where, \[\nu =X-\]rays frequency
Z= atomic number of the metal
\[a\And b\] are constant.
(iii) Atomic number = Number of positive charge on nucleus = Number of protons in nucleus = Number of electrons in nutral atom.
(iv) Two different elements can never have identical atomic number.
(2) Mass number
(i) The sum of proton and neutrons present in the nucleus is called mass number.
Mass number (A) = Number of protons + Number of neutrons or Atomic number (Z)
or Number of neutrons = A – Z .
(ii) Since mass of a proton or a neutron is not a whole number (on atomic weight scale), weight is not necessarily a whole number.
(iii) The atom of an element X having mass number (A) and atomic number (Z) may be represented by a symbol,
Note : q A part of an atom up to penultimate shell is a kernel or atomic core.
q Negative ion is formed by gaining electrons and positive ion by the loss of electrons.
q Number of lost or gained electrons in positive or negative ion =Number of protons \[\pm \]charge on ion.
(3) Different Types of Atomic Species
| Atomic species | Similarities | Differences | Examples | |||||||||||||
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Atomic Models and Planck's Quantum Theory
Atomic spectrum - Hydrogen spectrum.
Atomic spectrum
(1) Spectrum is the impression produced on a photographic film when the radiation (s) of particular wavelength (s) is (are) analysed through a prism or diffraction grating. It is of two types, emission and absorption.
(2) Emission spectrum : A substance gets excited on heating at a very high temperature or by giving energy and radiations are emitted. These radiations when analysed with the help of spectroscope, spectral lines are obtained. A substance may be excited, by heating at a higher temperature, by passing electric current at a very low pressure in a discharge tube filled with gas and passing electric current into metallic filament. Emission spectra is of two types,
(i) Continuous spectrum : When sunlight is passed through a prism, it gets dispersed into continuous bands of different colours. If the light of an incandescent object resolved through prism or spectroscope, it also gives continuous spectrum of colours.
(ii) Line spectrum : If the radiations obtained by the excitation of a substance are analysed with help of a spectroscope a series of thin bright lines of specific colours are obtained. There is dark space in between two consecutive lines. This type of spectrum is called line spectrum or atomic spectrum..
(3) Absorption spectrum : When the white light of an incandescent substance is passed through any substance, this substance absorbs the radiations of certain wavelength from the white light. On analysing the transmitted light we obtain a spectrum in which dark lines of specific wavelengths are observed. These lines constitute the absorption spectrum. The wavelength of the dark lines correspond to the wavelength of light absorbed.
Hydrogen spectrum
(1) Hydrogen spectrum is an example of line emission spectrum or atomic emission spectrum.
(2) When an electric discharge is passed through hydrogen gas at low pressure, a bluish light is emitted.
(3) This light shows discontinuous line spectrum of several isolated sharp lines through prism.
(4) All these lines of H-spectrum have Lyman, Balmer, Paschen, Barckett, Pfund and Humphrey series. These spectral series were named by the name of scientist discovered them.
(5) To evaluate wavelength of various H-lines Ritz introduced the following expression,
\[\bar{\nu }=\frac{1}{\lambda }=\frac{\nu }{c}=R\left[ \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right]\]
Where R is universal constant known as Rydberg’s constant its value is 109, 678\[c{{m}^{-1}}\].
Thomson's model.
(1) Thomson regarded atom to be composed of positively charged protons and negatively charged electrons. The two types of particles are equal in number thereby making atom electrically neutral.
(2) He regarded the atom as a positively charged sphere in which negative electrons are uniformly distributed like the seeds in a water melon.
(3) This model failed to explain the line spectrum of an element and the scattering experiment of Rutherford.
(2) From the above observations he concluded more...
Dual Nature of Electron
(1) In 1924, the french physicist, Louis de Broglie suggested that if light has both particle and wave like nature, the similar duality must be true for matter. Thus an electron, behaves both as a material particle and as a wave.
(2) This presented a new wave mechanical theory of matter. According to this theory, small particles like electrons when in motion possess wave properties.
(3) According to de-broglie, the wavelength associated with a particle of mass \[m,\] moving with velocity \[v\] is given by the relation
\[\lambda \,=\,\frac{h}{mv},\] where h = Planck?s constant.
