Solutions Manual for Chemical Principles the Quest for Insight 7th Edition Atkins

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Chapter 1 ATOMS: THE QUANTUM WORLD INVESTIGATING ATOMS (Sections 1 . 1-1 .3) 1.1 The Nuclear Atom • Subatomic particles � Electron: mass (m = 9. 1 094 x 1 0- 31 kg) � Proton: mass (m = 1 .6726 x 1 0- 27 kg) � Neutron: mass (m = 1 .6749 x 1 0- 27 kg) • Nucleus charge (-e = -1.602 1 77 x 1 0- 19 C) charge ( e = 1 .602 1 77 x 1 0- 19 C) charge = 0 � Nucleons (protons and neutrons) occupy a small volume at the center of the atom. The binding energy of the nucleus is attributed to a strong force (nuclear) acting over a very short distance. � The radius of the nucleus (assumed to be spherical) is given roughly by rnue = ro A lI3 , where ro:::; 1 .3 x 1 0- 15 m = 1 .3 fm. • Atom � Atomic number: Z = Np = number of protons in the nucleus � Atomic mass number: A = Np + Nn = number of protons and neutrons in the nucleus � Uncharged atom: Np = Ne (number of protons equals the number of electrons) � Electrons occupy a much larger volume and define the "size" of the atom itself. The binding energy of the electrons is attributed to a weak force (coulomb) acting over a much longer distance. 1.2 The Characteristics of Electromagnetic Radiation • Oscillating amplitude of electric and magnetic field � Wave characterized by wavelength and frequency Distance behavior (fixed time t) Oscillating Wave -x direction wavelength A +-----'-r----f-------'l- .. x Time behavior (fixed position x) Oscillating Wave -I time period frequency v = (1/period) Speed of light (distance/time) = wavelength / period = wavelength x frequency I c = AV I SI units: speed of light = wavelength x frequency (m·sI ) (m) (S-I ) [1 Hz (hertz) = 1 S-I ] Note: The speed of light c (co in vacuum:::; 3.00 x 1 08 m·s-I) depends on the medium it travels in. Medium effects on wavelength in the visible region are small (beyond three significant fi gures). SG-l 1.3 Atomic Spectra • Spectral lines � Discharge lamp of hydrogen H2 + electrical energy - H + H * [* == asterisk denotes excited atom] H* - H(* ) + hv [ (* ) == denotes a less excited atom] • Lines form a discrete pattern � Discrete energy levels • Hydrogen atom spectral lines � Johann Rydberg' s general equation � n2 = nupper and n) = n)ower 91 = 3.29 X 1 0)5 Hz = Rydberg constant Rydberg expression reproduces pattern of lines in H atom emission spectrum. The value of 91 is obtained empirical ly. Note: Lines with a common n) can be grouped into a series and some have special names: n) = I (Lyman), 2 (Balmer), 3 (Paschen), 4 (Brackett), 5 (Pfund). QUANTUM THEORY (Sections 1.4-1.7) 1.4 Radiation, Quanta, and Photons • Black body � Perfect absorber and emitter of radiation � Intensity of radiation for a series of temperatures � Stefan-Boltzmann law: Power emitted (watts) = constant x T4 Surface area (meter2 ) � Wavelength corresponding to maximum intensity = "max � Wien's law: I T"-rnax = constant I where constant = 2.88 K-mm At higher temperature, maximum intensity of radiation shifts to lower wavelength. • Energy of a quantum (packet) of light (generally called a photon) � Postulated by Max Planck to explain black body radiation � Resolved the "ultraviolet catastrophe" of classical physics, which predicted intense ultraviolet radiation for all heated objects (T> 0) � Quantization of electromagnetic radiation photon energy = Planck constant x photon frequency SI units: (J) (h = 6.626 1 x 1 0- :14 J.s) SG-2 • Photoelectric effect � Ejection of electrons from a metal su rface exposed to photons of su fficient energy � I ndicates that light behaves as a particle EK = kinetic energy of the ejected electron, = threshold energy (work fu nction) requ ired for electron ejection from the metal su rface, and hv = photon energy � hv � requ ired for electron ejection • Bohr frequency condition I hv = Eupper - Elower I Relates photon energy to energy difference between two energy levels in an atom • Wave behavior of light � Diffraction and interference effects of su perimposed waves (constructive and destructive) 1.5 The Wave-Particle Duality of Matter • Matter has wave properties � Proposed by Lou is de Broglie � Consider matter with mass m and velocity v � Su ch matter behaves as a wave with a characteristic wavelength I A � h I de Brogl;e wavelength fo' a part; cle w; th h ne", momentu m p � mv mv 1.6 The Uncertainty Principle • Complementarity of location (x) and momentum (P) � U ncertainty in x is fu; u ncertainty in p is t:..p � Limitation of knowledge I i3.ptu � h/2 I Heisenberg u ncertainty principle, where Ii = hl2n � I 17 is called "h bar" I Ii = 1.0546 x 10-34 J· s � Refu tes classical physics on the atomic scale 1.7 Wavefunctions and Energy Levels • Classical trajectories � Precisely defined paths • Wavefunction \jJ � Gives probable position of particle with mass m • Born interpretation � Probability of fi nding particle in a region is proportional to \jJ 2 SG-3 • SchrOdinger equation � Allows calcu lation of \1' by solving a differential equ ation � H\If = E\If; H is called the Hamiltonian • Particle in a box � Mass m confi ned b etween two rigid walls a distance L apart � \1' = 0 outside the b ox and at the walls (b oundary condition) ( 2 ) 1/2 . (nnx) \/f17(x) = L Sin L n= l, 2, ... \If n(x) = wavefunction t hat satisfies the Schro dinger equation b etween the b ox l imits. n is a quantum number. Note: A node is a poi nt in the b ox where \If = 0 and \jJ changes sign. I \lf2 � 0, always I En = allowed energy val ues of a particle in a b ox Note: n = I gives the zero-point energy EI • EI *' 0 implies residual motion. t::"E = E - E = (2n+l)h IHI n 8117L2 2 E nergy difference b etween two neighb oring levels • Probability as a function of position in the box � Plot is shown for the first three levels E 11( 2 Particle in a box 't' n -> \lf 2 as a function of the dimensionless variab le xlL o //�� ________________________ '" � ___ ///"'-�"'''' ____________________ '' . n = 2 ..-.-_ .. _ ....... _---.- .. -. .-­ -"'''''- __ "�·:,·:�: __________________________________ �_:·:-;cco,�- n = 1 0.0 0.2 0.4 x/L 06 THE HYDROGEN ATOM (Sections 1.8-1.11) 1.8 The Principal Quantum Number • Charge on an electron � q = -e = -1.602 177 33 x 10-19 C • Vacuum permittivity � Eo = 8.854 187 817 x 10-12 C2. N-I. m-2 • Coulomb potential energy ( ) product of charges on pm1icles V r =--------�----------�--�-------------- 411: x permittivity of free space x distance between charges SG-4 (SI system) 0. 1.0 c 2 (k -2) k ? -2 Units of V(r): ]= 2 I 2 =N · m= g·m·s m= g·m-· s (C ·N - · m- ) m • H atom and one-electron ion energy levels (He+ , Lr, Be3+ , etc.) � Solut ions to the Schro dinger equat ion 2=1 & n=I,2,3, ... 4 9"l = ny 2 = 3.289842 x 1015 Hz 8h EO Units: (n is dimensionless) � All quant um numbers are dimensionless. With 91 in units of frequency, the H atom ener gy-level equat ion has t he same for m as the Planck equat ion, E = hv. � En = ener gy levels (st at es) of t he H at om. N ote t he negative sign. � 91 = Rydberg constant, calculated exact ly using Bohr theory or t he Schro di nger equat ion � Z = at omic number, equal to I for hydr ogen � n = pr incipal quantum number � As n incr eases, ener gy incr eases, t he at om becomes less stable, and ener gy stat es become more closely spaced (more dense). � Integer n var ies fr om 1 (gr ound stat e) to higher int egers (excit ed st ates) to 00 (ionizat ion). � Ener gies of H at om stat es vary fr om -h91 (n = I) to 0 (n = 00). Stat es with E> 0 are possible and correspond to an ionized atom in which the ener gy > 0 equals t he kinetic ener gy of t he electr on. 1 .9 Atomic Orbitals (AOs) • Definition of AO � Wavefunct ion (\jJ, psi) descr ibes an electr on in an at om. � Or bit al (\jJ 2) holds 0, 1, or 2 electr ons. � Or bit al can be viewed as a cloud wit hin which t he point density r epresents t he probability of finding the electr on at that point. � Or bital is specified by three quant um numbers (n, f!, n1(; see below). • Wavefunction � Fills all space � Depends on t he three spher ical coor dinates: r, 8, � � Wr it ten as a pr oduct of a r adial [R(r)] and an angular [Y(8,�)] wavefunct ion; mat hemat ical ly, \jJ(r, 8, �) = R(r)Y(8, �) SG-S • Wavefunction for the H atom 2s orbital � n = 2, I! = 0, and me = ° R(r) = (2_*}-rI2 ao (2ao )312 ao= 4TIEofi2 = { 5.29177X10-11 m } mee2 (Bohr radius) Units: R(r) = m-312 Y(8,$) = none (C2 ·W1 .m-2 ) (J·S) 2 a - =m 0- kg. C 2 Three Quantum Numbers [11, e, mel Specify an Atomic Orbital Symbol Name n Principal quantum number I! Orbital angular momentum quantum number me Magnetic quantum number • Terminology (nomenclatu re) shell: AOs with the same n value Allowed Values = 1 , 2, 3, ... =0, 1, 2, ... , n-J = I!, I!-J, 1!-2, .. . ,-I! = 0, ± 1 , ± 2, .. . , ±I! subshell: AOs with the same n and e values; Constraints Positive integer Each value of n corresponds to n allowed values of I!. Each value of I! corresponds to (21!+ 1) al lowed values of me. I! = 0, 1 , 2,3 equivalent to S-, p-, d-, .f subshells, respectively, or s-orbital => I! = ° me = ° p-orbital => e= 1 me=-i,O,or+i d-orbital => e = 2 me = -2, -I, 0, +1 , or +2 .f orbital => e = 3 me = -3, -2, -I, 0, +1, +2, or +3 • Physical significance of the wavefu nction \l/(r,e,� ) F or all atoms and molecules, \II 2(r, e, �) is proportional to the probability of finding the electron at a point r, e ,�. We can also regard \112 (r, e ,�) as the electron density at point r, e ,�. • Plot of electron dens ity for the 2s-orbital of hydrogen Computer-generated electron density dot diagram for the hydrogen atom 2s-orbital. The nucleus is at the center of the square and the density of dots is proportional to the probability of finding the electron. Notice the location of the spherical node.