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# Chem 301 Week 5 Chem 301

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This 12 page Class Notes was uploaded by Nhi Notetaker on Sunday October 2, 2016. The Class Notes belongs to Chem 301 at University of Texas at Austin taught by Dr. Jones in Fall 2015. Since its upload, it has received 3 views. For similar materials see Principles of Chemistry I in Chemistry at University of Texas at Austin.

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Date Created: 10/02/16

Light is electromagnetic radiation with wavelight properties Amplitude is height of wave Wavelength is measure of the distance covered by the wave (nm or m) The frequency (v) is the number of waves that pass a point in a given period of time (1/s or Hz) Periodic changes of electric and magnetic fields in time and space. Two fields are always perpendicular to each other Speed of light: c= 3x10^8 m/s C is a constant in a vacuum (wavelength)(frequency)=c Inverse relationship between wavelength and frequency Very short wavelength = very high frequency Gamma rays X-rays UV Visible light Infrared Microwaves radio Very long wavelength = very short frequency Ideas about modern atomic theory stemmed from observations of the interactions between light and matter Observations not explained well by classical mechanics or electromagnetism 1. Discrete spectra of atoms 2. Blackbody problem 3. Photoelectric effect Light experiments: 1. Flame tests a. Putting metal salt in a hot flame-different colors 2. Electrical discharges in gas a. Neon gas under low pressure excited by an electric arc 3. Blackbody radiation a. Heating a bar of iron to red heat then hotter to white hot Calcium chloride- brick red Potassium- lilac Sodium- yellow Copper- blue Barium- apple green Strontium- red Exciting gas atoms causes light to be emitted- “bright light spectra” There is different emission spectra for different gases Radiation emitted-light given out as electron returns to atom Blackbody radiation Electromagnetic radiation emitted by an opaque or non-reflective body Spectrum and intensity which depend on ly the temperature of the body Room temp-... A method for determining the temperature of stars A continuous spectrum-not discreet lines as in excited gas spectra or flame tests Planck Vibrating atoms led to emission of light As they vibrated faster…. Light is composed of particles called photons Energy is proportional to the frequency E=hv H is planck's constant (6.6260755x10^-34 J) Combined: E= (hc)/wavelength 1/wavelength is also known as a wavenumber or reciprocal centimeter Photoelectric Effect Intensity of light (brightness, number of photon striking a given area per second) 1. Proportional to current 2. Does not affect kinetic energy of ejected electrons as long as the wavelength is short enough Frequency of incident photon (color of light) 1. Is proportional to kinetic energy of ejected photons 2. Does not affect number of electrons ejected (current) Energy of radiation striking metal surface (E) (E=hv) is converted into energy needed to just allow the electron to escape the surface atom plus the kinetic energy of the electron as it flies off into space hv= work function of the metal Atom Wavefunction and Energy Electrons in atoms (and electrons in general) are governed by the laws of Quantum Mechanics (QM). When we solve quantum mechanical problems we get two things: a wavefunction and the energy. wavefunction This is a mathematical function (given the symbol psi, ψψ) that describes the system of interest. For us the important part of the wavefunction is that it can help to understand the spatial extent of the system -- it tells us where the electron is. Technically, the square of the wave function (ψ2ψ2) is related to the probability of finding the particle in a particular point in space. energy Every wave function has a particular energy. This is a key idea of quantum mechanics: There are only particular energies that an electron in a hydrogen atom can possess. Ground State and Degenerate Energy Solutions We can order them in terms of their energies. In particular, we are most interested in the lowest energy solution, as this is the ground state or most stable state of the electron in an H-atom. If we look at the next highest energy solutions we discover that there are many solutions with the same energy. We call these degenerate solutions. As we move up in energy we find another group of degenerate solutions with a different higher energy. And we can keep moving up in energy with more and more solutions. Quantum Mechanics: Solutions of the Hydrogen Atom The Schrödinger Equation The Schrödinger equation is a differential equation that we solve to get all the wavefunctions that will describe (model) the electron energy levels within the atom. The following is a simplified version (one dimension, x) of the Schrödinger equation. You can see how kinetic and potential energies are split into two parts and combine for the total energy. The reason this is simplified is because it is only being shown for one dimension, the x axis, which works great for particle in a box but we have to use all three dimensions for the atom. This is best done with spherical coordinates (r,θ,ϕr,θ,ϕ). Wavefunctions are Defined by Their Quantum Numbers We can classify our wavefunctions with sets of quantum numbers. These help us to