(4) This can be derived as follows according to Planck?s equation, \[E=\,h\nu =\frac{h.c}{\lambda }\] \[\left( \because \ \ \nu =\frac{c}{\lambda } \right)\]
energy of photon (on the basis of Einstein?s mass energy relationship), \[E=\,mc{}^{2}\]
equating both \[\frac{hc}{\lambda }=\,\,mc{}^{2}\,\,or\,\,\lambda =\frac{h}{mc}\] which is same as de-Broglie relation. \[\left( \because \ \ mc=p \right)\]
(5) This was experimentally verified by Davisson and Germer by observing diffraction effects with an electron beam. Let the electron is accelerated with a potential of V than the Kinetic energy is
\[\frac{1}{2}mv{}^{2}=\,\,eV\]; \[m{}^{2}v{}^{2}=\,\,2eVm\]
\[mv=\sqrt{2eVm}=\,\,P\]; \[\lambda =\frac{h}{\sqrt{2eVm}}\]
(6) If Bohr?s theory is associated with de-Broglie?s equation then wave length of an electron can be determined in bohr?s orbit and relate it with circumference and multiply with a whole number
\[2\pi r=n\lambda \,\,or\,\,\lambda =\frac{2\pi r}{n}\]
From de-Broglie equation, \[\lambda =\frac{h}{mv}\]. Thus \[\frac{h}{mv}=\frac{2\pi r}{n}\] or \[mvr=\frac{nh}{2\pi }\]
 Note : q For a proton, electron and an \[\alpha \]-particle moving with the same velocity have de-broglie wavelength in the following order : Electron > Proton > \[\alpha \]- particle.
(7) The de-Broglie equation is applicable to all material objects but it has significance only in case of microscopic particles. Since, we come across macroscopic objects in our everyday life, de-broglie relationship has no significance in everyday life.
Example: 34 An electron is moving with a kinetic energy of 4.55 × 10\[{}^{-25}\,\] J. What will be de-Broglie wavelength for this electron
(a) 5.28 × \[10{}^{-7}\,m\] (b) 7.28 × \[10{}^{-7}m\] (c) 2 × 10\[{}^{-10}m\] (d) 3 × \[10{}^{-5}m\]
Solution : (b) KE\[=\frac{1}{2}\,mv{}^{2}=\,4.55\,\,\times \,\,10{}^{-25}\] J
\[v{}^{2}=\frac{2\,\times 4.55\times 10{}^{-25}}{9.1\times 10{}^{-31}}=\,1\,\times \,10{}^{6}\]; \[v=10{}^{3}\,m/s\]
De-Broglie wavelength \[\lambda =\frac{h}{mv}=\frac{6.626\,\,\times \,\,10{}^{-34}}{9.1\,\times \,10{}^{-31}\times \,10{}^{3}}=\,\,7.28\,\,\times \,\,10{}^{-7}\,m\,\,\]
Example: 35 The speed of the proton is one more...
Uncertainty Principle and Schrodinger Wave Equation
Heisenberg’s uncertainty principle.
(1) One of the important consequences of the dual nature of an electron is the uncertainty principle, developed by Warner Heisenberg.
(2) According to uncertainty principle “It is impossible to specify at any given moment both the position and momentum (velocity) of an electron”.
Mathematically it is represented as, \[\Delta x\,.\,\Delta p\ge \frac{h}{4\pi }\]
Where \[\Delta x=\]uncertainty is position of the particle, \[\Delta p=\]uncertainty in the momentum of the particle
Now since \[\Delta p=m\,\Delta v\]
So equation becomes, \[\Delta x.\,m\Delta v\ge \frac{h}{4\pi }\] or \[\Delta x\,\times \,\Delta v\ge \frac{h}{4\pi m}\]
The sign \[\ge \] means that the product of \[\Delta x\]and \[\Delta p\](or of \[\Delta x\]and\[\Delta v\]) can be greater than, or equal to but never smaller than \[\frac{h}{4\pi }.\]If \[\Delta x\]is made small, \[\Delta p\]increases and vice versa.
(3) In terms of uncertainty in energy, \[\Delta E\]and uncertainty in time \[\Delta t,\]this principle is written as, \[\Delta E\,.\,\Delta t\ge \frac{h}{4\pi }\]
 Note :q Heisenberg’s uncertainty principle cannot we apply to a stationary electron because its velocity is 0 and position can be measured accurately..
