be measurable by any instrument and hence no significance. (b) For a small mass Let us now consider an electron in a hydrogen atom. It has a mass = 9.1091 × 10– 28 g and moves with a velocity 2.188 × 10– 8 cm/sec. The de Broglie’s wavelength λ is given as 6.6256 × 10−27 λ = 9.1091 × 10−28 × 2.188 × 10−8 = 3.32 × 10– 8 cm This value is quite comparable to the wavelength of X-rays and hence detectable. It is, therefore, reasonable to expect from the above discussion that everything in nature possesses both the properties of particles (or discrete units) and also the properties of waves (or continuity). The properties of large objects are best described by considering the particulate aspect while properties of waves are utilized in describing the essential characteristics of extremely small objects beyond the realm of our perception, such as electrons. THE WAVE NATURE OF ELECTRON de Broglie’s revolutionary suggestion that moving electrons had waves of definite wavelength associated with them, was put to the acid test by Davison and Germer (1927). They demonstrated the physical reality of the wave nature of electrons by showing that a beam of electrons could also be diffracted by crystals just like light or X-rays. They observed that the diffraction patterns thus obtained were just similar to those in case of X-rays. It was possible that electrons by their passage through crystals may produce secondary X-rays, which would show diffraction effects on the screen. Thomson ruled out this possibility, showing that the electron beam as it emerged from the crystals, underwent deflection in the electric field towards the positively charged plate. Davison and Germers Experiment In their actual experiment, Davison and Germer studied the scattering of slow moving electrons by reflection from the surface of nickel crystal. They obtained electrons from a heated filament and passed the stream of electrons through charged plates kept at a potential difference of V esu. Due to the electric field of strength V × e acting on the electron of charge e, the electrons emerge out with a uniform velocity v units. The kinetic energy 12 mv 2 acquired by an electron due to the electric field shall be equal to the electrical force. Thus, 1 2 mv 2 = Ve 2Ve or v = m Multiplying by m on both sides, 2Ve mv = m = 2mVe...(i) m But according to de Broglie’s relationship h mv =...(ii) λ  46 2 PHYSICAL CHEMISTRY Comparing (i) and (ii) h = 2mVe λ h h2 ∴ λ = = 2mVe 2mVe Substituting for h = 6.6256 × 10– 27 erg-sec, m = 9.1091 × 10– 28 g, e = 4.803 × 10– 10 esu, and changing V esu to V volts by using the conversion factor 1 3 × 10−2 , we have λ = ( 6.6256 × 10 ) −27 2 × 0.33 × 10−2 2 × 9.1091 × 10−28 × 4.803 × 10−10 V volts 150 150 = × 10−8 cm = Å...(iii) V volts V volts If a potential difference of 150 volts be applied, the wavelength of electrons emerging out is λ = 1 Å. Similarly if a potential difference of 1500 volts be created, the electrons coming out shall have a wavelength 0.1 Å. It is clear, therefore, that electrons of different wavelengths can be obtained by changing the potential drop. These wavelengths are comparable with those of X-rays and can undergo diffraction. Figure 2.1 Schematic representation of the apparatus used by Davison and Germer. The electrons when they fall upon the nickel crystal, get diffracted. Electrons of a definite wavelength get diffracted along definite directions. The electron detector measures the angle of diffraction (say θ) on the graduated circular scale. According to Bragg’s diffraction equation, the wavelength λ of the diffracted radiation is given by λ = d sin θ, where d is a constant (= 2.15 for Ni crystal) and θ the angle of diffraction. By substituting the experimental value of θ in Bragg’s equation (λ = d sin θ), the wavelength of electrons may be determined. This wavelength would be found to agree with the value of λ, as obtained from equation (iii). Since diffraction is a property exclusively of wave motion, Davison and Germer’s ‘electron diffraction’ experiment established