describe the mathematical form of the wavefunction with a simple set of integers. For a one-electron system there are three quantum numbers: 1. Principal Quantum Number (n): distance from the nucleus 2. Angular Momentum Quantum Number (ℓ): shape 3. Magnetic Quantum Number (mℓ): orientation in space Principal Quantum Number, n The most important categorization of the solution to the hydrogen atom is the principal quantum number, n. We regard this as the most important because this number is related to the energy associated with that particular wavefunction. For hydrogen-like atoms (one electron and a nucleus) all solutions with the same n value are degenerate (they have the same energy). The energy of the level in joules is given by where Z is the nuclear charge. For hydrogen (Z=1), the energy levels start at n=1 with a value of -2.18 x 10-18J. Then energies are negative as we have defined E=0 as the energy of the electron and the nucleus at a distance of infinity. The electron is more stable (lower in energy) when it is closer to the nucleus. Since we have defined the separated nucleus and electron as zero, the more stable energies must all be negative. The more negative, the more stable it is. Then they move up in energy (closer to zero) with a spacing that decreases with each level until they all approach an energy of zero (n=infinity). The energy also depends on the square of the nuclear charge, Z. The ground state energy (n=1) for helium (1s2, Z=2) is 4 times lower than the ground state energy (n=1) for hydrogen (1s1, Z=1). We can now use this formula to find the energy difference between any two states in the hydrogen atom. This will come out to be identical to the trends that were found for the Rydberg equation. Angular Momentum Quantum Number, ℓ The next quantum number is the angular momentum quantum number. This gets the symbol ℓ (that's a cursive ell). This quantum number is related to the "shape" of the wavefunction. ℓℓ is an integer and can have any value starting at zero and going up n−1n−1. Since the ℓℓ values are also related to some historical experiments, they maintain other notations. ℓ=0 are called s, ℓ=1 are p, ℓ=2 are d, ℓ=3 are f, ℓ=4 are g, ℓ=5 are h.... Magnetic Quantum Number, mℓmℓ For a given n and ℓ, there are a number of degenerate solutions. The number of degenerate levels is equal to 2ℓ+1. There are three p solutions (ℓ=1, 2(1)+1 = 3). There are seven ff solutions (ℓ=3ℓ=3, 2(3)+1 = 7). Each of these solutions gets a unique quantum number mℓ. mℓ is also an integer and can range from mℓ=−ℓ,...0...,+ℓ So if ℓ=4ℓ=4 the nine possible mℓmℓ values are -4,-3,-2,-1,0,+1,+2,+3,+4. Orbital Notations Orbital notation is simply a different means that chemists use to describe the wavefunction for a hydrogen atom. Rather than using the term wavefunction, instead we use the word "orbital". The orbital notation uses only the n and ℓ quantum numbers. In this notation we simply state the principal quantum number n as a number. A letter is used to denote the ℓ term as letters s,p,d... Orbital Shapes The hydrogen atoms orbitals are the "wavefunction" portion of the quantum mechanical solution to the hydrogen atom. The wavefunctions tell us about the probability of finding the electron at a certain point in space. Thus the orbitals offer us a picture of the electron in a hydrogen atom. However, this picture is not a simple one. First of all, the electron is spread out over space. We cannot pin-point its location. If this seems odd (even impossible), it should. There are two key features for an orbital. The distribution of the electron away from the nucleus. This is known as the radial distribution. The other is the "shape" of the orbital and is the angular distribution. The radial distribution is mostly dependent on the principle quantum number n. The angular distribution depends on ℓ and mℓ s orbitals are wavefunctions with ℓℓ = 0. They have an angular distribution that is uniform at every angle. That means they are spheres. p orbitals are wavefunctions with ℓℓ = 1. They have an angular distribution that is not uniform at every angle. They have a shape that is best described as a "dumbbell". d orbitals are wavefunctions with ℓℓ = 2. They have an even more complex angular distribution than the p orbitals. For most of them it is a "clover leaf" distribution (something like 2 dumbbells in a plane). D orbitals have two angular nodes (two angles at which the probability of electron is always zero. Radial Distribution Functions When the wavefunction, ψψ, is squared the result is a number that is directly proportional to the probability of finding an electron at specific coordinates in three-dimensional space. The radial portion of the wavefunction really only tells us if there is high or low probability at various distances from the nucleus (possible radii for the electrons). Multiplying this probability by the area available at that distance will give us the Radial Distribution Function for the given electron. The concentric spherical shells have areas equal to the surface area of a sphere which is 4πr24πr2. What you need to notice: Note that the greatest probability for the 3 curves progresses to distances further away from the nucleus (nucleus is at zero). So you conclude that a 3s3s-orbital is slightly larger than a 2s2s-orbital which is slightly larger than a 1s1s-orbital. And, even though we don't show more orbitals, you can conclude that the trend will continue for the 4s4s-orbital all the way through to the 7s7s-orbital. What We Know About the Hydrogen Atom We use what we know about the electrons in hydrogen as the basis for our model for electrons in all the elements. There are several key ideas that we can take away from studying the hydrogen atom that we can apply to all atoms. 