Chapter 12: Phenomena Phenomena: Different wavelengths of electromagnetic radiation were directed onto two different metal sample (see picture). Scientists then recorded if any particles were ejected and if so what type of particles as well as the speed of the particle. What patterns do you see in the results that were collected? K Wavelength of Light Where Particles Ejected Speed of Exp. Intensity Directed at Sample Ejected Particle Ejected Particle -7 - 4 푚 1 5.4×10 m Medium Yes e 4.9×10 푠 -8 - 6 푚 2 3.3×10 m High Yes e 3.5×10 푠 3 2.0 m High No N/A N/A 4 3.3×10-8 m Low Yes e- 3.5×106 푚 Electromagnetic 푠 Radiation 5 2.0 m Medium No N/A N/A Ejected Particle -12 - 8 푚 6 6.1×10 m High Yes e 2.7×10 푠 7 5.5×104 m High No N/A N/A Fe Wavelength of Light Where Particles Ejected Speed of Exp. Intensity Directed at Sample Ejected Particle Ejected Particle 1 5.4×10-7 m Medium No N/A N/A -11 - 8 푚 2 3.4×10 m High Yes e 1.1×10 푠 3 3.9×103 m Medium No N/A N/A -8 - 6 푚 4 2.4×10 m High Yes e 4.1×10 푠 5 3.9×103 m Low No N/A N/A -7 - 5 푚 6 2.6×10 m Low Yes e 1.1×10 푠 -11 - 8 푚 7 3.4×10 m Low Yes e 1.1×10 푠 Chapter 12: Quantum Mechanics and Atomic Theory Chapter 12: Quantum Mechanics and Atomic Theory o Electromagnetic Radiation o Quantum Theory o Particle in a Box Big Idea: The structure of atoms o The Hydrogen Atom must be explained o Quantum Numbers using quantum o Orbitals mechanics, a theory in o Many-Electron Atoms which the properties of particles and waves o Periodic Trends merge together. 2 Electromagnetic Radiation Spectroscopy: The analysis of electromagnetic radiation emitted/absorbed by substances. Electromagnetic Radiation: Consists of oscillating (time-varying)electric and magnetic fields that 8 푚 travel through space at 2.998 × 10 푠 (c = speed of light) or just over 660 million mph. Note: All forms of radiation transfer energy from one region of space into another. Examples of Electromagnetic Radiation: • Visible light • Radio Waves • X-Rays 3 Chapter 12: Quantum Mechanics and Atomic Theory Electromagnetic Radiation Wavelength (λ): Is the peak-to-peak distance. Note: Changing the wavelength changes the region of the spectrum (i.e. x-ray to visible). Amplitude: Determines brightness of the radiation. Frequency (ν): The number of cycles per 1 second (1 퐻푧 = 1 푠). 4 Chapter 12: Quantum Mechanics and Atomic Theory Electromagnetic Radiation 5 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory If white light is passed through a prism, a continuous spectrum of light is found. However, when the light emitted by excited hydrogen atoms is passed through a prism the radiation is found to consist of a number of components or spectra lines. 6 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory Black Body: An object that absorbs and emits all frequencies of radiation without favor. Heated Black Bodies 푾 푻흀 = ퟐ. ퟗ푲풎풎 푰 = ퟓ. ퟔퟕ × ퟏퟎ−ퟖ 푻ퟒ 풎풂풙 풕풐풕 풎ퟐ푲ퟒ Note: λmax is the most prevalent wavelength, not the longest wavelength. 7 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory Theory Scientists used classical physics arguments to derive an expression for the energy density (Rayleigh Jens Law). 푇 퐸푛푒푟푦 퐷푒푛푠푡푦 ∝ 휆4 The Problem With Explaining Black Body Radiation Using Classical Mechanics Classical mechanics put no limits on minimum wavelength therefore at very small 휆, 퐸 would be a huge value even for low temperatures. 8 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory Max Plank: Proposed that the exchange of energy between matter and radiation occurs in quanta, or packets of energy. His central idea was that a charged particle oscillating at a frequency 휈, can exchange energy with its surroundings by generating or absorbing electromagnetic radiation only in discrete packets of energy. Quanta: Packet of energy (implies minimum energy that can be emitted). Charged particles oscillating at a frequency, 휈, can only exchange energy with their surroundings in discrete packets of energy. 9 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory Photoelectric Effect: Ejection of electrons from a metal when its surface is exposed to ultra violet radiation. 10 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory Findings of the Photoelectric Effect 1) No electrons are ejected unless the radiation has a frequency above a certain threshold value characteristic of the metal. 2) Electrons are ejected immediately, however low the intensity of the radiation. 