beyond doubt the wave nature of electrons. We have described earlier in this chapter that electrons behave like particles and cause mechanical motion in a paddle  STRUCTURE OF ATOM—WAVE MECHANICAL APPROACH 47 wheel placed in their path in the discharge tube. This proves, therefore, that electrons not only behave like ‘particles’ in motion but also have ‘wave properties’ associated with them. It is not easy at this stage to obtain a pictorial idea of this new conception of the motion of an electron. But the application of de Broglie’s equation to Bohr’s theory produces an important result. The quantum restriction of Bohr’s theory for an electron in motion in the circular orbit is that the angular momentum (mvr) is an integral multiple (n) of h/2π. That is, h mvr = n...Bohr Theory 2π On rearranging, we get h 2π r = n mv h Putting the value of from equation (i), we have mv ⎛ h ⎞ 2π r = nλ ⎜∵ λ = ⎟ ⎝ mv ⎠ Figure 2.2 de Broglie's wave accommodated in Bohr's orbits. For these two wave trains the value of n is different. Now the electron wave of wavelength λ can be accommodated in Bohr’s orbit only if the circumference of the orbit, 2πr, is an integral multiple of its wavelength. Thus de Broglie’s idea of standing electron waves stands vindicated. However, if the circumference is bigger, or smaller than nλ, the wave train will go out of phase and the destructive interference of waves causes radiation of energy. SOLVED PROBLEM. Calculate the wavelength of an electron having kinetic energy equal to 4.55 × 10– 25 J. (h = 6.6 × 10– 34 kg m2 sec– 1 and mass of electron = 9.1 × 10– 31 kg). SOLUTION 1 Kinetic energy of an electron = mv 2 2 = 4.55 × 10– 25 J (given) = 4.55 × 10– 25 kg m2 sec– 2 2 × 4.55 × 10−25 or ν2 = m 2 × 4.55 × 10−25 kg m 2 sec −2 = 9.1 × 10−31 kg or ν2 = 1 × 106 m2 sec– 2  48 2 PHYSICAL CHEMISTRY or ν = 1 × 103 m sec– 1 h We know λ = m × ν (de Broglie equation) 6.6 × 10−34 kg m 2 sec−1 ( ) ( = 9.1 × 10−31 kg × 1 × 103 m sec−1 ) = 7.25 × 10– 7 m = 7.25 × 10– 7 × 109 nm = 725 nm SOLVED PROBLEM. Calculate the wavelength of an α particle having mass 6.6 × 10– 27 kg moving with a speed of 105 cm sec– 1 (h = 6.6 × 10– 34 kg m2 sec– 1) h SOLUTION. We know λ= (de Broglie equation) mv Given h = 6.6 × 10– 34 kg m2 sec– 1 m = 6.6 × 10– 27 kg ν= 1 × 105 cm sec– 1 = 1 × 103 m sec– 1 On substitution, we get 6.6 × 10−34 kg m 2 sec −1 λ = 6.6 × 10−27 kg × 103 m sec−1 = 1 × 10– 10 m HEISENBERG’S UNCERTAINTY PRINCIPLE One of the most important consequences of the dual nature of matter is the uncertainty principle developed by Werner Heisenberg in 1927. This principle is an important feature of wave mechanics and discusses the relationship between a pair of conjugate properties (those properties that are independent) of a substance. According to the uncertainty principle, it is impossible to know simultaneously both the conjugate properties accurately. For example, the position and momentum of a moving particle are interdependent and thus conjugate properties also. Both the position and the momentum of the particle at any instant cannot be determined with absolute exactness or certainty. If the momentum (or velocity) be measured very accurately, a measurement of the position of the particle correspondingly becomes less precise. On the other hand if position is determined with accuracy or precision, the momentum becomes less accurately known or uncertain. Thus certainty of determination of one property introduces uncertainty of determination of the other. The uncertainty in measurement of position, Δx, and the uncertainty of determination of momentum, Δp (or Δmv), are related by Heisenberg’s relationship as h Δx×Δp ≥ 2π h or Δ x × m Δν ≥ 2π where h is Planck’s constant. It may be pointed out here that there exists a clear difference between the behaviour of large objects like a stone and small particles such as electrons. The uncertainty product is negligible in case of large objects.  