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- 1. Unit II - Structure of Atom I. II. Atomic Theory A. Modern Atomic Theory 1. All matter is made up of very tiny particles called atoms 2. Atoms of the same element are chemically alike 3. Individual atoms of an element may not all have the same mass. However, the atoms of an element have a definite average mass that is characteristic of the element 4. Atoms of different elements have different average masses 5. Atoms are not subdivided, created, or destroyed in chemical reactions. Sizes of Atoms A. Atomic radius 1. 40 to 270 Pico meters (pm) 1pm = 10-12m 2. Most of the atomic radius is due to the electron cloud B. Nuclear radius 1. 0.001 pm 2. density is 2x108 metric tons/cm3 1metric ton = 1000kg “An insight into the structure of atom was obtained from the experiments on electrical discharge through gases i.e., through discharge tube experiments.” Structure of the Nuclear Atom I. The Electron Discovery of Electron In 1830, Michael Faraday showed that if electricity is passed through a solution of an electrolyte, chemical reactions occurred at the electrodes, which resulted in the liberation and deposition of matter at the electrodes. He formulated certain laws. These results suggested the particulate nature of electricity. As we know “Like charges repel each other and unlike charges attract each other”. In mid 1850s many scientists mainly Faraday began to study electrical discharge in partially evacuated tubes, known as cathode ray discharge tubes. It is depicted Fig. A cathode ray tube is made of glass containing two thin pieces of metal, called electrodes, sealed in it. The electrical discharge through the gases could be observed only at very low pressures and at very high voltages. The pressure of different gases could be adjusted by evacuation. When sufficiently high voltage is applied across the electrodes, current starts flowing through a stream of particles moving in the tube from the negative electrode (cathode) to the positive electrode (anode). These were called cathode rays or cathode ray particles. The flow of current from cathode to anode was further checked by making a hole in the anode and coating the tube behind anode with phosphorescent material zinc sulphide. When these rays, after passing through anode, strike the zinc sulphide coating, a bright spot on the coating is developed (same thing happens in a television set) Rutherford alpha ray scattering experiment Important: 1. Almost all the particles pass through the metal and are un -deflected. 2. Some of these particles are deflected at small angles. 3. Very few of these particles are deflected through as much as 90o or even larger angles.
- 2. 1. Joseph John Thomson (1897) The characteristic properties of cathode rays: (i) The cathode rays start from cathode and move towards the anode. (ii) These rays themselves are not visible but their behaviour can be observed with the help of certain kind of materials. (iii) In the absence of electrical or magnetic field, these rays travel in straight lines. (As they cast shadow if we place an opaque object in their path.) (iv) In the presence of electrical field, they move towards +ve electrode, suggesting that the cathode rays consist of negatively charged particles, called electrons. (v) The characteristics of cathode rays (electrons) do not depend upon the material of electrodes and the nature of the gas present in the cathode ray tube. Thus, we can conclude that electrons are basic constituent of all the atoms. Fig: Cathode ray discharge tube experiment Cathode ray tubes are in most televisions and computer monitors. THOMSON'S EXPERIMENTS: Cathode ray tubes with a fluorescent screen at one end would glow. Thomson measured the deflection of the beam - Magnet deflected in one direction and plate attracted the beam - therefore, negative particles (electrons). Calculate the ratio: Charge vs. Mass Fig: Cathode ray discharge tube experiment in presence of magnetic field Regardless of the type of gas or metal used in cathode-ray tube - the ratio of charge to mass remained the same. Therefore, the particles in the cathode-ray tube were identical to one another. Electron, symbol is e- with a negative one charge. PROTONS Thomson also observed in cathode-ray tube, when he used Hydrogen gas and high voltage with low pressure - he noticed that two beams (one negative) and another beam moving in the opposite direction - toward the cathode - a positive beam. Thomson found that the deflection of the beam varied with different gases. Hydrogen ions had the greatest deflection - therefore the smallest mass. A Proton is a positively charged particle found in all atoms and each proton possess a plus one charge. DISCOVERY OF PROTON: POSITIVE RAYS OR CANAL RAYS Atoms are electrically neutral. Hence after the discovery of the negatively charged constituent (electron) of an atom, attempts were made to discover the positively charged counterpart of electrons. By using a discharge tube containing Fig: Cathode ray discharge tube experiment in presence of electric field
- 3. a perforated cathode. Goldstein (1886) found that some rays passed through these holes in a direction opposite to that of the cathode rays. These are called the positive rays or canal rays. J.J. Thomson (1910) measured their charge by mass ratio from which he was able to deduce that these contain positive ions. Their properties are: They are positively charged. The positive charge is either equal to or whole number multiple of the charge on an electron. When hydrogen gas was filled in the discharge tube the positive charge on the positive rays was equal to the negative charge on an electron, and the mass was less than the hydrogen atom. Fig: Cathode ray discharge tube experiment with perforated cathode Unlike cathode rays the properties of positive rays are characteristics of the gas in the tube. The deflection of positive rays under the influence of an electric or magnetic field is smaller than that of the cathode rays for the same strength of field. This shows that the positive rays have a greater mass than that of electrons. The mass of the