Quantum Mechanics and Atomic Structure 1

Quantum Mechanics and Atomic Structure 1 INTRODUCTION The word atom is derived from the Greek word, atomos, which means uncut or indivisible. It was Dalton (1808) who established that elementary constituents of elements are atom and are indivisible. In 1930s, it seemed that protons, neutrons, and electrons were the smallest objects into which matter could be divided and they were termed elementary particles which means having no smaller constituent parts, or indivisible. Again, later knowledge changed our understanding and over 100 other elementary particles were discovered between 1930 and the present time. Number of models was proposed from time to time to explain the structure of atoms but none of them could explain satisfactorily till the devolvement of quantum theory. QUANTUM THEORY The birth of quantum mechanics replaced classical mechanics which was used for the description of motion on an atomic scale. In 1900, Planck showed that the description of the distribution of energies of electromagnetic radiation in a cavity requires the quantization of energy. This was quickly followed by the application of quantization of atomic and molecular phenomena. Modern chemistry relies on quantum mechanics for the description of most phenomena. The Origins of Quantum Theory During the latter part of the 19th century, the classical mechanics had successfully explained most of the phenomena in which the physicists were interested. But the, problems arose when the classical laws did not explain certain phenomena on the atomic or molecular level. These problems were the lowtemperature heat capacities of solids, the photoelectric effect, the distribution of radiation from a black body, and the spectra of atoms. By 1925, a new mechanics called quantum mechanics had been invented which successfully explained all these phenomena. Particles A particle has the attributes of mass, momentum, and thus possess kinetic energy. A particle has a definite location in space, and thus is discrete and countable. A particle may carry an electric charge. Wave A wave is a periodic variation of some quantity as a function of location. For example, the wave motion of a vibrating guitar string is defined by the displacement of the string from its center as a

2 COMPREHENSIVE PHYSICAL CHEMISTRY function of distance along the string. A sound wave consists of variations in the pressure with location. A wave is characterized by its wavelength l and frequency u, which are related by l = u/u where u is the velocity of propagation of the disturbance in the medium. Wave Theory of Light In 1860s James Clerk Maxwell developed his electromagnetic theory and showed theoretically that when an electric charge is accelerated (by being made to oscillate within a piece of wire, for example), electrical energy will be lost, and an equivalent amount of energy is radiated into space, spreading out as a series of waves extending in all directions. These waves consist of periodic variations in the electrostatic and electromagnetic field strengths. These variations occur at right angles to each other. Each electrostatic component of the wave induces a magnetic component, which then creates a new electrostatic component, so that the wave, once formed, continues to propagate through space, essentially feeding on itself. Black Body Radiation When radiation falls on an object, a part of it is reflected, a part is absorbed and the remaining part is transmitted. This is due to the fact that no object is a perfect absorber. In contrast to this, we may visualize a black body which completely absorbs all radiations that falls on it and retains all the radiant energy that strikes it. For experimental purposes, a black body is generally blackened metallic surface of a hollow sphere and blackened from inside with a hole. All radiations that enter through the hole will be absorbed completely by successive reflections inside the enclosure. Such a material is called a black body and a good example of it is an empty container with a pin-hole maintained at a constant temperature. Thus a black body may be defined as one that absorbs and emits all radiations (frequencies). A black body behaves not only a perfect absorber of radiant energy, but is also an idealized radiator. The radiation emerging from the hole will also be very nearly equal to that of the black body and is called black body radiation. It has been shown that the energy which radiates is dependent on the temperature of the enclosure but independent of the nature of the interior material. A plot of the intensity of black body radiation versus wavelength at different temperatures are shown in Fig. 1.1. Fig. 1.1 Emission of radiation from a blackbody at different temperatures. The area under the curve between two different wavelengths gives the energy in calories emitted by 1 cm 2 surface of a black-body in the specified range of wavelengths The shape of the curves could not be explained on the basis of classical electromagnetic theory in which it was assumed that the body radiates energy continuously. The experimental observations are contrary to the classical view. For each temperature there is a maximum in the curve corresponding to a particular wavelength, indicating the maximum radiation of energy. At higher temperatures, the position of the maximum in the curve shifts towards shorter wavelength and becomes more pronounced. To explain the distribution of black body radiation as function of frequency or wavelength of radiation,