STRUCTURE OF ATOM—WAVE MECHANICAL APPROACH 49 For a moving ball of iron weighing 500 g, the uncertainty expression assumes the form h Δx × m Δν ≥ 2π h or Δx × Δν ≥ 2π m 6.625 × 10−27 ≥ ≈ 5 × 10−31 erg sec g –1 2 × 3.14 × 500 which is very small and thus negligible. Therefore for large objects, the uncertainty of measurements is practically nil. But for an electron of mass m = 9.109 × 10– 28 g, the product of the uncertainty of measurements is quite large as h Δx × Δv ≥ 2π m 6.625 × 10−27 ≥ ≈ 0.3 erg sec g –1 2 × 3.14 × 9.109 × 10−28 This value is large enough in comparison with the size of the electron and is thus in no way negligible. If position is known quite accurately i.e., Δx is very small, the uncertainty regarding velocity Δv becomes immensely large and vice versa. It is therefore very clear that the uncertainty principle is only important in considering measurements of small particles comprising an atomic system. Physical Concept of Uncertainty Principle The physical concept of uncertainty principle becomes illustrated by considering an attempt to measure the position and momentum of an electron moving in Bohr’s orbit. To locate the position of the electron, we should devise an instrument ‘supermicroscope’ to see the electron. A substance is said to be seen only if it could reflect light or any other radiation from its surface. Because the size of the electron is too small, its position at any instant may be determined by a supermicroscope Figure 2.3 employing light of very small wavelength The momentum of the electron changes when a photon of light strikes it, so does its position. (such as X-rays or γ-rays). A photon of such a radiation of small λ, has a great energy and therefore has quite large momentum. As one such photon strikes the electron and is reflected, it instantly changes the momentum of electron. Now the momentum gets changed and becomes more uncertain as the position of the electron is being determined (Fig. 2.3). Thus it is impossible to determine the exact position of an electron moving with a definite velocity (or possessing definite energy). It appears clear that the Bohr’s picture of an electron as moving in an orbit with fixed velocity (or energy) is completely untenable. As it is impossible to know the position and the velocity of any one electron on account of its small size, the best we can do is to speak of the probability or relative chance of finding an electron with a probable velocity. The old classical concept of Bohr has now been discarded in favour of the probability approach.  50 2 PHYSICAL CHEMISTRY SOLVED PROBLEM. Calculate the uncertainty in position of an electron if the uncertainty in velocity is 5.7 × 105 m sec– 1. SOLUTION. According to Heisenberg’s uncertainty principle h Δx × Δp = 4π h or Δx × m Δν = 4π h or Δx = 4π m × Δν Here Δν = 5.7 × 105 m sec– 1 h = 6.6 × 10– 34 kg m2 sec– 1 m = 9.1 × 10– 31 kg On substitution we get 6.6 × 10−34 kg m 2 sec −1 ( Δx = 4 × 3.14 × 9.1 × 10−31 kg ) ( 5.7 × 10 5 m sec−1 ) 6.6 × 10−8 = m 4 × 3.14 × 9.1 × 5.7 = 1 × 10– 10 m SOLVED PROBLEM. The uncertainty in the position and velocity of a particle are 10– 10 m and 5.27 × 10– 24 m sec– 1 respectively. Calculate the mass of the particle. h SOLUTION. We know Δx × Δp = 4π h or Δx × m Δν = 4π h or m = 4π × Δx × Δν Here h = 6.6 × 10– 34 kg m2 sec– 1 Δx = 1 × 10– 10 m Δν = 5.27 × 10– 24 m sec– 1 Substituting the values, we get 6.6 × 10−34 kg m 2 sec−1 ( ) ( 5.27 × 10 ) m = 4 × 3.14 × 1 × 10−10 m −24 m sec−1 = 0.10 kg = 100 g SCHRÖDINGER’S WAVE EQUATION In order to provide sense and meaning to the probability approach, Schrödinger derived an equation known after his name as Schrödinger’s Wave Equation. Calculation of the probability of finding the electron at various points in an atom was the main problem before Schrödinger. His equation is the keynote of wave mechanics and is based upon the idea of the electron as ‘standing wave’ around the nucleus. The equation for the standing wave*, comparable with that of a stretched string is * For the derivation of equation for a ‘standing wave’ in a stretched string, the reader may refer to a book on Physics (Sound).  