positive rays depends on the atomic weights or molecular weights of the gases in the discharge tube. The charge/mass ratio also varies because the change in positive charge on the rays. It may be either equal to or integral multiple of the charge on an electron. The lightest of all particles identified in positive rays from different elements was one with a mass very slightly less than that of hydrogen atom (or nearly equal to H-atom). The lightest positively charged particle is called a proton (P or P+). Positive rays are atomic or molecular resides from which some electrons have been removed. The removed electrons constitute the cathode rays and the positive residues form the positive or canal rays. In 1932 Discovery of Neutrons by James Chadwick (1891 - 1974) In 1932, James Chadwick bombarded beryllium (Be) with alpha particles. He allowed the radiation emitted by beryllium to incident on a paraffin wax. It was found that protons were shot out form the paraffin wax. People began to look for what was in the "beryllium radiations". The radiation consisted of neutral particles of mass approximately equal to that of proton. This neutral particle was named neutron. J.J.Thomson
- 4. Fundamental particles: Name of the fundament al particle Electron Proton Neutron Symbo l Absolut e charge in C -1.6022*10-19 +1.6022*10-19 0 Relativ e Mass in kg Charg e -1 +1 0 M as s in a m u Appr ox mass 9.10939*10-31 1.67262*10-27 James Chadwick 1.67493*10-27 J.J.Thomson’s plum pudding model: Thomson suggested a model of the atom called the Plum Pudding model. ● Its name is supposed to make you think of a lump of stuff with little pieces floating inside it. ● this model of the atom is that small negatively charged electrons are floating around inside a lump of positively charged material. Drawbacks of this model 1. It could not explain the stability of atom. 2. It could not give the theoretical explanations of a no. of experiments. Rutherford Nuclear Model: On the basis of the scattering experiment, Rutherford described the structure of the atom as follows: 1. An atom consists of a positively charged nucleus and is surrounded by electrons that revolve around it. The nucleus' positive charge is due to the presence of protons. 2. Electrons and neutrons are held together by columbic force of attraction. 3. The nucleus' effective volume is found to be extremely small as compared to the effective volume of the atom. The nucleus occupied a volume of about 10-12 times the volume of the atom. 4. The atom's entire mass is concentrated only at the nucleus. 5. As each atom is found to be electrically neutral, the number of protons in the nucleus of an atom is exactly equal to that of the electrons in it. J.J.Thomson’s plum pudding model
- 5. Rutherford's model suffered some drawbacks, as 1. It could not explain the stability of an atom in-spite of the revolving electrons around the nucleus. 2. The electrons revolving around should emit radiation and subsequently lose energy. The loss of energy should slow down the electrons, which should gradually move towards the nucleus in a spiral path and then fall into the nucleus. This should result in the collapse of the atom and make it unstable, which is found to be untrue. Millikan's Oil Drop Experiment: In this experiment a charged oil drop was suspended vertically between the two electrodes. Then the electric field is applied to such an extent that the free fall of the oil drop, under the influence of gravitational pull, was stopped. With the known electric field and other parameters Millikan was succeeded to calculate the charge on the drop. He published his findings in 1913 and was awarded Nobel Prize for physics in 1923 for his work on finding the charge of the electron. With the known electric field applied and other parameters it is easy to calculate the charge on the oil drop. The Millikan using this apparatus was able to find the electron charge as precious as 1.5924 × 10−19 C, which is within 1% accuracy range of the modern corrected value of the electron (1.6 x 10-19 C). Postulates of Niels Bohr Atomic Theory The important postulates in his theory are, 1. 2. 3. 4. 5. 6. 7. 8. Electrons revolve round the nucleus with definite velocities in concentric circular orbits situated at definite distances from the nucleus. The energy of an electron in a certain orbit remains constant. As long as it remains in that orbit, it neither emits nor absorbs energy. These are termed stationary states or main energy states. Bohr proposed that the angular momentum of an electron is quantized. Thus, the motion of an electron is restricted to those orbits where its angular momentum is an integral multiple of h/2π where h is Planck’s constant. Thus we have the relationship mvr = nh/2π where m is mass of electron, v is velocity of electron of said orbit, r is radius of that orbit, n is a simple integer. The stationary states or allowed energy levels are only those where n = 1, 2, 3 … This is called Bohr quantum condition. The energy of an electron changes only when it moves from one orbit to another. An electronic transition from an inner orbit to outer orbit involves absorption of energy. Similarly, when an electron jumps from an outer orbit to inner orbit it releases energy, which is equal to the difference between the two energy levels. The energy thus released in the form of a radiation of a certain frequency appears in the form a line in the atomic spectrum. If the energy of an electron in the outer orbit (n2) is E2 and energy of electron in the inner orbit (n1) is E1 then E2 - E1 = ΔE = hν. The value of n could be small integers 1, 2, 3 and these correspond to the first, second, third, and so on. Quantum states are shells for the electron; n is termed as principal quantum number. Based on the Bohr Theory, he calculated the radii of the various orbits and the energies associated with the electrons present in those shells. Millikan’s oil drop experiment apparatus