3) The kinetic energy of the ejected electrons increases linearly with the frequency of the incident radiation. Albert Einstein Proposed that electromagnetic radiation consists of massless “particles” (photons) Photons are packets of energy Energy of a single photon = ℎ · 휈 11 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory Work Function (횽): The minimum amount of energy required to remove an electron from the surface of a metal. If 퐸푝ℎ표푡표푛 < Φ no electron ejected If 퐸푝ℎ표푡표푛 > Φ electron will be ejected 12 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory With what speed will the fastest electrons be emitted from a surface whose threshold wavelength is 600. nm, when the surface is illuminated with light of − wavelength 4.0 × 10 7 푚? 13 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory H atoms only emit/absorb certain frequencies. What is causing these properties? Electrons can only exist with certain energies. The spectral lines are transitions from one allowed energy level to another. 14 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory Finding energy levels of H atom ∆퐸 = 퐸푢푝푝푒푟(2) − 퐸푙표푤푒푟 1 = ℎ휈푒푚푡푡푒푑 15 1 1 휈푒푚푡푡푒푑 = 3.29 × 10 퐻푧 2 − 2 (Found experimentally) 푛1 푛2 Note: In this equation n2 is always the large number which corresponds to the higher energy level. 15 1 1 −34 15 1 1 −ℎ 3.29 × 10 퐻푧 2 − 2 = − 6.626 × 10 퐽 ∙ 푠 3.29 × 10 퐻푧 2 − 2 푛2 푛1 푛2 푛1 Note: The negative sign appeared because a negative sign was taken out of parenthesis. −18 1 1 -2.178 × 10 퐽 2 − 2 =퐸푢푝푝푒푟(2) − 퐸푙표푤푒푟(1) 푛2 푛1 Energy level of H atom 1 퐸 = −2.178 × 10−18퐽 n= 1, 2, 3, … 푛2 Energy level of other 1e- systems 푍2 퐸 = −2.178 × 10−18퐽 n= 1, 2, 3, … 푛2 15 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory Student Question What is the wavelength of the radiation + emitted by a 퐿2 ion when an electron transitions between 푛 = 4 to the 푛 = 2 levels? 8 푚 Helpful Hint: c = 2.998×10 푠 a) -1.664×1024 m b) -5.399×10-8 m c) 1.621×10-7 m d) 4.864×10-7 m e) None of the Above 16 Chapter 12: Quantum Mechanics and Atomic Theory m s Quantum Theory Constructive Interference: When the peaks of waves coincide, the amplitude of the resulting wave is increased. Destructive Interference: When the peak of one wave coincides with the trough of another wave the resulting wave is decreased. 17 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory When light passes though a pair of closely spaced slits, circular waves are generated at each slit. These waves interfere with each other. Where they interfere constructively, a bright line is seen on the screen behind the slits; where the interference is destructive, the screen is dark. 18 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory Student Question Imagine an alternate universe where the value of Planck constant is 6.62607×10-9 J·s. In that universe, will the following require quantum mechanics or classical mechanics to adequately describe it? A bacterium with a mass of 3.0 pg, 9.0 μm long, 휇푚 moving at 6.00 푠 . a) classical b) quantum c) You could use either 19 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory Particles have wave like properties, therefore, classical mechanics are incorrect. Classical Mechanics: Particles have a defined trajectory. Location and linear momentum can be specified at every moment. Quantum Mechanics: Particles behave like waves. Cannot specify the precise location of a particle. Note: For the hydrogen atom, the duality means that we are not going to be able to know the speed of an electron orbiting the nucleus in a definite trajectory. 20 Chapter 12: Quantum Mechanics and Atomic Theory Quantum Theory Wavefunction (흍): A solution of the Schrödinger equation; the probability amplitude. Note: This gives you a representation of the particle’s trajectory. Examples : 휓 = sin(푥) Probability Density (흍ퟐ): A function that, when multiplied by volume of the region, gives the probability that the particle will be found in that region of space. Note: This value must be between 0 and 1 21 Chapter 12: Quantum Mechanics and Atomic Theory Particle in a Box Particle in a 1-D Box Boundary Conditions 푥 = 0휓 = 0 푥 = 퐿휓 = 0 What is the wavefunction? sin(푥) (sin(0) = 0 푎푛푑 sin(휋) = 0) 1 2 Τ2 푛휋푥 휓 푥 = 푠푛 퐿 퐿 Note: n is a quantum number. Quantum numbers are integers or ½ integers that label a wavefunction. 1Τ2 Note: The term 2 is there for normalization in order to get 2 퐿 휓 = 1 22 Chapter 12: Quantum Mechanics and Atomic Theory Particle in a Box What are the allowed energy levels? Get the allowed energy levels from Schrödinger’s equation 2 1 ℏ 2 Τ2 푑2 푛휋푥 ℏ2 푑2휓 − 2 sin =퐸휓 − +푉 푥 휓=퐸휓 2푚 퐿 푑푥 퐿 2푚 2 푑푥 풏흅풙 Take 1st derivative of 풔풊풏 Note: V(x)=0 for the particle in a box 푳 푑 푑 푛휋푥 푑 푛휋 푛휋푥 ℏ2 푑2휓 푠푛 = 푐표푠 − =퐸휓 푑푥 푑푥 퐿 푑푥 퐿 퐿 2푚푑푥2 풏흅풙 1 Take 2nd derivative of 풔풊풏 ℏ2 푑2 2 ൗ2 푛휋푥 푳 − 2 sin =퐸휓 2푚푑푥 퐿 퐿 푑 푛휋 푛휋푥 2휋 2 푛휋푥 cos =− sin 푑푥 퐿 퐿 퐿 퐿 23 Chapter 12: Quantum Mechanics and Atomic Theory Particle in a Box Plug 2nd derivative back into wavefunction 1 1 2 Τ2 2 Τ2 ℏ 2 푛휋 푛휋푥 2 푛휋푥 2푚 퐿 퐿 sin 퐿 = 퐸 퐿 sin 퐿 Cancel out 2 2 ℏ 푛휋 2푚 퐿 = 퐸 Simplify ℎ ℏ = 2휋 2 ℎ 2 2 2 2푛 휋 ℎ2푛2 퐸 = 2 휋 = 2푚퐿2 8푚퐿2 24 Chapter 12: Quantum Mechanics and Atomic Theory The Hydrogen Atom Atomic Orbitals: A region of space in which there is a high probability of finding an electron in an atom.