Example: 36 What is the maximum precision with which the momentum of an electron can be known if the uncertainty in the position of electron is \[\pm 0.001{\AA}?\] Will there be any problem in describing the momentum if it has a value of \[\frac{h}{2\pi {{a}_{0}}},\]where \[{{a}_{0}}\]is Bohr’s radius of first orbit, i.e., 0.529Å?
Solution : \[\Delta x\,.\,\Delta p=\frac{h}{4\pi }\]
\[\because \] \[\Delta x=0.001{\AA}={{10}^{-13}}m\]
\[\therefore \] \[\Delta p=\frac{6.625\times {{10}^{-34}}}{4\times 3.14\times {{10}^{-13}}}=5.27\times {{10}^{-22}}\]
Example: 37 Calculate the uncertainty in velocity of an electron if the uncertainty in its position is of the order of a 1Å.
Solution : According to Heisenberg’s uncertainty principle
\[\Delta v\,.\,\Delta x\approx \frac{h}{4\pi m}\]
\[\Delta v\approx \frac{h}{4\pi m.\Delta x}\] \[=\frac{6.625\times {{10}^{-34}}}{4\times \frac{22}{7}\times 9.108\times {{10}^{-31}}\times {{10}^{-10}}}\] \[=5.8\times {{10}^{5}}m\ \,{{\sec }^{-1}}\]
Example: 38 A dust particle having mass equal to \[{{10}^{-11}}g,\]diameter of \[{{10}^{-4}}cm\] and velocity \[{{10}^{-4}}cm\,{{\sec }^{-1}}.\]The error in measurement of velocity is 0.1%. Calculate uncertainty in its positions. Comment on the result.
Solution : \[\Delta v=\frac{0.1\times {{10}^{-4}}}{100}=1\times {{10}^{-7}}cm\,{{\sec }^{-1}}\]
\[\because \] \[\Delta v\,.\,\Delta x=\frac{h}{4\pi m}\]
\[\therefore \] \[\Delta x=\frac{6.625\times {{10}^{-27}}}{4\times 3.14\times {{10}^{-11}}\times 1\times {{10}^{-7}}}=5.27\times {{10}^{-10}}cm\]
The uncertainty in position as compared to particle size.
\[=\frac{\Delta x}{diameter}=\frac{5.27\times {{10}^{-10}}}{{{10}^{-4}}}=5.27\times {{10}^{-6}}cm\]
The factor being small and almost being negligible for microscope particles.
Schrödinger wave equation.
(1) Schrodinger wave equation is given by Erwin Schrödinger in 1926 and based on dual nature of electron.
(2) In it electron is described as a three dimensional wave in the electric field more...
Quantum Number, Electronic Configuration and Shape of Orbitals
Quantum numbers and Shapes of orbitals.
Quantum numbers
(1) Each orbital in an atom is specified by a set of three quantum numbers (n, l, m) and each electron is designated by a set of four quantum numbers (n, l, m and s).
(2) Principle quantum number (n)
(i) It was proposed by Bohr’s and denoted by ‘n’.
(ii) It determines the average distance between electron and nucleus, means it is denoted the size of atom.
\[r=\frac{{{n}^{2}}}{Z}\times 0.529{\AA}\]
(iii) It determine the energy of the electron in an orbit where electron is present.
\[E=-\frac{{{Z}^{2}}}{{{n}^{2}}}\times 313.3\,Kcal\,per\,mole\]
(iv) The maximum number of an electron in an orbit represented by this quantum number as \[2{{n}^{2}}.\] No energy shell in atoms of known elements possess more than 32 electrons.
(v) It gives the information of orbit K, L, M, N------------.
(vi) The value of energy increases with the increasing value of n.
(vii) It represents the major energy shell or orbit to which the electron belongs.
(viii) Angular momentum can also be calculated using principle quantum number
\[mvr=\frac{nh}{2\pi }\]
(3) Azimuthal quantum number (l)
(i) Azimuthal quantum number is also known as angular quantum number. Proposed by Sommerfield and denoted by ‘l’.
(ii) It determines the number of sub shells or sublevels to which the electron belongs.
(iii) It tells about the shape of subshells.
(iv) It also expresses the energies of subshells \[s<p<d<f\] (increasing energy).
(v) The value of \[l=(n-1)\] always where ‘n’ is the number of principle shell.
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