- 6. Bohr Model of Hydrogen Bohr used the concept of quantization of energy and laws of classical physics. He gave many mathematical expressions like radius of orbits, velocity and energy of electrons. These expressions satisfactorily explain the spectrum of hydrogen. Let’s calculate all these expressions to explain the spectrum of hydrogen (1H1). 1. Radius of nth orbit: According to bohr model, the attraction force between electron and nucleus is balanced by centrifugal force of electron which is due to motion of electron and tend to take electron away from nucleus. Let’s take, Atomic number of atom = Z Charge on electron = e Charge on nucleus = Z e Radius of nth orbit = r n The electrostatic force of attraction = Z e 2 /r n 2 -------------- (1) The centripetal force = mV 2 / r ----------------(2) Since both forces balanced each other. Hence from equation 1 and 2 Ze 2 /rn 2 = mV 2/ r (or) V2 = Ze 2 /rn m ---------------(3) Form Bohr postulate mVrn = h /2π or m2V2rn2 = h 2/4π 2 Plug the value of V2 from equation (3) m2 * Ze 2 rnm * rn2 = h 2/4π 2 (or) m x Ze2 x rn = n 2 h 2/ 4π 2 (or) rn = n 2 h 2 /mZe 24π 2 Since h = 6.62 x 10-27 erg.sec π = 3.142 m = 9.109 x 10-28gm e = 4.803 x 10-10esu so h 2/me 2 4π 2 = 0.529 Å Plug value of constants in (4) rn = n 2 /Z x 0.529Å --------------(4)
- 7. Since the atomic number for hydrogen is one, so, the radius of nth orbit of hydrogen will be r1= n 2 * 0.529 Å. Since the value of Z is constant for an atom, r n α n 2 , so radius increases with increasing the value of n. If the value of n is constant , rn α 1 /Z Hence, radius of a particular orbit decreases with increasing the atomic number. 2. Calculation of the velocity of electron in Bohr orbit From Bohr postulate Ze 2 /rn 2 = mV 2 /r (Or) V2 = Ze V = Ze 2 /rn m V=Z n 2 2π /nh x 2.188 x 108 cm/sec for hydrogen atom, Z = 1 Since rn = n 2 h 2 /mZe 2 4π 2 Hence V2 = Z 2 e 4 4π 2 /n 2 h 2 (Or) V = Ze 2 2π /nh ------------- (5) Plug all constants (e, r, h, π) values in equation (5) V=Z n x 2.188 x 108 cm/sec for hydrogen atom, Z = 1 Hence V = 2.188 x 108 /n cm.sec-1 3. Energy of electron in nth orbit According to Bohr atomic model, the maximum energy value of electron at infinite is zero because of negligible attraction force between electron and nucleus at infinite distance. Hence, as electron comes closer to nucleus, the energy becomes negative. Hence V = 2.188 x 108 /n cm.sec-1
- 8. Energy of electron is the sum of its potential energy because the electron lies in the field of the positive nucleus and kinetic energy which is due to motion of electron. The potential energy of electron is negative and equals to –Ze 2 /r , while the kinetic energy is positive and equals to 1/ 2 mv 2. Hence the total energy of electron En = –Ze 2/ r + 1 /2 mv 2 Since mv2 = Ze 2 / rn Hence En = −1/ 2 Ze 2 ×mZe 2 4π 2 /n 2 h2 Electromagnetic radiations travel in wide range of wavelengths. Lower the wave length higher will be the energy. En = −Z 2/ n 2 x 13.60 ev/atom En = −Z 2 / n 2 x 2.179 erg/atom En = - 1/2Ze 2 /rn We know that rn = n 2 h2 /mZe 24π 2 So En = 1/ 2 Ze2 * mZe 2 4π 2 /n 2 h 2 = −Z 2 /n 2 * 2π 2 me 2/ h 2 Since 2π 2 me 2 /h 2 is a constant value, which is equals to 13.60 ev/atom. So En = −Z 2/ n 2 x 13.60 ev/atom En = −Z 2 / n 2 x 2.179 erg/atom En = −Z 2/ n 2 x 313.6 KCal/mol En = −Z 2/n 2 x 21.79 x 10-19 J/atom For hydrogen Z = 1 so for So En = - 1 /n 2 x 13.60 eV/atom Atomic Spectra According to Rutherford model of atom the electrons revolve round the nucleus containing protons and neutrons. Bohr improved upon Rutherford theory by suggesting the fixed orbits for the movement of electrons. The subsequent improvements fixed the movement of electrons in limited number in orbits and are called orbitals with different energy potential. The farther the electron is in the orbit from the nucleus and lesser the number in an orbital, the electron can be removed easily. The energy that is required to excite an electron from normal position is called ionization energy. By absorbing energy from outside source the electron gets excited from the designated position in the orbit. The energy it absorbs to reach the excited state is given back when it reaches the normal position in the atom. This emitted energy is recorded on spectrometer. En = −Z 2/ n 2 x 313.6 KCal/mol En = −Z 2/n 2 x 21.79 x 10-19 J/atom For hydrogen Z = 1 so for So En = - 1 /n 2 x 13.60 eV/atom
- 9. The equipment that records the wavelengths is called the spectrometer. In the visible range, it is a well-known fact that the white light is a combination of 7 colours VIBGYOR, and white light is spread over a range of 3,800Å - 7600Å. A spectrum is an assembly of energy levels in the form of radiations emitted by an atom in its excited state. Every atom gives discontinuous line spectra. Each line in the spectra corresponds to a specific wavelength and it is unique to a given element. No two elements give same pattern of lines in their spectra. White light is a combination of 7 colours VIBGYOR. White light is spread over a range of 3,800Å - 7600Å. Spectroscopy Spectroscopy is the branch of physical chemistry which involves the observation and interpretation of radiation emitted and absorbed by atoms and molecules and used to identify their structures. When a monochromatic light passes through the prism, the electromagnetic radiation gets separated in a different number of colors. This image of color is called as spectrum. The spectrum obtained from white light shows seven different colors and this is also called as simple spectrum. Each color in spectrum is associated with a certain wave length. In 1864, Maxwell explained the term electromagnetic radiation. According to Maxwell, theory of radiation, the alternating current of high frequency emitted radiant energy. This radiant energy travels in space with the velocity of light. These radiations are the form of energy emitted and absorbed by charged particles. It can exhibit wave-like behaviour with both electric and magnetic field components which oscillate in phase perpendicular to each other and perpendicular to the direction of energy and wave propagation. The energy of electromagnetic radiation is E = hc /λ Where, h = Plank’s constant c = velocity of light λ = wave length of light Atomic Spectra Definition Each element can produce a unique set of spectral lines