Lecture 7 - Early Quantum Theory

Wednesday, January 31, 2024

10:11 AM

This week's overview Assignments this week: Activity 1: ﷟HYPERLINK "https://boisestatecanvas.instructure.com/courses/28699/assignments/985392"Energy Source Investigation Write-up due yesterday (1/31) Finish this by Friday at the very latest (a late penalty will apply) ﷟HYPERLINK "https://boisestatecanvas.instructure.com/courses/28699/discussion_topics/654027"Peer reviews due Sunday (2/4) Read Chapter 4 and begin chapter 5 Keep posted for an announcement Enjoy your weekend Office Hours: Today after class in ILC 118 Friday 11-1 CIC (Educ 107) ﷟HYPERLINK "https://calendly.com/bricejurban/office-hours"By appointment Email (﷟HYPERLINK "mailto:BriceJurban@boisestate.edu"BriceJurban@boisestate.edu) Midterm 1: The first midterm will be graded by myself and a few other graders tomorrow and Saturday. Tests will be reviewable starting Monday using the Gradescope software which can be found on Canvas. We will be using a rubric so that you can know exactly how you were graded. I recommend printing a copy then reviewing the answer key in the big display case outside SCNC 336 next to the chemistry computer lab. See map below for location of answer key . After reviewing, if you notice any discrepancies you can submit a regrade request in Gradescope. Today's Schedule (2/1): Ionization energies Electromagnetic Radiation Emission Spectra Blackbody Radiation and the Photoelectric Effect Tuesday's Schedule (2/6): Wave-Particle Duality of Light and Matter Quantization of Energy and the Bohr Model Quantum Theory AnswerKey.png SCALE: REV: / 04 Ill 2 3 CAMPUS LN 1 0 20 0 工 ○ C 二 丄 冖 凵 工 一 工 」 , qen Jalndtuoo s 一 go 山 00 」 00 ; ; S ! 山 0 0 0 O•h•ue 」 。 」 。 。 56 冖 冖 凵 POOuedPV ﷟HYPERLINK "https://ptable.com/#Properties/Ionization/1st"First Ionization Energy Energy required to remove the first electron from a neutral atom A higher value means it is harder (more difficult) to occur Untitled picture.emf Energy
Untitled picture.emf Energy Untitled picture.emf 375.7 kJ/mol 23723 kJ/m01 Li 520.2 Na 495.8 K 418.8 Rb cs 375.7 Be 899.5 Mg 737.7 ca 589.8 sr Ba 502.9 Ra 509.3 sc 633.1 Y 599.9 La 538.1 4988 Lanthanides Ti 6588 zr 640.1 658.5 Rf Ce 534,4 608.5 v 6509 652.1 Ta 7284 Pr 528.1 Pa cr 652.9 Mo w 758.8 Sg Nd 533.1 5976 717.3 Tc Re 755.8 Pm 5386 604.5 Fe 762.5 710.2 Os 814.2 Sm 544.5 Pu 5814 co 7604 Rh 719.7 865.2 rvlt 547.1 Am 576.4 Ni 737.1 Pd Pt 864,4 Ds 593 A Cm 578.1 cu 745.5 Ag Au Rg 565.8 598.0 zn 906.4 8678 (Jub Dy 573.0 Cf 606.1 B 800.6 Al 577.5 Ga 578.8 In 5583 TI 589A Uut Ho 5810 Es c 786.5 762.2 Sn 7086 715.6 U uq Er 589.3 14023 p 1011.8 As 830.6 Bi 703.0 Uup Tm 596.7 1313.9 s 9996 Se 9410 Te 869.3 Po 812.1 603.4 No 1681.0 Cl 1251.2 Br 1139.9 1008*4 At Lu 5233 Lr 4728 2372.3 Ne 20807 1350.8 Xe 1037.1 Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings
Untitled picture.emf 375.7 kJ/mol 23723 kJ/m01 Li 520.2 Na 495.8 K 418.8 Rb cs 375.7 Be 899.5 Mg 737.7 ca 589.8 sr Ba 502.9 Ra 509.3 sc 633.1 Y 599.9 La 538.1 4988 Lanthanides Ti 6588 zr 640.1 658.5 Rf Ce 534,4 608.5 v 6509 652.1 Ta 7284 Pr 528.1 Pa cr 652.9 Mo w 758.8 Sg Nd 533.1 5976 717.3 Tc Re 755.8 Pm 5386 604.5 Fe 762.5 710.2 Os 814.2 Sm 544.5 Pu 5814 co 7604 Rh 719.7 865.2 rvlt 547.1 Am 576.4 Ni 737.1 Pd Pt 864,4 Ds 593 A Cm 578.1 cu 745.5 Ag Au Rg 565.8 598.0 zn 906.4 8678 (Jub Dy 573.0 Cf 606.1 B 800.6 Al 577.5 Ga 578.8 In 5583 TI 589A Uut Ho 5810 Es c 786.5 762.2 Sn 7086 715.6 U uq Er 589.3 14023 p 1011.8 As 830.6 Bi 703.0 Uup Tm 596.7 1313.9 s 9996 Se 9410 Te 869.3 Po 812.1 603.4 No 1681.0 Cl 1251.2 Br 1139.9 1008*4 At Lu 5233 Lr 4728 2372.3 Ne 20807 1350.8 Xe 1037.1 Untitled picture.png 4.0 20 1.0 0.0 10 15 20 25 30 35 40 45 50 55 Atomic runnter 60 65 70 75 80 85 90 95 Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings
Untitled picture.png 4.0 20 1.0 0.0 10 15 20 25 30 35 40 45 50 55 Atomic runnter 60 65 70 75 80 85 90 95 Successive Ionization Energies suggest a shell structure to the atom If you add more energy you can continue to remove electrons from an atom. I2, I3, I4 . . . up to the number of electrons that atom has Ionization energies are on the order of 10-18 J so tabulated values in our text our given in attojoules (aJ) You can estimate the total energy needed to ionize an atom into a certain state by looking up tabulated values of ionization energies and summing them Untitled picture.png Ionİzatİon energy/aJ Z 2 3 4 5 6 7 8 9 10 11 Element H He Be c N o Ne 11 2.18 3.94 0.86 1.49 1.33 1.81 2.32 2.17 2.78 3.45 0.83 12 8.72 12.1 2.92 4.04 3.90 4.75 5.63 5.60 6.56 7.57 19.6 24.7 6.08 767 7.60 8.80 10.0 10.2 11.5 14 34.9 41.5 10.3 12.4 12.4 14.0 15.6 15.8 15 54.5 62.8 15.7 18.2 18.3 20.2 22.2 16 78.5 88.4 22.7 25.2 25.3 27.6 107 118 29.7 33.2 33.4 18 140 153 38.3 42.3 177 192 48.0 110 218 238 264 We should note that at some point the energy needed to ionize increases dramatically. This seems to suggest electrons are found in shells around the atom. When plotted as the natural log(In) against the number of electrons removed (n),you get a graph that's super obvious. Here's the data for sodium: Untitled picture.emf Successive Ionizations of a Sodium Atom Number of electrons removed (n) The outer most occupied electrons are the valence electrons Lewis Electron-Dot Formulas: Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings
The outer most occupied electrons are the valence electrons G. N. Lewis at the early part of the twentieth century recognized these electrons were involved in bonding Lewis Electron-Dot Formulas: Test your Understanding Calculate the energy required to form a single C4+ ion from a neutral carbon atom in the gas phase. 1.81 + 3.90 + 7.67 + 10.3 = 23.68 aJ/atom What is the energy required to convert 1 mol of C atoms into C4+ ions in the gas phase? What is the energy in terms of kJ/mol 23.68 aJ/atom * 6.022 * 10^23 atom/mol =1.43 *10^25 aJ/mol 1.43*10^25 aJ/mol *1 J/10^18 aJ = 1.43 *10^7 J 14.26 MJ/mol or 14260 kJ/mol Write the Lewis electron-dot formula for Al and Cl Write the Lewis electron-dot formula for Al3+ and ﷐𝐶𝑙﷮−﷯ Light is just a portion of the Electromagnetic (EM) Spectrum Untitled picture.png matte r. Emitted by energetic nuclei; very penetrating Penetrates human tissue and other 1012 The colors Of light perceived by the human eye. Wavelength (nm) Can be felt as heat. Includes radar. Used in microwave Ovens, and in cellular telephones. radiation. 0-2 Gamma ray 1020 100 X-ray 101B 102 Ultra- n violet 1016 104 Infrared 1014 106 Microwave 1010 1010 1012 10B Radio frequency 106 104 Frequen%' (s- Causes sunburn; kills bacteria. AM, shortwave, FM, and television transmissions. Untitled picture.png field Magnetic field Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings
Untitled picture.png matte r. Emitted by energetic nuclei; very penetrating Penetrates human tissue and other 1012 The colors Of light perceived by the human eye. Wavelength (nm) Can be felt as heat. Includes radar. Used in microwave Ovens, and in cellular telephones. radiation. 0-2 Gamma ray 1020 100 X-ray 101B 102 Ultra- n violet 1016 104 Infrared 1014 106 Microwave 1010 1010 1012 10B Radio frequency 106 104 Frequen%' (s- Causes sunburn; kills bacteria. AM, shortwave, FM, and television transmissions. Regions of the EM Spectrum can be classified by their wavelength 𝜆 and frequency 𝜈 Untitled picture.png field Magnetic field EM radiation is wave propagating in one direction with oscillating electric and magnetic fields that are perpendicular to the direction of travel and each other. Untitled picture.png Long wavelength Low frequency Short wavelength High frequency Untitled picture.png 400 nm 780 nm Wavelength and frequency are inversely proportional. Wavelength is represented by the Greek letter lambda 𝜆 and has units of meters Frequency by the Greek letter nu 𝜈 and has units of Hz or ﷐𝑠﷮−1﷯ Regions of high energy (gamma rays) have extremely fast oscillations (high frequency) which corresponds to very short wavelengths In a vacuum, EM radiation travels at the speed of light c = 2.998 ×﷐10﷮8﷯ m ∙ ﷐𝑠﷮−1﷯ c = 𝜆𝜈 The visible spectrum is just a very small portion of the EM spectrum