STRUCTURE OF ATOM—WAVE MECHANICAL APPROACH 51 x ψ = A sin 2π...(a) λ where ψ (pronounced as sigh) is a mathematical function representing the amplitude of wave (called wave function) x, the displacement in a given direction, and λ, the wavelength and A is a constant. By differentiating equation (a) twice with respect to x, we get dψ 2π x = A cos 2π...(1) dx λ λ d 2ψ 4π 2 x and 2 = − A 2 sin 2π...(2) dx λ λ x But A sin 2π =ψ λ d 2ψ 4π2 ∴ = − ψ...(3) dx 2 λ2 The K.E. of the particle of mass m and velocity ν is given by the relation 1 2 1 m2 v 2 K.E. = mv =...(4) 2 2 m According to Broglie’s equation h λ = mv h2 or λ2 = m2 v 2 h2 or m 2 v2 = λ2 Substituting the value of m2 v2, we have 1 h2 K.E. = ×...(5) 2 mλ 2 From equation (3), we have 4π2 ψ − λ2 = d 2ψ...(6) dx 2 Substituting the value of λ in equation (5) 2 1 h2 d 2ψ K.E. = −. 2. 2 2m 4π ψ dx h2 d 2ψ = −. 8π2 mψ dx 2 The total energy E of a particle is the sum of kinetic energy and the potential energy i.e., E = K.E. + P.E. or K.E. = E – P.E. h2 d 2ψ = −. 8π m ψ dx 2 2 d 2ψ 8π 2 m or = − ( E − P.E.) ψ dx 2 h2  52 2 PHYSICAL CHEMISTRY d 2ψ 8π 2 m + ( E − P.E.) ψ = 0 dx 2 h2 This is Schrödinger’s equation in one dimension. It need be generalised for a particle whose motion is described by three space coordinates x, y and z. Thus, d 2ψ d 2ψ d 2ψ 8π 2 m + + ( E − P.E.) ψ = 0 + dx 2 dy 2 dz 2 h2 This equation is called the Schrödinger’s Wave Equation. The first three terms on the left-hand side are represented by Δ2ψ (pronounced as del-square sigh). 8π2 m Δ2 ψ + ( E − P.E.) ψ =0 h2 Δ2 is known as Laplacian Operator. The Schrödinger’s wave equation is a second degree differential equation. It has several solutions. Some of these are imaginary and are not valid. If the potential energy term is known, the total energy E and the corresponding wave function ψ can be evaluated. The wave function is always finite, single valued and continuous. It is zero at infinite distance. Solutions that meet these requirements are only possible if E is given certain characteristic values called Eigen-values. Corresponding to these values of E, we have several characteristic values of wavefunction ψ and are called Eigen-functions. As the eigen-values correspond very nearly to the energy values associated with different Bohr-orbits, the Bohr’s model may be considered as a direct consequence of wave mechanical approach. Significance of ψ and ψ2 In Schrödinger’s wave equation ψ represents the amplitude of the spherical wave. According to the theory of propagation of light and sound waves, the square of the amplitude of the wave is proportional to the intensity of the sound or light. A similar concept, modified to meet the requirement of uncertainty principle, has been developed for the physical interpretation of wave function ψ. This may be stated as the probability of finding an electron in an extremely small volume around a point. It is proportional to the square of the function ψ2 at that point. If wave function ψ is imaginary, ψψ* becomes a real quantity where ψ* is a complex conjugate of ψ. This quantity represents the probability ψ2 as a function of x, y and z coordinates of the system, and it varies from one space region to another. Thus the probability of finding the electron in different regions is different. This is in agreement with the uncertainty principle and gave a death blow to Bohr’s concept. In Schrödinger’s Wave Equation, the symbol ψ represents the amplitude of the spherical wave. For hydrogen atom, Schrödinger’s Wave Equation gives the wave function of the electron (with energy = – 2.18 × 10–11 ergs) situated at a distance ‘r’, ψ = C1e – C2r where C1 and C2 are constants. The square of the amplitude ψ2 is proportional to the density of the wave. The wave of energy or the cloud of negative charge is denser in some parts than in others. Max Born interpreted the wave equations on the basis of probabilities. Even if an electron be considered as a particle in motion around the nucleus, the wave equation may be interpreted in terms of probability or relative