which will be quite different from any other atom. Due to this property, any two elements can be identified on the basis of their line spectrum. The spectrum from the radiation emitted by an atom is known as atomic spectrum. According to Bohr’s atomic model, there are certain energy levels in an atom, where electrons revolve around positively charged nucleus. When an atom gets excited, it emits light of certain wavelengths which correspond to different colors. This emitted light can be observed as a series of color lines with dark spaces in between. This image of series of color lines is known as a line or atomic spectra. Each energy level is associated with a certain amount of energy. For the transition of electron form lower energy level to higher energy level it absorbs some amount of energy. The transition of electron from higher level to lower level emits some amount of energy in the form of radiation. The energy of electromagnetic radiation is E = hc /λ Where, h = Planck’s constant c = velocity of light λ = wave length of light
- 10. The energy level is associated with different energy values. Hence each transition involves a different amount of energy and emits a certain amount of energy. Excitation is the process of an electron in an atom absorbing a photon and after absorbing a photon the electron, atom is said to be in an excited state. Since electron in excited state is not stable, it radiates some amount of energy in the form of radiation and back to ground state. A photon emitted or absorbed is equal to the difference E2 - E1 between the energy levels is released or absorbed in the process. The relation between frequency and energy change of the spectral line is called as Bohr's frequency. E2 – E1 =hυ An electron located in the lowest energy level (n=1) is the closest to the nucleus. As well as electron occupying its lowest energy level is known to be in the ground state. The energy of an electron in a certain energy level is given by, En = - RH/n2 Where RH = Rydberg constant n = Energy level of the electron The energy of photon involve in transition of electron in two level is equals to, When an atom gets excited, it emits light of certain wavelengths which correspond to different colors. This emitted light can be observed as a series of color lines with dark spaces in between. This image of series of color lines is known as a line or atomic spectra. The energy of an electron in a certain energy level is given by, En = - RH/n2 Where RH = Rydberg constant n = Energy level of the electron Ephoton = RH (1/ni2 – 1/nf2) Where ni = Initial energy level of the electron nf = final energy level of the electron Since energy is directly proportional to the frequency of the photon emitted, hence νphoton = (Ei - Ef)/h Where Ei = Initial energy of the electron Ef = Final energy of the electron. Atomic spectroscopy involves the interaction of light with gaseous atoms. A device converts a sample into a gaseous atoms and is called as atom cell. There are three basic types of atomic spectroscopy. 1. Atomic emission 2. Atomic absorption 3. Atomic fluorescence The energy of photon involve in transition of electron in two level is equals to, Ephoton = RH (1/ni2 – 1/nf2) Where ni = Initial energy level of the electron nf = final energy level of the electron Atomic Line Spectra Spectrometers or spectroscopes are instruments which record the emitted radiations in their wavelengths. These recordings are done in bands which are a range of wavelengths. More sensitive spectroscopes to record a particular wavelength are used for the sensitive studies like atomic radiations. Such recordings are done as lines on a graph. Each line is a specific wavelength. This is called Atomic Line Spectrum. With a prism we can spread out the light from a bulb to give a continuous spectrum that is a spectrum containing light of all wavelength, like that of a rainbow. The light emitted by a heated gas, however yields different results, rather than a continuous spectrum with all colours or specific wavelength of light. When a light from a hydrogen gas discharge tube is separated into its components by a prism, it gives a spectrum of lines, each line corresponding to the light of a given wavelength. The relation between frequency and energy change of the spectral line is called as Bohr's frequency. E2 – E1 =hυ
- 11. Atomic Emission Spectrum When we provide some amount of energy to a substance, its atoms get excited and transit to higher energy level. This excited state of atoms is not a stable condition, hence atoms emit some photon of energy and drop to the lower energy level. Each photon corresponds to a particular wavelength and energy. If many electrons emit the same wavelength of photons it will form a spike in the spectrum at that particular wavelength and form a banding pattern. For example, in emission spectrum of hydrogen the Hydrogen atoms present inside the lamp are excited by using an electric current. The emitted light then passed through a prism to get spectrum. If there are n energy levels in hydrogen atom, we can predict the frequencies of the spectral lines of Hydrogen using an equation discovered by Johann Balmer. ν = 3.2881 x 1015s-1 (1/22 - 1/n2) Where n must be a number greater than 2. This is because Balmer’s formula only applies to visible light and some longer wavelengths of ultraviolet. Hence by using Rydberg’s equation, we can calculate the wave number of different spectrum lines. In spontaneous emission, the electron “spontaneously", i.e., without any outside influence, transit from a higher energy level to a lower one. While in case of Stimulated emission, an electron is induced for the transition from a higher energy level to a lower one in the presence of electromagnetic radiation at the frequency of the transition. Hence this type of atomic emission is also termed as induced emission. Photo electric effect is an effect caused by the electromagnetic radiation on the surface of metals. If a radiation of sufficient energy is made to fall on the surface of a metal, the surface electrons absorb some energy and get excited. These excited electrons return to their normal