1. The electron in a hydrogen atom is only found in particular fixed energy levels. This is odd -- it's unlike any of the macroscopic objects we deal with in our everyday lives -- and it's a consequence of quantum mechanics. This is a critical idea for understanding atoms, molecules, and chemistry. 2. Small mass particles like electrons obey a different set of rules than the macroscopic objects of our everyday lives. They are not the same, and we should not try to imagine them as being the same. We must always remember that they are not macroscopic particles, so they will never fit in with our macroscopic pictures. Electrons are simply not small particles orbiting like planets around nuclei. 3. Electrons follow the rules of quantum mechanics. Quantum mechanics provide us two useful ideas about electrons: their energy and their wavefunction. Energies come in discrete or "quantized" units. Wavefunctions tell us something about the "spatial distribution" of the electron (i.e. "where the electron is"). 4. We find that the energies and wavefunctions that describe an electron in a hydrogen atom can be classified by a set of "quantum numbers". These define the actual mathematical function that is the wavefunction. They effectively describe the "size and shape" of the electron's wavefunction. 5. We use the ideas we get from the hydrogen atom to describe all the rest of the elements. So these solutions that we obtain from quantum mechanics for hydrogen are used repeatedly in chemistry to describe atoms and bonding. Energy of radiation striking metal surface (E) (E=hv) is converted into energy needed to “just” allow the electron to escape the surface atom plus the kinetic energy of the electron as it flies off into space Plot of KE vs. frequency is a straight line Slope is the same for any metal 6.626x10^-34, planck’s constant Work function: energy to just liberate an electron E=hv0 Excited atoms only emit light at discrete wavelengths Electrons must only make transitions b/w discrete energy level Energy levels are “quantized” Bohr’s Theory Allowed calculation of an energy level The calculations of the emitted wavelength upon the release of an energy when an electron transitions from higher to lower energy E=h(c/wavelength)=hv E is proportional to 1/n^2 N is a simple integer E=-Rh(1/n^2) Rh=2.8x10^-18 1. Electrons have specific (quantized) energies 2. Light is emitted as e^- moves from one level to another Rydberg Formula E=R(1/n^2f-1/n^2i) [pay attention to units] [hydrogen atom only, z=1,] For other elements: E=z^2 R(1/n^2f-1/n^2i) z= atomic number Louis de Broglie proposed: the wavelength associated with a moving particle is inversely proportional to its momentum (mass x velocity) Wavelength = h/mv Particles have wavelike properties Wavelength and mass have an inverse relationship 1. H is very very small 2. Large objects make wavelength so small that it's negligible 3. But very small objects (electrons, protons) will have a significant wavelength associated with them Heisenberg’s Uncertainty Principle The product of the uncertainties in both the position and speed of a particle is inversely proportional to its mass It is fundamentally impossible to determine simultaneously and exactly both the momentum of a particle. Pauli Exclusion In a given atom, no two electrons can have the same set of four quantum number An orbital can hold only two electrons and they must have opposite spin Wavelength The wavelength of a light wave (or any other wave) is the distance between two peaks of the wave. Since it is a physical distance, it has units of length. Frequency The frequency of light is the number of peaks that pass by a given point in space per second. For all waves, this depends on both the speed and the wavelength. The relationship for light is: c=λν where c is the speed of light, λλ is the wavelength, and νν is the frequency. Since speed has units of distance per time, and wavelength has units of distance, frequency has units of inverse time. We will use units of inverse seconds, which are typically called Hertz (Hz). Speed of Light The speed of light in a vacuum is another Universal constant because it is the same for all types of light. However, the speed of light actually varies (slightly) depending on the medium that it is travelling through -- that's why we say that the speed of light in a vacuum is constant. For our purposes, we will consider the speed of light to be the speed in vacuum. This value is the same for all frequencies and is: c=2.998×108ms−1 We typical use 7 categories ranging from the shortest wavelength to the longest. They include: 1. Gamma Rays 2. X-Rays 3. Ultraviolet Radiation 4. Visible Light 5. Infrared Radiation 6. Microwaves 7. Radio waves You might notice that we call some categories "waves," others "light," and other "radiation." However they're actually all radiation, and they're all waves, and they're all light. It is also handy to remember the order of color in the visible region. The easiest (and oldest) way to remember