QUANTUM MECHANICS AND ATOMIC STRUCTURE 3 Max Planck in 1900 resolved this discrepancy by postulating the revolutionary assumption that the black body radiated energy not continuously but discontinuously in the form of energy packets called quanta, given by E= hn...(1.1) and the energy (E n ) can only have integral values of a quantum, i.e., E n = n hn (n = 0,1,2...) where E is quantum of energy radiated, n is the frequency of radiation and h is the Planck s constant having a value 6.264 10 27 erg sec or 6.264 10 34 J s, and n is a positive integer. On the basis of this equation, Planck obtained an expression (Eqn 1.2) which correctly gives the distribution of energy in black body radiation: 8πhc 1 de = r dl where r = 5 hc / λkt...(1.2) λ e 1 where e means energy divided by volume. This expression fits the experimental curve at all wavelengths. Equation (1.1) is the fundamental relation of the quantum theory of radiation. Planck s theory of quantized radiation of black body led Einstein to propose a generalization of the quantum theory. Einstein stated that all radiations absorbed or emitted by a body must be in quanta and their magnitude depends on the frequency according to eqn. (1.1) or multiple thereof. Heat Capacities of Solids French scientists Pierre-L. Dulong and Alexis T. Petit determined the heat capacities of a number of monatomic solids and, in 1819, proposed their law that atoms of all simple bodies have exactly the same heat capacity. In other words, the molar heat capacities of all monatomic solids are nearly equal to 3R (25 JK 1 mol 1. This value could be predicted from the relation de C v,m = dt = 3R = 24.9 JK 1... (1.3) V where the molar internal energy is given by E = 3N A kt = 3RT (since N A k = R)... (1.4) where N A is Avagadro s number, k is Boltzmann constant and R is the Gas constant. This law could easily be explained in terms of classical mechanics. But marked deviations from this law was noticed when the heat capacities were measured at low temperatures. The value of molar heat capacities of all metals were found to decrease with decreasing temperature and even becomes zero at T = 0. Therefore, important advancements in the calculation of atomic heat capacities was made by A. Einstein in 1907 on the basis of quantum theory. Quantum Theory Einstein Model According to classical theory the specific heat should be independent of temperature but it markedly depends on temperature. All specific heats increase with temperature. According to Planck, a harmonic oscillator does not have a continuous energy spectrum, as assumed in the classical theory, but can accept only energy values equal to integer times h n, where h is Planck s constant and n is the frequency. The possible energy levels of an oscillator may thus be represented by e n = n hv, n = 0,1,2,3... In their model it is supposed that all the atoms vibrate with the same frequency, but with different amplitudes, i.e., with different amount of vibrational energy.

4 COMPREHENSIVE PHYSICAL CHEMISTRY According to Planck, the average energy of an oscillator ε oscillating in one direction in space, at a temperature T, is given by ε =...(1.5) hv / kt e 1 The vibrational energy of a solid element containing N atoms oscillating in three directions is equal to E = 3Nε = 3N...(1.6) e / ht 1 The specific heat at constant volume is therefore, obtained by differentiating E with respect to T. i.e., C v, m = 2 / kt de e 3R dt = kt ( e 1) / kt 2...(1.7) when kt >> hn i.e., hn /kt is small in comparison with unity, then C v = 3R (Classical result) when T is very low, C v decreases and tends to zero. To discuss this behaviour, it is convenient to express the frequency of oscillation of the atoms in terms of Einstein temperature, q E defined as q E = h ν...(1.8) K Thus, Eqn. (1.7) may be written in the form θe / T θe e C v = 3R T θe / T e 1...(1.9) or C ν 3R θ T = E f E where f E = θe / T θ E e T θe / T e 1 f E is called Einstein function. It determines the ratio of the specific heat at a temperature T and the classical value (high temperature value) 3R. In the high temperature range, when T << q E, the observed and theoretical specific heat values are in good agreement. However, at very low temperatures there is deviation. At very low temperatures, majority of the atoms will have small or zero energy and the contribution to the heat capacity will be / small. At T << q E, eqn. (1.9) suggests that specific heat is proportional to E T i.e., / C e θ E T ν α...(1.10) It means Einstein function will fall more rapidly at low temperature and at T = 0, the value of f E = 0. This failure arises from Einstein s assumption that all the atoms oscillate with the same frequency. It is not correct. In fact they oscillate over a range of frequencies from zero up to a maximum, n m. Debye solved this problem by averaging overall the frequencies. e θ