state by releasing the same amount of energy that it absorbed while getting excited. Each electron in an atom has a definite minimum energy required to get excited, which is called the threshold energy or work function or ionization energy. The energy of an electromagnetic radiation is measured in its wavelengths. Spectrometer is an instrument that can record these wavelengths. Atomic Absorption Spectrum As the name suggests the atomic absorption spectra are the spectral reading obtained when the electromagnetic radiations are absorbed by the atoms. These spectra are used for the study of certain atomic reaction where the process require some energy to activate an atom. The transition of an electron form lower energy level to higher energy level takes place by the absorption of photons of certain energy. The magnitude of absorbed energy must be equal to the difference between those two energy levels. Î.E = E2-E1 Where; E1= energy of lower energy level E2= Higher energy level
- 12. When a beam of white light is passed through the given sample solution, the sample atoms absorb some of the light and their electrons get excited. Since the electrons only absorb certain frequencies of photon, there are some black bands in continuous spectrum of white light. This type of spectrum is called as atomic absorption spectrum. A range of wavelength radiations of known wavelengths are passed through the medium under examination and the wavelengths that pass through are recorded. The missing wavelength lines give the value of the absorbed radiation for the required excitement. Since the study is with absorption it is called Atomic Absorption Spectra. Types of Atomic Spectrum Atomic absorption and emission spectrum can be of two types, 1. Atomic continuous spectrum 2. Atomic discontinuous spectrum Atomic continuous spectrum If there is no boundary between the spectrum lines in a spectrum, it is called as continuous spectrum. When a white light passes through a prism, it divides into a seven colour spectrum. Each colour corresponds to a certain frequency and wave length. There is no separation line between all these seven colours. Hence it not possible to decide the end line of one colour and starting line of another colour. Such type of spectrum is known as atomic continuous spectrum. Atomic discontinuous spectrum This type of spectrum is divided in certain number of segments. They can be of two types 1. Atomic line spectrum There are many lines associated with a certain value of wave length and wave number. Generally line spectrum are obtained from atoms in gaseous or vapor state. Since each electron can show various transitions between different energy levels, hence there are many lines for one electron system. For example, line spectrum of hydrogen atom. If the line spectrum is absorption spectrum, black lines observed on a bright surface, while emission line spectrum shows bright lines on a dark surface. 2. Atomic band spectrum Such type of atomic spectrum is given by the substance if it is present in a liquid state while in a solution; such spectrums are given off if the substance is present in its molecular state. Hence, they are also called as molecular spectrum. In this type of spectrum a group of bands are observed for certain frequency and wave number. Atomic absorption and emission spectrum can be of two types, 1. Atomic continuous spectrum 2. Atomic discontinuous spectrum
- 13. Atomic Spectra of Hydrogen Atom Hydrogen is the simplest element with its atom having only one electron. Hence, the atomic spectrum of hydrogen has played a significant role in the development of atomic structure. In the emission spectrum of hydrogen, when an electric discharge is passed through hydrogen gas, the molecules of hydrogen break into atoms. The hydrogen atoms get energized and go into an excited state. The excited atoms then return to the ground state by emitting light. Hydrogen atoms emit bluish light. On passing this light through a prism, a discontinuous line spectrum consisting of several sharp lines is obtained. This is the line spectrum of hydrogen. Electromagnetic radiations are emitted when an electric discharge is passed through a discharge tube filled with hydrogen gas at a very low pressure. These emitted radiations produce various lines in the emission spectrum. The spectra obtained by recording various electromagnetic radiations emitted by hydrogen are called the Atomic spectra of hydrogen. Different series of lines are recorded by different scientists and they are named after their discoverers. J.J.Balmer first discovered the spectral series of hydrogen and named it as Balmer series. This series is in visible region of the spectrum and consists of a number of lines named as I 1, I2, I3, etc., at different wavelengths. Balmer showed that if spectral lines are expressed in wave numbers (1/ƛ) then all the visible lines of hydrogen spectrum can be fitted in the formula 1/ƛ cm-1 = R [ 1/22 - 1/n2 ] where n is an integer equal to or greater than 3. Rydberg later gave a more general equation 1/ƛ = ῦ cm-1 = R [ 1/n12 - 1/n22 ] where for Balmer's spectral lines n1 is 2 and n2 = 3, 4,5, etc., to âˆž. R is called Rydberg constant and has a value 197,677 per centi meter. Four sharp coloured lines were observed in the visible region of this spectrum by Balmer, in the ultra violet region by Lyman, in the infrared region by Paschen, Brackett and Pfund. These series of lines are named after these scientists who discovered them. Balmer expressed these lines in terms of inverse of their wavelength ( ) by a mathematical relation, which was later modified by Rydberg. where 'RH' is the Rydberg's constant and 'n1', 'n2' are integers with values equal to or greater than 3 and 'l' is the wavelength. Later on Lymann found hydrogen series starting at UV range and the value of n 1 = 1 and n2 = 2, 3, 4........ And Paschen found another series in Infra-red range with value of n1 = 3 and n2 = 4, 5, 6........ Bracket series are also in Infra-red range and has values of n1 = 4 and n2 = 5, 6, 7...... Pfund series are the next in IR range and it has the values of n1 = 5 and n2 = 6, 7, 8 ...