this is to think "Roy G. Biv" (Red, Orange, Yellow, Green, Blue, and Violet). Don't worry about specific wavelengths within the visible region, other than knowing that 400 nm is the "blue" end of the spectrum and 700 nm is the "red" end. Matter Interaction with EM Radiation . As the radiation gets more and more energetic, matter will have more and more excited responses to the exposure. Here is a brief summary of the types of interaction between EM radiation and matter. radio waves - Radio waves are such low energy that there is effectively no interaction between matter and radio waves. This is a good thing seeing as we humans are constantly bombarded by all kinds of radio waves (AM/FM radio, television, wireless phones, etc.) microwaves waves - Microwaves have just enough energy to get molecules to start rotating. Rotational excitation is probably the lowest of all the interactions. It can cause a great response however if the microwaves are powerful enough and directed precisely. Microwaves are used in microwave ovens. Molecules in food (mostly water) rotate rapidly enough to cause heat generation through the friction between molecules. The molecules remain intact however. infra-red radiation - Infra red radiation is enough energy to get molecules to start vibrating. The bonds within a molecule are pushed and pulled such that the bonds stretch and bend. This too, can generate heat. Scientists can also learn what type of bonds are within the molecule by measuring the wavelength of the absorbed radiation. Specific bonds between specific atoms have very specific absorptions. IR Spectroscopy is essential for an organic chemist. visible light - Once the visible range is reached, molecules first start having electronic excitations. The electrons within the molecule are excited to higher energy states. This is a key factor in the perception of color. Certain wavelengths of the visible range are absorbed and other wavelengths transmit through or fully reflect. We observe this a color. Once electrons are excited to higher states, the molecule becomes a bit more susceptible to further chemical reaction. ultra violet radiation - Ultra violet is like visible in that the radiation causes electronic excitations. However, UV radiation is at the onset of full ionization of the molecule. This means that UV has enough energy to actually promote certain electrons all the way out of the molecule. This much energy will often lead to bonds breaking and molecular decomposition. x-rays, gamma-rays, and cosmic rays - All of these radiation types are more than enough energy to completely ionize matter. Radiation is absorbed and electrons are ejected from the atoms and molecules. Knocking out electrons will leave an ion behind and that is why all of these are called ionizing radiation. Photoelectric Effect Photons The energy of light is different than energies we are used to dealing with in our everyday world. This is because we can think of the energy of light as being packaged up into small pieces with a particular energy. Although these "pieces" are not really little pieces or particles, it is just easy to think of them that way because of the way they interact with things such as electrons. The energy of these photons is proportional to the frequency of the light, and the proportionality constant is called Planck's constant. Thus, the energy is given by Ephoton=hν where E is the energy, νν is the frequency and h is Planck's constant. It is useful to note that Planck's constant is an incredibly small number h=6.626×10−34Js So even very high frequency EM radiation like x-rays have photons that are tiny amounts of energy. However, everything is relative -- tiny to you and me can be enormous relative to the energy of an electron. Why do we think of the energy as being packaged like this? Because we find that the interactions of light and electrons consist of one photon for each electron. So, if we have bigger and brighter light sources, they have more total energy. But, the energy per photon is only determined by the frequency and is the only value that matters for the electron. The photoelectric effect was key to the development of the idea of a "photon" or the relationship of the energy of light to its frequency. The photoelectric effect is simply the effect that sometimes when you shine light on a metal, electrons are ejected. There are several key findings that we can investigate: 1. Unless light of sufficient frequency is used, then no electrons are ejected. That is there is a threshold below which no matter how intense the light source is, no electrons leave the metal. 2. If you are using light of a sufficient frequency, then as the light source is increased in intensity (brightness), the number of electrons ejected increases. 3. As the frequency is increased above the threshold, the velocity of the ejected electrons increases. 4. From this we can conclude that energy is proportional to frequency, and that the proportionality constant is Planck's constant, h E=hν We can also examine the relationship between the kinetic energy (Ek) of the electron and the frequency of the light used in the experiment. The maximum kinetic energy of the electron is the energy of the photon minus the threshold energy. This threshold energy we call the "work function" and we give it the symbol Φ.

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