QUANTUM MECHANICS AND ATOMIC STRUCTURE 5 Debye Model Close, examination of the Einstein s model has shown that at low temperature, the specific heat fall off more rapidly than do the experimental values. It is not correct, therefore, that all the N atoms in 1g. atom of a crystal oscillate at the same frequency and a more reasonable postulate would be that the g. atom involves a coupled system of 3N oscillators i.e. one can write r= 3N r E =...(1.11) r kt r= 1 e / 1 where n r is the frequency for a particular value of r which can vary from 1 to 3N. By the theory of elasticity the number of such vibrations per unit volume (dz) in between the frequency n and n + dn is found to be 2 dν d z = 9Nν...(1.12) ν 3 m where n m represents the maximum of 3N vibration frequencies. 9N νm 2 E = ν dν 2 0 / kt ν...(1.13) m ( e 1) So on differentiating E with respect to temperature, T the Debye heat capacity equation is written as 9R νm 2 C n = ν dv 2 0 / kt ν (where R = kn)...(1.14) ( e 1) m Further, the quantity kt can be replaced by the variable x, therefore one can write 3 kt m/ kt exdx C n = 3R...(1.15) h 0 x 2 νm ( e 1) The quantity hn m /k is called the characteristic temperature and is represented by the symbol q D. Thus m q D =...(1.16) k h e ν / kt when the temperature is large, m is small and therefore, C v = 3R The Debye equation at low temperatures is written as C v = a T 3...(1.17) where a is a constant for a particular substance. Thus it is clear that the principle of quantization must be introduced in order to explain the thermal properties of solids. x 4 The Photoelectric Effect The first important application of the quantum theory of radiation was the explanation of the photoelectric effect given by Einstein in 1905 which put the quantum theory on a sound footing.

6 COMPREHENSIVE PHYSICAL CHEMISTRY When a beam of visible or ultraviolet light falls on a clean metal surface in a vacuum, the surface emits electrons. This effect is known as the photoelectric effect and could not be explained on the basis of classical theory of electromagnetic radiation. The important observations made are: (i) No matter how great the intensity of light is, electrons would not be emitted unless the frequency of light exceeds a certain critical value, u 0, known as the threshold frequency. This is different for different metals. (ii) The kinetic energy of the emitted electron is independent of the intensity of incident light but varies linearly with its frequency. (iii) Increase of intensity of the incident radiation increases the number of electrons emitted per unit time. According to the wave theory, radiant energy is independent of the frequency of radiation; hence it cannot explain the frequency dependence of kinetic energy and the existence of the threshold frequency, u 0. Furthermore, the wave theory predicted that the energy of electrons should increase with the increase of the intensity which is contrary to the experimental fact. Einstein pointed out that the photoelectric effect could be explained by considering that light consisted of discrete particles or photons of energy hu. When a photon of frequency u strikes the metal surface, it knocks out the electrons. In doing so, a certain amount of energy is used up in extricating the electron from the metal. The remaining energy, which will be the difference between the energy hu imparted by the incident photon and the energy used up at the surface W, would be given to the emitted electron as kinetic energy. Hence we have hu W = 1 2 mu2...(1.18) It is apparent from this equation that W accounts for the threshold frequency by the relation W = hu 0. Thus equation (1.18) becomes equal to or hu hu 0 = 1 2 mu2 1 2 mu2 = hu hu 0...(1.19) From the equation (1.19) it is clear that if the energy of the incident photon is less than the energy required by an electron to escape from the surface, no emission can take place regardless of the intensity of the incident light, i.e., the number of photons that strikes the surface per second. Now, if the kinetic energy of the ejected electrons is plotted as a function of frequency, a straight line with slope equal to Planck s constant h and intercept equal to hu 0 is obtained. This is shown in Fig. 1.2. This clearly proves the correctness of Einstein s theory of photoelectric emission and incidentally gives a proof in favour of the quantum theory. Fig. 1.2 Variation of energy with frequency of incident light