- 14. The Photoelectric Effect There are three ways in which electrons eject out of a material. They are (i) Thermionic emission (ii) Field emission (iii) Photo electric emission In all the above cases, energy is given to the material but in different forms. If given in the form of heat it is called as Thermionic emission, if in the form of electrical energy, it is field emission and if in the form of light (photons), then it is photoelectric effect. What is photoelectric effect? When light of sufficiently small wavelength is incident on a metal surface, electrons are ejected from the metal. This phenomenon is called as 'photoelectric effect' and the ejected electrons are called as 'photoelectrons'. A systematic study of this effect can be done using the following experiment. Experimental study Two metal plates are sealed in a vacuum tube. If light of reasonably short wavelength is made to fall on the plate (emitter). If a high potential, difference is applied to the plates, making the emitter negative and the collector positive a photocurrent is registered in the ammeter connected in series in the circuit as long as the emitter is irradiated with light. Let us increase the potential difference between the plates continuously. We find that the photocurrent also increases but reaches a saturation. This shows that the number of electrons attracted by the collector is becoming more because of higher potential difference. After a certain potential, the number of electrons reaching the collector remains the same irrespective of the attracting potential. This must be due to the fact that the number of electrons ejected out remains the same and all the ejected electrons are attracted by the collector thereby producing a 'saturation region'. The adjacent graph (V-I) shows the above observation. Now if the potential difference between the plates is made zero still there is a photocurrent, as shown by the intercept on the y-axis. If the potential of the collector is made negative with respect to the cathode. Even then, a photocurrent is registered and if the negative potential is increased, the photocurrent decreases. For some negative potential of the collector, the photocurrent becomes zero and it is called as 'stopping potential'. The above can be explained as follows. When the collector potential is negative, the electrons are repelled by the anode. Some electrons go back to the anode. Some of them with high kinetic energy are still able to reach the anode (collector). As the negative potential increase, less and less electrons reach the
- 15. collector and finally when none can reach the photocurrent becomes zero. The stopping potential is related to the maximum kinetic energy of the ejected electrons. The fastest photoelectron as it reaches the anode has kinetic energy given by [K.Emax - eV0] ..... (1) where K.Emax - energy of the electron when it leaves the emitter. eV0 - increase in the potential energy of the electron as it moves from emitter to collector. If the intensity of the light increases the photocurrent also increases proportionally but the stopping potential remains the same. This is depicted in the graph. At the same time, if we decrease the intensity of light the amount of the photocurrent decreases but for even the weakest signal 'photoelectric phenomenon' takes place. (K.E)max + (f) = hu Einstein's photoelectric equation Where f= work function of the surface If classical theory is true, the lesser the intensity the energy imparted to the electron must be less and hence there should have been a 'threshold intensity' below which there is no photoelectric effect. But this is not true. Instead, it was found that the frequency of the light is the key parameter. When the frequency of light is increased, the stopping potential also increases and there lies a threshold frequency below which there is no photoelectric effect. The correct explanation for photoelectric effect was given by Albert Einstein in 1905. Einstein postulated that a beam of light consisted of small bundles of energy called as photons. The energy 'E' of the photon is proportional to the frequency ''u". Mathematically expressed as E = hu ..... (2) where 'h' is Planck's constant. The value of 'h' is 6.626 x 10-34 Js. When a photon collides with an electron, it may transfer its energy to the electron. This transfer is an "all or none" process, the electron getting all the photon's energy or none at all. The energy received by the electrons helps it to escape from the surface of the metal and to do this the electron loses an amount of energy called as the work function of the surface of the metal (f). Therefore, if the energy received by the electron from photon is hu [equation (2)] and uses an energy (f) (work function of the surface) to escape the surface then the remaining is in the form of kinetic energy. (K.E)max + (f) = hu ..... (3). this is called as Einstein's photoelectric equation. But the stopping potential V0 provides a direct measurement of the maximum kinetic energy with which electrons leave the cathode. (Or) We can write it as eV0 = hu – f Since
- 16. Equation (3) can be written as eV0 = hu – f The graph shows the variation of stopping potential with frequency in accordance to the above equation. The above experimental study can be summarized as follows: Before that, let us recollect some of the parameters and their symbols. V0 - Stopping potential u0 - Threshold potential f - Work function h - Planck's constant ϑ - Frequency of the incident light e - Electronic charge (i) The metal emits electrons when light of sufficiently small wavelength falls on it and the emission is instantaneous. (ii) There exists a threshold frequency g0 for a given metal, below which there is no photoelectric effect. (iii) The photocurrent can be made zero by applying a negative potential to the collector and the minimum negative potential to produce zero photocurrent is called as stopping potential (iv) The stopping potential depends on the frequency of the light falling on the metal and not on the intensity of light. (v) The photocurrent increases with the intensity of the incident light. Millikan made the first accurate measurement of cut-off voltage for sodium metal by using monochromatic light of known frequencies. QUANTUM MECHANICAL MODEL OF ATOM The atomic model which is based on the particle and wave nature of the electron is known as wave or quantum mechanical model of the atom. This was developed by Erwin Schrodinger in 1926. This model describes the electron as a three dimensional wave in the electronic field of positively charged nucleus. Schrodinger derived an equation which describes wave motion of an electron. The differential equation is ∂2ψ /∂x2 + ∂2ψ/∂y2 + ∂2ψ/∂z2 + 8π2m/h2 ( E - v ) ψ = o where x, y, z are certain coordinates of the electron, m = mass of the electron E = total energy of the electron. V = potential energy of the electron; h = Planck’s constant and ψ (psi) = wave function of the electron.
