Understanding the Structure of Atoms: Electrons, Quantum Theory, and Atomic Orbitals - Pro, Study notes of Chemistry

Download Understanding the Structure of Atoms: Electrons, Quantum Theory, and Atomic Orbitals - Pro and more Study notes Chemistry in PDF only on Docsity! e- e- e- CHE 115 Structure of the Atom 1 V. Structure of the Atom A) Early Models of the Atom By the begin of the twentieth century experimental evidence indicated that the atom was composed of positive and negative electricity. Most of the mass of the atom is associated with the positive electricity. The negative electricity was found to be composed of negative charged particles. The charge on one particle was determined to be 1.60@10 coulombs or 4.80@10 esu-19 -10 (electrostatic units) and the mass of one particle was found to be 9.1@10 g. These particles were-28 named electrons. 1) Thomson Model, 1904 A spherical sea of positive electricity in which a sufficient number of electrons are embedded to neutralize the positive charge. Thomson suggested that high energy particles (He ) upon striking an atom would pass2+ through the atom rather than be deflected. 2) Geiger-Marsden Experiment, 1911 The 0.1 % of the particles that were deflected through large angles encountered centers of high positive charge. 3) Rutherford (Planetary) Model, 1911 C Most of the atom is empty space. C Positive electricity is concentrated in a very small volume called the nucleus. Mass Number of an Atom ' Atomic Number of the Element % Number of Neutrons Mass Number Atomic Number Ne, Ne, Ne 20 21 22 10 10 10 CHE 115 Structure of the Atom 2 C Electrons sufficient in number to equal the positive charge on the nucleus reside around the nucleus and determine the effective volume of the atom. C The unit of positive charge on the nucleus is called a proton. The number of protons on the nucleus is equal to the atomic number of the element. Element Atomic No. No. of Protons No. of Electrons H 1 1 1 B 5 5 5 Cl 17 17 17 The protons account for only about half the mass of the atom. Most of the remaining mass is associated with the neutrons. A neutron is a neutrally charged particle on the nucleus. The mass number of an atom is the sum of the number of protons and the number of neutrons. Atoms of the same element with different masses are called isotopes. Example: Some Isotopes of Neon The atomic weights (molar masses) listed in the periodic table are the weighted average of the masses of the various isotopes. Example: Atomic Weight of Chlorine Isotope Natural Abundance AW (g/mole)ò Cl 75.4 34.9735 Cl 24.6 36.9737 The natural abundance is the percentage of the natural occurring atoms of an element that areò this isotope. Ephoton ' Et % KEmax (3) Relationships: Ephoton ' hc and Ephoton ' Et % KEmax hc ' Et % KEmax (6.626@10&27erg&sec) (2.998@1010 cm sec ) (4500D)( 10 &8 cm 1D ) ' Et % 3.36@10 &12 erg Et ' 1.05@10 &12erg White Light Spectroscope Hydrogen Lamp Tube containing hydrogen gas (10 torr) 6563 4861 4340 41018 Line Spectrum CHE 115 Structure of the Atom 5 The maximum kinetic energy, KE , of the ejected electron was experimentally found to bemax independent of the intensity (energy/area-time) of the monochromatic light and dependent on the wavelength, , of the monochromatic light. Einstein’s explanation where the threshold energy, E , is the minimum energy required to remove the electront from the surface of the metal. Example: When monochromatic light of 4500 D strikes the surface of metallic sodium, electrons with KE = 3.36@10 erg are ejected. What is the threshold energy for sodium?max -12 Unknown: E for sodiumt Knowns: = 4500 D and KE = 3.36@10 ergmax -12 Concepts: Quantum theory , photoelectric effect C) Line Spectra of the Elements When an electric current is passed through a discharge tube that contains hydrogen gas (10 torr), a white light is emitted. Analysis of the white light reveals that only certain wavelengths of light are emitted. Conclusion: atoms emit only certain, discrete energies. ' c ' (3.289@1015 sec&1)ð 1 (n1) 2 & 1 (n2) 2 ë (4) ' c ' (3.289@1015 sec&1)ð 1 (n1) 2 & 1 (n2) 2 ë 2.998@1010 cm sec ' (3.289@1015 sec&1)ð 1 (2)2 & 1 (6)2 ë ð 10&8cm 1D ë ' 4102D H+ e- Infinity E = 21.79.10-12 erg E1 = ? E = 0 OrbitRadius of Orbit = 0.529 A o oo E = 21.79.10-12 erg = E - E1 = 0 - E1 E1 = - 21.79 .10-12 erg oo CHE 115 Structure