QUANTUM MECHANICS AND ATOMIC STRUCTURE 7 COMPTON EFFECT The Compton effect (also called Compton scattering) is the result of a high-energy photon colliding with a target, which releases loosely bound electrons from the outer shell of the atom or molecule. The scattered radiation experiences a wavelength shift that cannot be explained in terms of classical wave theory, thus supports to Einstein s photon theory. The effect was first demonstrated in 1923 by Arthur Holly Compton (for which he received a Nobel Prize in 1927). In Compton scattering, the incoming photon scatters off an electron that is initially at rest. The electron gains energy and the scattered photon have a frequency less than that of the incoming photon (Fig. 1.3). The effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. Fig.1.3 Compton scattering Differences between Classical Laws and Quantum Mechanics Classical mechanics is called Newtonian mechanics and normally assume bodies to be rigid and continuous. Quantum mechanics, however is different and explains the way subatomic particles behave. It provides a mathematical way to describe an atomic system. This new theory also provide a set of rules to determine the behaviour of the quantum system in the same way as Newton s laws determine the behaviour of a classical system. However, this new theory of quantum mechanics is by no means equivalent to Newton s laws. Some major differences between classical and quantum mechanics which are given below: 1. In classical mechanics a particle can have any energy and any speed. In quantum mechanics these quantities are quantized. This means that a particle in a quantum system can only have certain values for its energy, and certain values for its speed (or momentum). These special values of the energy or momentum are called eigenvalues of the quantum system. Associated with each eigenvalue there is a special state called an eigenstate. The eigenvalues and eigenstates of a quantum system are the most important features for characterizing the behaviour of that system s. In contrast, there are no eigenvalues or eigenstates in classical mechanics. 2. Newton s laws allow one, in principle, to determine the exact location and velocity of a particle at some future time. Quantum mechanics, on the other hand, only determines the probability for a particle to be in a particular location with a certain velocity at some future time. The probabilistic nature of quantum mechanics makes it very different from classical mechanics.

8 COMPREHENSIVE PHYSICAL CHEMISTRY 3. Quantum mechanics incorporates the Heisenberg uncertainty principle. This principle states that one cannot know the location and velocity of a quantum particle with infinite accuracy. 4. Quantum mechanics permits superpositions of states. This means that a quantum particle can be in two different states at the same time. For instance, a particle can actually be located in two different places at one time. This phenomenon is not possible at all in classical mechanics. 5. Quantum mechanical systems can exhibit a number of other very interesting features, such as tunneling and entanglement. These features are also not observed in classical mechanics. Electromagnetic Radiation A beam of light has oscillating electric and magnetic fields associated with it. It is characterized by the properties such as frequency, wavelength and wave number. We can understand all these properties by considering a wave propagating in one dimension (Fig. 1.4). Fig. 1.4 A beam of electromagnetic radiation, showing the electric (E) and magnetic (M ) components Electromagnetic theory of light depicts propagation of light through space, as oscillating electric and magnetic fields; these fields are mutually perpendicular and also perpendicular to the direction of propagation of light. Wavelength (l) is the distance between two successive crests or troughs. The length between X and Y in Fig. 1.4 is equal to the wavelength. The frequency (u) is the number of waves passing per second. Its unit is hertz (Hz). In fact, one hertz is equal to second 1 (s 1 ). Wavelength and frequency are related as (Eqn. 1.20) c l =...(1.20) υ where c is the velocity of light and in vacuum (c = 2.998 10 8 ms 1 ). The reciprocal of wavelength is called the wave number ( υ ) and is defined as 1 υ υ = =...(1.21) λ c The SI unit of wave number, υ is m 1 although most of the literature values are in cm 1. The peak height (P) or trough depth (Q) is called the amplitude of the wave. Spectra Dispersion of visible radiation from prisms is called spectra. Characteristic spectra can be obtained from substances by causing them to emit radiation. This can be done by heating a substance or by