Untitled picture.png 400 nm 780 nm Test your Understanding What is the frequency of violet light at 400 nm? What is the wavelength of a radio station broadcasting at 89.6 MHz? Electromagnetic Radiation is Quantized (comes in discrete packets) The above image of the visible spectrum is continuous and has no gaps, but if we use a prism to examine radiation emitted from a chemical sample such as sodium burning in a flame or incandescent bulbs we see discrete lines. Why is this? The emission spectra of hydrogen produces 5 discrete lines at 397.0 nm, 410.2 nm, 434.0 nm, 486.1 nm, and 656.3 nm. The human eye can only see between 400 nm and 750 nm so usually see 3 or 4 lines. Untitled picture.png 434.0 nm 410.2 nm 397.0 nm 400 450 486.1 nm 500 550 600 656.3 nm 650 700 750 nm Johann Balmer came up with an empirical equation to represent these lines: ﷐1﷮𝜆﷯=(1.097×﷐10﷮7﷯﷐𝑚﷮−1﷯)(﷐1﷮4﷯−﷐1﷮﷐𝑛﷮2﷯﷯) n = 3, 4, 5 This equation is known as the Rydberg-Balmer equation and the constant 1.097 ×﷐10﷮7﷯﷐𝑚﷮−1﷯ is the Rydberg constant (﷐𝑅﷮𝐻﷯) The Ultraviolet Catastrophe Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings
The Ultraviolet Catastrophe The ultraviolet catastrophe was a theoretical problem in the late 19th and early 20th centuries involving the emission of electromagnetic radiation by a black body. A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. According to classical physics, specifically Rayleigh-Jeans Law, which applies classical mechanics to electromagnetic radiation, the energy emitted by a black body at high frequencies (such as the ultraviolet region) should increase indefinitely with frequency. This implies that a black body would radiate infinite energy, which is not physically observed. Max Planck resolve this paradox by proposing that EM radiation can only be emitted or absorbed in discreted quantities called "quanta" with the energy proportional to the frequency of radiation. Untitled picture.png le6 1000 Blackbody Radiation 1500 Wavelength (nm) 5000 K 7000 K Classical Theory (Rayleigh-Jeans) 2000 2500 3000 𝐸=ℎ𝜈=﷐ℎ𝑐﷮𝜆﷯ ℎ= Planck's constant = 6.626 × ﷐10﷮−34﷯𝐽∙𝑠 Test your Understanding Calculate the energy of an X-ray with a wavelength of 80 nm. Is it of sufficient energy to ionize an argon atom with a first ionization energy of 2.52 aJ? Calculate the minimum number of photons of green light that can be detected by the human eye if 2.35 × ﷐10﷮−18﷯𝐽 of green light (510nm) is the least amount the eye can detect Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings
Calculate the minimum number of photons of green light that can be detected by the human eye if 2.35 × ﷐10﷮−18﷯𝐽 of green light (510nm) is the least amount the eye can detect Calculate the frequency and wavelength of photons capable of just ionizing 1.0 × ﷐10﷮16﷯ sodium atoms in the gas phase. Assuming an ionization efficiency of 16%, how many such photons are required to ionize the whole sample? Use the ionization energies from before (energy in aJ) Untitled picture.png Machine generated alternative text: Z Ionization energy/aJ Element Untitled picture.png 1.0 × ﷐10﷮16﷯ sodium atoms ×﷐100 photons﷮16 ions﷯=6.3×﷐10﷮16﷯ photons needed Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings

 

 

 

 

 

 

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