chance of finding the electron at any given distance from the nucleus. The space characteristic of an electron is best described in terms of distribution function given by D = 4πr2 ψ2 The numerical value of ‘D’ denotes the probability or chance of finding the electron in a shell of radius r and thickness dr, or of volume 4πr2 dr. Substituting for ψ we have, D = 4πr 2 (C1 e − C2 r )2  STRUCTURE OF ATOM—WAVE MECHANICAL APPROACH 53 The probability of finding the electron is clearly a function of ‘r’. When r = 0 or ∝ , the probability function D becomes equal to zero. In other words, there is no probability of finding the electron at the nucleus or at infinity. However, it is possible to choose a value of r such that there is 90-95 percent chance of finding the electron at this distance. For the hydrogen atom, this distance is equal to 0.53 × 10– 8 cm or 0.53 Å. If the probability distribution be plotted against the distance r from the nucleus, the curve obtained is shown in Fig. 2.4. The probability distribution is maximum at the distance 0.53 Å and spherically symmetrical. This distance corresponds to Bohr’s first radius a0. The graph can be interpreted as representing a contour that encloses a high-percentage of charge. When the electron gets excited and it is raised from n to higher energy levels (say n = 2 or n = 3), the solution of wave equation gives sets of value of ψ2 which give different shapes to the space distribution of the electron. CHARGE CLOUD CONCEPT AND ORBITALS The Charge Cloud Concept finds its birth from wave mechanical theory of the atom. The wave equation for a given electron, on solving gives a three-dimensional arrangement of points where it can possibly lie. There are regions where the chances of finding the electron are relatively greater. Such regions are expressed in terms of ‘cloud of negative charge’. We need not know the specific location of the electrons in space but are concerned with the negative charge density regions. Electrons in atoms are assumed to be vibrating in space, moving haphazardly but at the same time are constrained to lie in regions of highest probability for most of the time. The charge cloud concept simply describes the high probability region. The three-dimensional region within which there is higher probability that an electron having a certain energy will be found, is called an orbital. An orbital is the most probable space in which the electron spends most of its time while in constant motion. In other words, it is the spatial description of the motion of an electron corresponding to a particular energy level. The energy of electron in an atomic orbital is always the same. Electron density o 0.53 A (b) 0 0.5 1.0 1.5 2.0 o Distance from nucleus (A) (a) Figure 2.4 Shows the probability distribution of electron cloud : (a) gives the graphical representation while (b) depicts cross-section of the cloud. Each energy level corresponds to a three-dimensional electron wave which envelopes the nucleus. This wave possesses a definite ‘size’, ‘shape’ and ‘orientation’ and thus can be represented pictorially. QUANTUM NUMBERS Bohr’s electronic energy shells or levels, designated as Principal Quantum Numbers ‘n’, could  54 2 PHYSICAL CHEMISTRY hardly explain the hydrogen spectrum adequately. Spectra of other elements that are quite complex, also remained unexplained by this concept. Many single lines of the spectra are found to consist of a number of closely related lines when studied with the help of sophisticated instruments of high resolving power. Also the spectral lines split up when the source of radiation is placed in a magnetic field (Zeeman Effect) or in an electrical field (Stark Effect). To explain these facts, it is necessary to increase the number of ‘possible orbits’ where an electron can be said to exist within an atom. In other words, it is necessary to allow more possible energy changes within an atom (or a larger number of energy states) to account for the existence of a larger number of such observed spectral lines. Wave mechanics