- 17. Significance of ψ: The wave function may be regarded as the amplitude function expressed in terms of coordinates x, y and z. The wave function may have positive or negative values depending upon the value of coordinates. The main aim of Schrodinger equation is to give solution for probability approach. When the equation is solved, it is observed that for some regions of space the value of ψ is negative. But the probability must be always positive and cannot be negative, it is thus, proper to use ψ2 in favour of ψ. Where nth frequency of the wave and ‘h’ is is Planck’s constant Schrodinger’s Equation: Significance of ψ2: ψ2 is a probability factor. It describes the probability of finding an electron within a small space. The space in which there is maximum probability of finding an electron is termed as orbital. The important point of the solution of the wave equation is that it provides a set of numbers called quantum numbers which describe energies of the electron in atoms, information about the shapes and orientations of the most probable distribution of electrons around nucleus. Where x, y, z are certain coordinates of the electron m = mass of the electron E = total energy of the electron. V = potential energy of the electron; h = Planck’s constant and ψ (psi) = wave function of the electron. ∂2ψ /∂x2 + ∂2ψ/∂y2 + 8π2m/h2 ( E - v ) ψ = o ∂2ψ/∂z2 + Nodal Points and Planes: The point where there is zero probability of finding the electron is called nodal point. There are two types of nodes: Radial nodes and angular nodes. The former is concerned with distance from the nucleus while latter is concerned with direction. Nodal point: The point where there is zero probability of finding the electron is called nodal point. No. of radial nodes = n – l – 1 No. of angular nodes = l No. of radial nodes = n – l – 1 No. of angular nodes = l Total number of nodes = n – 1 Nodal planes are the planes of zero probability of finding the electron. The number of such planes is also equal to l. Total number of nodes = n – 1 Nodal plane: DUAL CHARACTER (PARTICLE AND WAVE CHARACTER OF MATTER AND RADIATION) In case of light some phenomenon like diffraction and interference can be explained on the basis of its wave character. However, the certain other phenomenon such as black body radiation and photoelectric effect can be explained only on the basis of its particle nature. Thus, light is said to have a dual character. Such studies on light were made by Einstein in 1905. Louis de Broglie, in 1924 extended the idea of photons to material particles such as electron and he proposed that matter also has a dual character-as wave and as particle. Derivation of de-Broglie Equation: The wavelength of the wave associated with any material particle was calculated by analogy with photon. In case of photon, if it is assumed to have wave character, its energy is given by E = hυ (According to the Planck’s quantum theory) ------------- (I) Nodal planes are the planes of zero probability of finding the electron. The number of such planes is also equal to l.
- 18. If the photon is supposed to have particle character, its energy is given by E = mc2 … (ii) (according to Einstein’s equation) where ‘m’ is the mass of photon, ‘c’ is the velocity of light. λ = h/mv By equating (i) and (ii) Where mv = p, momentum of the particle hv = mc2 But v = c/λ -De- Broglie’s hypothesis h c/λ = mc2 (or) λ = h /mc The above equation is applicable to material particle if the mass and velocity of photon is replaced by the mass and velocity of material particle. Thus for any material particle like electron. λ = h/mv or λ = where mv = p is the momentum of the particle. HEISENBERG’S UNCERTAINTY PRINCIPLE All moving objects that we see around us e.g., a car, a ball thrown in the air etc., move along definite paths. Hence their position and velocity can be measured accurately at any instant of time. Is it possible for subatomic particle also? As a consequence of dual nature of matter, Heisenberg, in 1927 gave a principle about the uncertainties in simultaneous measurement of position and momentum (mass x velocity) of small particles. This Principle States: “It is impossible to measure simultaneously the position and momentum of a small microscopic moving particle with absolute accuracy or certainty” i.e., if an attempt is made to measure any one of these two quantities with higher accuracy, the other becomes less accurate. The product of the uncertainty in position ( x) and the uncertainty in the momentum ( p = m. v where m is the mass of the particle and v is the uncertainty in velocity) is equal to or greater than h/4π where h is the Planck’s constant. Thus, the mathematical expression for the Heisenberg’s uncertainty principle is simply written as x. p > h/4π “It is impossible to measure simultaneously the position and momentum of a small microscopic moving particle with absolute accuracy or certainty” - Heisenberg’s principle uncertainty
- 19. Explanation of Heisenberg’s uncertainty principle Suppose we attempt to measure both the position and momentum of an electron, to pinpoint the position of the electron we have to use light so that the photon of light strikes the electron and the reflected photon is seen in the microscope. As a result of the hitting, the position as well as the velocity of the electron are disturbed. The accuracy with which the position of the particle can be measured depends upon the wavelength of the light used. The uncertainty in position is ±λ. The shorter the wavelength, the greater is the accuracy. But shorter wavelength means higher frequency and hence higher energy. This high energy photon on striking the electron changes its speed as well as direction. But this is not true for macroscopic moving particle. Hence Heisenberg’s uncertainty principle is not applicable to macroscopic particles. QUANTUM NUMBERS An atom contains large number of shells and subshells. These are distinguished from one another on the basis of their size, shape and orientation (direction) in space. The parameters are expressed in terms of different numbers called quantum numbers. Quantum numbers may be defined as a set of four numbers with the help of which we can get complete information about all the electrons in an atom. It tells us the address of the electron i.e., location, energy, the type of orbital occupied and orientation of that orbital. Principal quantum number (n): It tells the main shell in which the electron resides, the approximate distance of the electron from the nucleus and energy of that particular electron. It also tells the maximum number of electrons that a shell can accommodate is 2n2, where n is the principal quantum number. Azimuthal or angular momentum quantum number (l): This represents the number of subshells present in the main shell. These subsidiary orbits within a shell will be denoted as 0, 1, 2, 3, 4, or s, p, d, f… This tells the shape of the subshells. The orbital angular momentum of the electron is given as √l (l +1) h/2π or √l (l+1) h for a particular value of ‘n’ (where h = h/2π). For a given value of n values of possible l vary from 0 to n – 1. The magnetic quantum number (m): An electron due to its