of the Atom 6 Spectroscope is an instrument that analyzes white light by separating it into its various wavelengths. Line spectrum consists of a limited number of lines and each line corresponds to a different wavelength of light. A mathematical equation that predicts the wavelengths of the bands in the hydrogen line spectrum was developed by J. J. Balmer. where n and n are dimensionless integers, i.e. n = 1, 2, 3, 4, 5, ã and n > n1 2 2 1 . Example: Calculate the wavelength of light in the hydrogen line spectrum that corresponds to n =1 2 and n = 6.2 D) Bohr Model of the Hydrogen Atom C The electron revolves around the nucleus with a fixed energy and does not emit or absorb energy spontaneously. For the hydrogen atom in its ground state, the energy of the electron is E .1 where E is the energy of the electron at an infinite distance from the nucleus. 4 En ' &21.79@10&12 erg n 2 (5) Radius of Orbit 5 4 3 2 1 n 13.22 8.46 4.76 2.12 0.529 6563 A o 4861 A o En Energy of electon increasing Energy Level Diagram for Hydrogen - 21.79.10-12 erg - 5.45.10-12 erg - 2.42.10-12 erg - 1.36.10-12 erg - 0.87.10-12 erg E (energy emitted or absorbed) ' En2 & En1 (6) where n2 > n1 E (energy emitted or absorbed) ' (&21.79@10&12 erg) ð 1 (n2 ) 2 & 1 (n1 ) 2 ë (7) where n2 > n1 CHE 115 Structure of the Atom 7 C The electron in the hydrogen atom may have other possible energies, namely where n called the Principal quantum number is a dimensionless integer i.e. 1, 2, 3, 4, ã. C Absorption of energy by the hydrogen atom corresponds to the transition of the electron from a lower energy level E with quantum number n to a higher energy level E with quantum n 1 n number n . Emission of energy by the hydrogen atom corresponds to the transition of the 2 electron from a higher energy E with quantum number n to a lower energy level E with n 2 n quantum number n .1 Note: The energy emitted or absorbed by an atom must equal the difference in energies of two energy levels. Substituting eq 5 into eq 6 yields the following equation for the hydrogen atom. Example: What wavelength of light will be emitted when an electron in a hydrogen atom drops ' A sin( x) n ' 2 a sinû n x a ƒ and En ' n 2 h 2 8ma 2 CHE 115 Structure of the Atom 10 equation has the form where A and are constants. The constraint that the particle must be inside the box is used to evaluate the constants and the solutions take the form where n, called the quantum number, is an integer, i.e. n = 1, 2, 3, 4, 5, 6, ã. Observations: C For each (solution to the wave equation) there is a corresponding E .n n C Only certain discrete values of the energy for the particle are possible, namely E = n n h /8ma .2 2 2 C The energy of the particle, E , is independent of the particle’s position, x.n 2 N r X Y Z r, 2, N (Spherical Polar) x, y, z (Cartesian) Shadow made by the r vector on the xy plane 2 N r X Y Z H+ e- V = - Ze2 r Z = charge on nucleus (+1 for H) e = charge on electron (4.8.10-10 esu) CHE 115 Structure of the Atom 11 The number of different types of quantum numbers is a consequence of the number of coordinates. Coordinates Quantum Numbers x n x, y, z n, l, m l x, y, z, time n, l, m , ml s Spherical Polar Coordinates G) Hydrogen Atom (Quantum Mechanical Model) 1) Solutions to the Schrödinger wave equation for the hydrogen atom have the following features. a) The wave functions are characterized by four quantum numbers n, l, m , and m .l s b) Acceptable solutions, , are obtained only when n, l, m , and m have the following l s values. Principal Qunatum Number, n n = 1, 2, 3, 4, 5, 6, ã Azimuthal Quantum Number, l l = 0, 1, 2, 3, ã, n - 1 Magnetic Quantum Number, m m = l, ã, 3, 2, 1, 0, -1, -2, -3, ã, - ll l Spin Quantum Number, m m = ½, - ½s s 2) Solution, , when n = 1, l = 0, m = 0n, l ,m l 1,0,0 ' 1s ' R ' ï2û 1 ao ƒ 3 2 expû &r ao ƒ„ ï 1 2 „ ï 1 2 „ E1s ' &21.79@10 &12 erg where ao'0.529D; R, called the radial part, is the part of that depends on r; , called the angular part, is the part of that depends on . CHE 115 Structure of the Atom 12 Note: the angular part of is a constant.1s If the electron is in state , then it will have energy E irrespective of its position in space,1s 1s i.e. r may be any value from o to 4. We know the energy of the electron exactly but we do not know its position (r, , ) in space. 