Bohr Model of Hydrogen (PDF)

Part 3A 1 (https://saintschemistry10.weebly.com/history-of-the-atom.html) Outline 3.1 Bohr Model of Hydrogen 3.2 Energy level of Hydrogen Atom 2 3.1 Bohr Model of Hydrogen Overview In ear...

Part 3A 1 (https://saintschemistry10.weebly.com/history-of-the-atom.html) Outline 3.1 Bohr Model of Hydrogen 3.2 Energy level of Hydrogen Atom 2 3.1 Bohr Model of Hydrogen Overview In early 1900s (before Bohr model): -Atom is known. + - -An atom of hydrogen contained positive charge (+e) at its center and negative charge (-e , an electron) outside that center. -No one understood why the electrical attraction between the electron and the positive charge did not simply cause the two to collapse together. - The energy of electron (particle-like property) in orbit around a nucleus is not considered discrete. 3 3.1 Bohr Model of Hydrogen Overview Bohr model plays a crucial role in the atomic theory and helps to explain many important phenomena. à quantization of energy levels (n) in atoms. à the discrete spectral lines observed in the emission and absorption spectra of hydrogen. à serving as a transitional step between classical physics and quantum mechanics. **Bohr model is a simplified model of the hydrogen atom and has limitations.** 4 3.1 Bohr Model of Hydrogen: Bohr Assumption Assumption Classical physics Bohr made two assumptions: 1. ‘𝐿’ describes a movement (rotation). (1) The electron in a hydrogen atom orbits the nucleus in a circle much like Earth orbits the Sun. 2. ‘𝐿’ could have any value. (2) The magnitude of the angular momentum L: of the electron in its orbit is quantized to the values of ‘n’. (https://en.wikipedia.org/wiki/Angular_momentum) 𝐿 = 𝑟⃑ 𝑥 𝑝⃑ Angular momentum is a fundamental property a single particle about a fixed point or axis. associated with the motion of the electron as it orbits the nucleus. 5 3.1 Bohr Model of Hydrogen: Bohr Assumption Discreate of angular momentum The magnitude of angular momentum L: of the electron in its orbit is quantized to the values of ‘n’. 𝑛=1 𝑛=2 𝑛=3 𝑛=4 (Note: not correctly give the angular momentum value of H atom) 6 3.1 Bohr Model of Hydrogen: 𝒔𝒊𝒏𝒈𝒍𝒆 𝒆𝒍𝒆𝒄𝒕𝒓𝒐𝒏 𝒔𝒚𝒔𝒕𝒆𝒎 Calculation of Bohr radius (𝒂𝟎 ) Hydrogen atom Bohr radius: Average distance between the electron and the nucleus in a hydrogen atom when the electron is in its ground state (Many electron system) (n=1). Newton’s Second Law (electron’s speed) " 𝐹 = 𝑚𝑎, (𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒) 1 𝑞" 𝑞# 𝑣# & = 𝑚( ) 1. Electron is in circular motion and thus experiences 4𝜋𝜀! 𝑟# 𝑟 a centripetal force, which causes a centripetal (centripetal acceleration) acceleration. (Coulomb force) (rest mass of electron) 2. Coulomb force between the electron (-e) and the proton (+e), separated by the orbital radius r. (𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒) 7  3.1 Bohr Model of Hydrogen: Calculation of Bohr radius (𝒂𝟎 ) 𝑓𝑟𝑜𝑚, 1) Bohr’s assumption of ‘L’ 1 𝑞" 𝑞# 𝑣# & =𝑚 (4) 4𝜋𝜀! 𝑟# 𝑟 (1) (3) à (4) 2) From classical physics, the magnitude of 𝐿 𝐿 = 𝑟𝑚𝑣 sin 𝜙 (2) 1 𝑒> 𝑛> ℎ> > =𝑚 (5) where 𝜙 (the angle between :r and :v ) is 90°. (sin 90 = 1) 4𝜋𝜀= 𝑟 𝑚>4𝑟 ?𝜋 > 1 = 2 , 𝑛ℏ = 𝑟𝑚𝑣 𝑛ℏ 𝑛,ℎ 𝑣= = (3) 𝑟𝑚 𝑟𝑚 , 2𝜋 8  3.1 Bohr Model of Hydrogen: Calculation of Bohr radius (𝒂𝟎 ) 1) The smallest possible orbital radius (for n = 1) is 𝑎=, which is 1 𝑒> 𝑛> ℎ> From Eq. 𝟓 , = 𝑚 ( > ? >) called the Bohr radius. 4𝜋𝜀= 𝑟 > 𝑚 4𝑟 𝜋 𝑒> 𝑛> ℎ> 2) Electron cannot get any closer to =( ) the nucleus than orbital radius 𝑎=. 𝜀= 𝑚𝑟𝜋 𝜀=ℎ> > 𝑟 = ( > )𝑛 𝑛 = 1, 2, 3 … 𝑚𝑒 𝜋 𝜺𝟎 𝒉𝟐 𝑛=1 𝒓 = 𝒂𝟎 = 𝟐 𝒎𝒆 𝝅 = 𝟓𝟐. 