angular motion around the nucleus generates an electric field. This electric field is expected to produce a magnetic field. Under the influence of external magnetic field, the electrons of a subshell can orient themselves in certain preferred regions of space around the nucleus called orbitals. The magnetic quantum number determines the number of preferred orientations of the electron present in a subshell. The values allowed depends on the value of l, the angular momentum quantum number, m can assume all integral values between –l to +l including zero. Thus m can be –1, 0, +1 for l = 1.Total values of m associated with a particular value of l is given by 2l+ 1. The spin quantum number (s): Just like earth which not only revolves around the sun but also spins about its own axis, an electron in an atom not only revolves around the nucleus but also spins about its own axis. Since an electron can spin either in clockwise direction or in anticlockwise direction, therefore, for any particular value of magnetic quantum number, spin quantum number can have two values, i.e., +1/2 and –1/2 or these are represented by two arrows pointing in the opposite directions, i.e., ↑ and ↓. When an electron goes to a vacant orbital, it can have a clockwise or anti clockwise spin i.e., +1/2 or –1/2. This quantum number helps to explain the magnetic properties of the substances. Shell K L M N Principal quantum number (n) 1 2 3 4 2 8 18 32 Maximum number electrons of The maximum number of electrons that a shell can accommodate = 2n2 Where n is the principal quantum number. For a given value of n values of possible l vary from 0 to n – 1. Total values of m associated with a particular value of l is given by 2l+ 1 values i.e. from – l to +l including zero. Spin quantum number can have a clockwise or anti clockwise spin i.e., +1/2 or –1/2
- 20. SHAPES AND SIZE OF ORBITALS An orbital is the region of space around the nucleus within which the probability of finding an electron of given energy is maximum (90–95%). The shape of this region (electron cloud) gives the shape of the orbital. It is basically determined by the azimuthal quantum number l, while the orientation of orbital depends on the magnetic quantum number (m). Let us now see the shapes of orbitals in the various subshells. S–orbital: These orbitals are spherical and symmetrical about the nucleus. The probability of finding the electron is maximum near the nucleus and keeps on decreasing as the distance from the nucleus increases. There is vacant space between two successive s–orbitals known as radial node. But there is no radial node for 1s orbital since it is starting from the nucleus. The size of the orbital depends upon the value of principal quantum number (n). Greater the value of n, larger is the size of the orbital. Therefore, 2s–orbital is larger than 1s orbital but both of them are non-directional and spherically symmetrical in shape. s- orbital – spherical shape p- Orbital – dumb bell shape d- orbital – double dumb bell shape f- orbital – complex shape P–orbital (l=1): The probability of finding the p–electron is maximum in two lobes on the opposite sides of the nucleus. This gives rise to a dumb–bell shape for the p– orbital. For p–orbital l = 1. Hence, m = –1, 0, +1. Thus, p–orbital have three different orientations. These are designated as px, py & pz depending upon whether the density of electron is maximum along the x y and z axis respectively. As they are not spherically symmetrical, they have directional character. The two lobes of p–orbitals are separated by a nodal plane, where the probability of finding electron is zero. The three p-orbitals belonging to a particular energy shell have equal energies and are called degenerate orbitals. d–orbital (l = 2): For d–orbitals, l = 2. Hence m = –2,–1, 0, +1, +2. Thus there are 5d orbitals. They have relatively complex geometry. Out of the five orbitals, the three (d xy, dyz, dzx) project in between the axis and the other two dz2 and dz2-y2 lie along the axis. FILLING OF ELECTRONS IN VARIOUS ORBITALS The atom is built up by filling electrons in various orbitals according to the following rules. Aufbau Principle: This principle states that the electrons are added one by one to the various orbitals in order of their increasing energy starting with the orbital of lowest energy. The increasing order of energy of various orbital is 1s,2s,2p,3s,3p,4s,3d,4p,5s,4d,5p,6s,4f,5d,6p,5f,6d,7p…………………… How to remember such a big sequence? To make it simple we are giving you the method to write the increasing order of the orbitals. Starting from the top, the direction of the arrow gives the order of filling of orbitals. Moeller’s Diagram
- 21. Alternatively, the order of increasing energies of the various orbitals can be calculated on the basis of (n + l ) rule. The energy of an orbital depends upon the sum of values of the principal quantum number (n) and the azimuthal quantum number (l). This is called (n +l) rule. According to this rule, “In neutral isolated atom, the lower the value of (n + l) for an orbital, lower is its energy. However, if the two different types of orbitals have the same value of (n +l), the orbitals with lower value of n has lower energy’’. Spins of an electron Pauli’s Exclusion Principle Pauli’s rule: No two electrons in an atom can have identical quantum numbers. This is an example of a general principle which applies not only to electrons but also to other particles of half-integer spin. It does not apply to particles of integer spin. Hund’s Rule When filling sublevels other than s, electrons are placed in individual orbitals before they are paired up. Electronic Configuration of Elements: Definition: A statement describing the populations of electronic energy sublevels of an atom. See the chart of electronic configurations to get the notation for all of the elements. Examples: The electronic configuration of the lithium atom is 1s22s1, which indicates there are two electrons in the 1s sublevel and one electron in the 2s energy sublevel. Anomalous electronic configurations: 1. Half-filled and completely filled degenerate orbitals give greater stability to atoms. 2. Chromium (Z = 24) and copper (Z = 29) have anomalous electronic configuration due to this reason. 3. Electronic configuration of chromium atom is 1s22s22p63s23p63d54s1 or [Ar] 3d54s1 but not 1s22s22p63s23p63d44s2. 4. Electronic configuration of copper atom is 1s22s22p63s23p63d104s1 or [Ar] 4s13d10 but not 1s22s22p63s23p63d94s2. Stability of atoms: • Theory of exchange forces will explain why Cr has (Ar) 3d5 4s1 but not (Ar) 3d44s2. • According to this theory, greater the number of unpaired electrons, greater is the number of possible exchange pairs of electrons and more is the exchange energy released and the atom is more stable. • For Cr → (Ar) 3d5 4s1, the possible number of exchange pairs = 15. • If energy released for each exchange pair is k, the total exchange energy is 15 k. • For Cr → (Ar) 3d4 4s2, the possible number of exchange pairs = 10 and total exchange energy is only 10k. Therefore Cr → (Ar) 3d5 4s1 is more stable than Cr: (Ar) 3d4 4s2 Hund’s Rule:

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