3) Probability Electron density, ( ) , is proportional to the probability per unit volume of findingn, ,ml 2 the electron at a point r, , in space. Units of ( ) : (volume)n, ,m l 2 -1 Graphical Representations of Probability a) Plot of versus r with and fixed2 CHE 115 Structure of the Atom 15 0.25 # ED # 0.50 + 0.50 # ED # 1.0 $ Contour Plot for Hydrogen 1s Atomic Orbital, 1s 1.73 . 1.60 . 1.47 . 1.33 . 1.20 . ::::: 1.07 . ::::::::::::::: 0.93 . ::::///////////:::: 0.80 . ::://///////////////::: 0.67 . :::///////////////////::: 0.53 . ::////////0000000////////:: 0.40 . ::://////00000000000//////::: 0.27 . :://////000+++++++000//////:: 0.13 . ::://///000+++$$$+++000/////::: 0.00 . ::://///000++$$$$$++000/////::: -0.13 . ::://///000+++$$$+++000/////::: -0.27 . :://////000+++++++000//////:: -0.40 . ::://////00000000000//////::: -0.53 . ::////////0000000////////:: -0.67 . :::///////////////////::: -0.80 . ::://///////////////::: -0.93 . ::::///////////:::: -1.07 . ::::::::::::::: -1.20 . ::::: -1.33 . -1.47 . -1.60 . -1.73 . -1.87 . The printed area represents the region in space in which there is a 99% probability of the finding the electron. e) Solid Representation Boundary surface of the electron density cloud for the hydrogen 1s atomic orbital, 1s The enclosed volume represents the region in which there is a 99% probability of 2s ' ïû 1 ao ƒ 3 2 û2 & r ao ƒ expû &r 2ao ƒ„ ï 1 4 2 „ CHE 115 Structure of the Atom 16 finding the electron. H) Atomic Orbitals The one electron wave function, , which is identified by the three quantum numbers n, l, and n, ,ml m is called an atomic orbital (AO).l A number and letter are used to designate the values of the n and l quantum numbers of an AO. Letter: s p d f g l = : 0 1 2 3 4 Example: A 3p AO is an AO with n = 3 and l = 1, i.e. = .3,1 3p 1) 2s AO for Hydrogen (n = 2, l = 0, m = 0) l Graphical Representations CHE 115 Structure of the Atom 17 Contour Plot for Hydrogen 2s Atomic Orbital 1.33 . 1.20 . ::::::: 1.07 . ::::::::::::::: 0.93 . ::::::::::::::::::: 0.80 . ::::::::::::::::::::::: 0.67 . ::::::::::::::::::::::::: 0.53 . ::::::::::: ::::::::::: 0.40 . ::::::::: ::::::::: 0.27 . :::::::: :///: :::::::: 0.13 . :::::::: //000// :::::::: 0.00 . :::::::: :/0+$+0/: :::::::: -0.13 . :::::::: //000// :::::::: -0.27 . :::::::: :///: :::::::: -0.40 . ::::::::: ::::::::: -0.53 . ::::::::::: ::::::::::: -0.67 . ::::::::::::::::::::::::: -0.80 . ::::::::::::::::::::::: -0.93 . ::::::::::::::::::: -1.07 . ::::::::::::::: -1.20 . ::::::: -1.33 . Boundary surface of the electron density cloud for the hydrogen 2s atomic orbital, 2s 2s 3s CHE 115 Structure of the Atom 20 Summary: The size, shape, and spatial orientation of the electron density (( ) ) cloudn, l, m 2 are related to the quantum numbers n, l, and m that define the atomic orbital ( ).l n, l, m C n is related to size . For the hydrogen atom the size of the electron density cloud increases as n increases. C l is related to the shape. C m is related to the spatial orientation.l e 3 - e 1 - e 2 - Li3+ Z = 3 Zeff = 1.3 Effective Nuclear Charge: Zeff = Z - S Actual Charge on Nucleus Screening Constant: the part of the nuclear charge, Z, "screened" from the "outer" electrons by the "inner" electrons CHE 115 Structure of the Atom 21 5) Polyelectronic Atoms Exact solutions to the Schrödinger wave equation for atoms with two or more electrons are not possible and we must settle for approximate solutions. The independent particle model is often utilized to obtain approximate solutions. In this model each electron in the atom is treated independently of the other electrons. Each electron is treated as if it were the only electron in an atom that has a nucleus with a charge Z . For example considereff the lithium atom To calculate the energy of Electron #3, we solve the Schrödinger wave equation for a hydrogenlike atom (one-electron atom) with a nuclear charge of Z = 1.3. To calculateeff the energy of Electron #2, we solve the Schrödinger wave equation for a hydrogenlike atom with a nuclear charge of Z = 2.7. Thus solutions to the three-electron problem areeff approximated by the solutions to three one-electron problems. Orbital Energy Increasing and Stability Decreasing Relative Orbital Energy Level Diagram 1s (n + l = 1) 2s (n + l = 2) 3s (n + l = 3) 4s (n + l = 4) 2px 2py 2pz (n + l = 3) 3px 3py 3pz (n + l = 4) 4px 4py 4pz (n + l = 