𝟗𝟐 𝒑𝒎 𝑟 = (𝑎8 )𝑛9 9  3.1 Bohr Model of Hydrogen: From previous slide: 1 𝑒 " 𝑚𝑣 " E = 4𝜋𝜀! 𝑟 " 𝑟 Quantization of Orbital Energy 𝐸 =𝐾+𝑈 Energy of the H-atom according to the Bohr 1 1 𝑒 > model. = 𝑚𝑣 > + (− , ) 2 4𝜋𝜀= 𝑟 1 𝒆𝟐 1 𝑒> = ( ) + (− , ) !=1 !=2 2 𝟒𝝅𝜀𝟎 𝒓 4𝜋𝜀= 𝑟 !=3 Kinetic energy of electron !=4 1 𝟏 1 2 𝑒> 𝐾 = 𝑚𝑣 # =( − , ), 2 𝟖𝝅𝜀𝟎 4𝜋𝜀= 2 𝑟 The electron–nucleus system has electric potential energy 𝟏 2 𝑒> 𝑞" 𝑞# (−𝑒)(+𝑒) =( − ), 𝑈= = 𝟖𝝅𝜀𝟎 8𝜋𝜀= 𝑟 4𝜋𝜀! 𝑟 4𝜋𝜀! 𝑟 1. Not show the ‘Quantized 𝟏 𝑒> form’ yet. , we will 𝐸=− , change the ‘r’. 𝟖𝝅𝜀𝟎 𝑟 2. Negative sign = attractive 10 interaction 3.1 Bohr Model of Hydrogen: Quantization of Orbital Energy 𝟏 𝑒# 𝑚𝑒 I 1 𝐸=− & 𝐸H = − > > , > From previous slide: 𝟖𝝅𝜀𝟎 𝑟 8𝜀= ℎ 𝑛 𝜀=ℎ> > 𝟏 𝑒> 𝑟 = ( > )𝑛 =− , Constant I 𝑚𝑒 𝜋 𝟖𝝅𝜀𝟎 𝜖=ℎ> > 𝑚𝑒 (( > )𝑛 ) = 13.61 𝑒𝑉 𝑟 = (𝑎=)𝑛> 𝑚𝑒 𝜋 > 8𝜀= ℎ > 𝟏 𝑚𝑒 >𝜋𝑒 > = 2.18 𝑥 10JKL 𝐽 =− , 𝟖𝝅𝜀𝟎 𝜖=ℎ>𝑛> IJ.LI 𝑚𝑒 I 1 𝐸H = − 𝑒𝑉 𝐸H = − > > , > H! 8𝜀= ℎ 𝑛 𝑛 = 1,2,3, … 11 3.1 Bohr Model of Hydrogen Quantization of Orbital Energy IJ.LI Here, Bohr was able to calculate the visible wavelengths 𝐸H = − H! 𝑒𝑉 emitted and absorbed by hydrogen. Note: - works well for hydrogen atom, which has only one electron orbiting a proton. !=1 !=2 !=3 - Hydrogen-like ions owning one electron, such as He⁺, Li²⁺ !=4 and other ions with a single electron orbiting a nucleus. Modified energy level equation for a hydrogen-like ion is 12 3.1 Bohr Model of Hydrogen: Quantization of Orbital Energy 1) The lowest level, for n = 1, is the ground state of hydrogen. 2) Higher levels correspond to excited states. Note: 1) Energy levels have negative values. 2) The greatest value of n (𝒏 = ∞) is 𝑬% = 𝟎. 3) Any energy greater than 𝐸% (zero), the electron and proton are not bound together (no hydrogen atom) 4) At the E > 0 region, it is nonquantized region An energy-level diagram for the hydrogen13 atom. 3.1 Bohr Model of Hydrogen: Quantization of Orbital Energy Part 3(A) Part 2(A/B) 𝑚𝑒 I 1 > M! 𝐸H = − > > , > 𝐸H = 𝑛 LNO! 8𝜀= ℎ 𝑛 ( kinetic energy only) IJ.LI 𝐸H = − H! 𝑒𝑉 !=1 !=2 !=3 !=4 An energy-level diagram Allowed energies for an for the hydrogen atom. electron confined to the infinite well of L = 100 pm. 14 3.2 Energy level of Hydrogen Atom Absorption & Emission The energy of a hydrogen atom changes when the atom absorbs or emits energy, e.g. light. ∆𝐸 = 𝐸&'(& − 𝐸)*+ (1) −13.6 𝑒𝑉 𝐸, = , 𝑛 = 1,2,3 … (2) 𝑛# 1 1 ∆𝐸 = −13.6 # − # 𝑒𝑉 (3) 𝑛&'(& 𝑛)*+ Spectrum Absorbed energy, electron Possible emission, after goes to higher state. electron back to ground state. 15 3.2 Energy level of Hydrogen Atom Absorption & Emission 1) Electron in H atom can jump between quantized energy levels by absorbing or emitting light at the specific wavelength. 2) This wavelength is called a line because of the way it is detected with a spectroscope; e.g. absorption lines and emission lines. 3) A collection of those lines in the visible range, is called a spectrum of the hydrogen atom. Which ’series’ is the shortest 16 𝝀? Emission lines 3.2 Energy level of Hydrogen Atom Absorption & Emission 1) If atom absorbs high enough energy and electron jumps upward to the nonquantized region, the electron is no longer trapped in the atom. à Free electron (Not confined particle) 2) This is called ionizations of hydrogen. (electron has been removed (Coulomb force is not applicable)) 3) The atom can be ionized if it absorbs enough energy. The free electron then has only kinetic energy K. !"#$%&'( *+%, !"#$%&'( *+%, free e- An energy-level diagram for the hydrogen atom. 17