makes a provision for three more states of an electron in addition to the one proposed by Bohr. Like the energy states of Bohr, designated by n = 1, 2, 3..., these states are also identified by numbers and specify the position and energy of the electron. Thus there are in all four such identification numbers called quantum numbers which fully describe an electron in an atom. Each one of these refers to a particular character. Principal Quantum Number ‘n’ This quantum number denotes the principal shell to which the electron belongs. This is also referred to as major energy level. It represents the average size of the electron cloud i.e., the average distance of the electron from the nucleus. This is, therefore, the main factor that determines the values of nucleus-electron attraction, or the energy of the electron. In our earlier discussion, we have found that the energy of the electron and its distance from the nucleus for hydrogen atom are given by 313.3 En = − kcals n2 and rn = 0.529 n2 Å where n is the principal quantum number of the shell. The principal quantum number ‘n’ can have non-zero, positive, integral values n = 1, 2, 3... increasing by integral numbers to infinity. Although the quantum number ‘n’ may theoretically assume any integral value from 1 to ∝ , only values from 1 to 7 have so far been established for the atoms of the known elements in their ground states. In a polyelectron atom or ion, the electron that has a higher principal quantum number is at a higher energy level. An electron with n = 1 has the lowest energy and is bound most firmly to the nucleus. The letters K, L, M, N, O, P and Q are also used to designate the energy levels or shells of electrons with a n value of 1, 2, 3, 4, 5, 6, 7 respectively. There is a limited number of electrons in an atom which can have the same principal quantum number and is given by 2n2, where n is the principal quantum number concerned. Thus, Principal quantum number (n =) 1 2 3 4 Letter designation K L M N Maximum number of electrons (2n = ) 2 2 8 18 32 Azimuthal Quantum number ‘l ’ This is also called secondary or subsidiary quantum number. It defines the spatial distribution of the electron cloud about the nucleus and describes the angular momentum of the electron. In other words, the quantum number l defines the shape of the orbital occupied by the electron and the angular momentum of the electron. It is for this reason that ‘l’ is sometimes referred to as orbital or angular quantum number. For any given value of the principal quantum number n, the azimuthal quantum number l may have all integral values from 0 to n – 1, each of which refers to an Energy sublevel or Sub-shell. The total number of such possible sublevels in each principal level is numerically equal to the principal quantum number of the level under consideration. These sublevels  STRUCTURE OF ATOM—WAVE MECHANICAL APPROACH 55 are also symbolised by letters s, p, d, f etc. For example, for principal quantum number n = 1, the only possible value for l is 0 i.e., there is only one possible subshell i.e. s-subshell (n = 1, l = 0). For n = 2, there are two possible values of l, l = 0 and l = 2 – 1 = 1. This means that there are two subshells in the second energy shell with n = 2. These subshells are designated as 2s and 2p. Similarly, when n = 3, l can have three values i.e. 0, 1 and 2. Thus there are three subshells in third energy shell with designations 3s, 3p and 3d respectively. For n = 4, there are four possible values of azimuthal quantum number l (= 0, 1, 2, and 3) each representing a different sublevel. In other words, the fourth energy level consists of four subshells which are designated as 4s, 4p, 4d and 4f. Thus for different values of principal quantum numbers we have n=1 n=2 n=3 n=4 n=5 l = 0 (1s) l = 0 (2s) l = 0 (3s) l = 0 (4s) l = 0 (5s) l = 1 (2p) l = 1 (3p) 1 = 1(4p) l =1 (5p) l = 1 (3d) l = 2 (4d) l = 2 (5d) l = 3 (4f) l = 3 (5f) l = 5 (5g) For a given value of principal quantum number the order of increasing energy for different subshells is s