QUANTUM MECHANICS AND ATOMIC STRUCTURE 9 subjecting it to electrical stimulation or excitations by using an electric arc or discharge. A variety of emission spectra can be obtained. (a) Continuous spectra: Continuous spectra show the presence of radiation of all wavelength over a wide range. Such spectra are given by incandescent solids, i.e., the filament in an electric light bulb. (b) Band spectra: Band spectra consist of a series of bands of overlapping lines. They are formed by the radiation emitted from excited molecules. (c) Line spectra: These consist of a series of sharply defined lines each corresponding to a definite wavelength. These are obtained when the atoms in a substance are excited so that they can emit radiation. The light from a mercury vapour lamp or from solid sodium chloride heated in a Bunsen flame provides a line spectrum. Because line spectra are caused by energy changes taking place within an atom they may also be called atomic spectra. They give informations about the energy changes taking place within an atom. These informations lead to the elucidation of the arrangement of electrons in the atoms. HYDROGEN SPECTRA Balmer in 1885 examined the radiations obtained from the hydrogen atom in the excited state. He observed number of lines in the spectrum (Fig. 1.5) and proposed an empirical relationship to explain these lines υ = 1 λ = R 1 1 2 2...(1.22) 2 n where υ is the frequency in wave number, n is an integer greater than 2 and R is a constant known as Rydgberg s constant (R = 109677.76 cm 1 for Hydrogen). By choosing a particular value of n, the wavelength of a line in the spectrum can be calculated. Fig. 1.5 Spectrum of hydrogen atom (Balmer series) However, the complete spectrum of hydrogen consist of a few more groups of lines. It was shown by J. J. Rydgberg that the wavelengths of the other series of lines may be expressed by general empirical relationship written as

10 COMPREHENSIVE PHYSICAL CHEMISTRY 1 l = υ = R 1 1 2 2...(1.23) n1 n2 where n 1 and n 2 are integers that may assume values 1, 2, 3,..., with the condition that n 2 is always greater than n 1. The values of n 1 and n 2 for various spectral series of hydrogen are given in Table 1.1. Table 1.1 The atomic spectrum of hydrogen Series n 1 n 2 Region of electromagnetic spectrum Lyman 1 2, 3, 4 Ultraviolet Balmer 2 3, 4, 5 Visible Paschen 3 4, 5, 6 Near infrared Bracket 4 5, 6, 7 Infrared Pfund 5 6, 7, 8 Infrared Humphrey 6 7, 8, 9 Far IR BOHR S ATOMIC MODEL In order to explain the atomic spectra of hydrogen, Niel Bohr in 1913 presented a simple picture of atom by using the concept of quantum theory. He made following assumptions: 1. Electrons revolve around the nucleus in a circular path only in certain allowed energy states called stationary states (also called orbits). 2. The motion of electron is restricted in such a manner that angular momentum (mvr) is quantized and is an integral multiple of h/2p. Thus nh mvr =...(1.24) 2π where m is the mass of the electron, v its tangential velocity, r the radius of circular path and h is Planck s constant. n is an integer having values 1, 2, 3 for the first, second, third etc. of Bohr orbits. 3. When electron jumps from one stationary state to another i.e., during electronic transition, the energy is being emitted or absorbed. This energy change occurs in a fixed amount, the lowest, being one quantum. If E 1 and E 2 are the energies of two states, then one can write E 2 E 1 = hu (E 2 > E 1 ) so that u = E 2 - E 1...(1.25) h where u is the frequency of emitted radiation.