5) 3dz2 3dxz 3dyz 3dxy 3dx2-y2 (n + l = 5) CHE 115 Structure of the Atom 22 With this model the electronic structure of a polyelectronic atom may be described in terms of atomic orbitals, , that are hydrogenlike and are defined by the quantum numbers n,n, l, m l, and m . The shape and spatial orientation of the electron density clouds for these AOsl are identical to shape and spatial orientation of the clouds for the hydrogen AOs. However, the size of the cloud will increase as n increases and Z decreases.eff I) n + l Rule and Electron Configuration 1) n + l Rule In the absence of a magnetic field, the energy of an AO for a polyelectronic atom is determined by the quantum numbers n and l. In a neutral, isolated atom the smaller the sum of the values of n and l (n + l), the more stable (lower energy) the AO described by n and l. If two AOs have the same n + l value, the AO with the smaller n value will be the more stable AO. The set of AOs with the same n value is called a shell. Example: 2s, 2p , 2p , and 2px y z The set of AOs with the same n value and the same l value is called a subshell. Example: 2p , 2p , and 2px y z Note: The AOs in a subshell have the same energy and are said to be degenerate. 2) Pauli Exclusion Principle No two electrons in an atom can have the same values for all four quantum numbers. CHE 115 Structure of the Atom 25 When a nitrogen atom in its ground electronic state absorbs monochromatic light of wavelength , it is promoted to an excited electronic state. The ground state is the lowest energy state of an atom, ion, or molecule. An excited state is any energy state other than the ground state. J) Periodic Table The properties of the elements are a periodic function of their atomic numbers. The horizontal divisions of the periodic table are called periods and the vertical divisions are called groups. The least-stable, “occupied” shell in the electron configuration of the atom is called the valence shell. Example: Valence shell is underlined. CHE 115 Structure of the Atom 26 Note: Elements in the same group of the periodic table have the same type of valence shell configuration. K) Size of the Atom 1) Atomic Radii a) van der Waals radius - distance from the center of the atom to a point where the electron density is effectively zero b) Covalent radius - one half the distance between the centers of two bonded atoms Contour Plot for Bonding Molecular Orbital in H2 1.97 . ::::: 1.84 . ::::::::::::::: 1.71 . :::::///////////::::: 1.57 . :::://///////////////:::: 1.44 . ::://///////////////////::: 1.31 . ::////////000000000////////:: 1.17 . ::///////0000000000000///////:: 1.04 . ::///////0000+++++++0000///////:: 0.91 . ::://////000++++$$$++++000//////::: 0.77 . :://////0000++$$$H$$$++0000//////:: 0.64 . :://////000+++$$$$$$$+++000//////:: 0.51 . ::://////000+++$$$$$$$+++000//////::: 0.37 . Covalent ::://///0000+++$$$7$$$+++0000/////::: 0.24 . 0.37 D ::://////000+++$$$ $$$+++000//////::: 0.11 . :://////000+++$$$5$$$+++000//////:: -0.03 . :://////0000++$$$H$$$++0000//////:: -0.16 . ::://////000++++$ $++++000//////::: -0.29 . ::///////0000+++ +++0000///////:: -0.43 . ::///////000000 000000///////:: -0.56 . :://///////000 000/////////:: -0.69 . van der Waals :::////////// //////////::: -0.83 . 1.2 D :::://////// ////////:::: -0.96 . :::::///// /////::::: -1.09 . ::::::: ::::::: -1.23 . ::5:: -1.36 2) Periodic Trends in Atomic Radii a) The atomic radii decrease with an increase in atomic number across a period. Explanation: Z increases while n remains constant across a period.eff b) The atomic radii increase with an increase in atomic number down a group. Explanation: n increases while Z increases only slightly down a group.eff CHE 115 Structure of the Atom 27 L) Ionization Energy The minimum energy required to remove an electron from a gaseous atom or ion in its ground state. Example: He Atom Periodic Trends in Ionization Energies Note: The periodic trends for ionization energies are the opposite of the trends for atomic radii. As Z increases across a period, more energy is required to remove the electron. As n increaseseff while Z increases only slightly down a group, less energy is needed to remove the electron.eff M) Electron Affinity The energy released when a gaseous atom or ion in its ground state adds an electron. Periodic Trends in Electron Affinities