Chemistry Exam 3 (Chapter 6)
Heisenberg Uncertainty Principle
(Δx)(Δpx) must be greater than or equal to ℏ/2 •Δx = uncertainty in the position of the particle •Δpx = uncertainty in the momentum of the particle in the x direction.
Speed of light (c)
2.998 × 10^8 m/s
Wave definition
A wave is an oscillation or periodic movement that can transport energy from one point in space to another. Common examples of waves are all around us. Shaking the end of a rope transfers energy from your hand to the other end of the rope, dropping a pebble into a pond causes waves to ripple outward along the water's surface, and the expansion of air that accompanies a lightning strike generates sound waves (thunder) that can travel outward for several miles. In each of these cases, kinetic energy is transferred through matter (the rope, water, or air) while the matter remains essentially in place
What characterizes waves?
All waves, including forms of electromagnetic radiation, are characterized by, a wavelength (denoted by λ, the lowercase Greek letter lambda), a frequency (denoted by ν, the lowercase Greek letter nu), and an amplitude.
Impact of scientist work with light
Although there were a few physical phenomena that could not be explained within this framework, scientists at that time were so confident of the overall soundness of this framework that they viewed these aberrations as puzzling paradoxes that would ultimately be resolved somehow within this framework. As we shall see, these paradoxes led to a contemporary framework that intimately connects particles and waves at a fundamental level called wave-particle duality, which has superseded the classical view
Example of standing waves
An example of two-dimensional standing waves is shown in Figure 6.8, which shows the vibrational patterns on a flat surface. Although the vibrational amplitudes cannot be seen like they could in the vibrating string, the nodes have been made visible by sprinkling the drum surface with a powder that collects on the areas of the surface that have minimal displacement. For one-dimensional standing waves, the nodes were points on the line, but for two-dimensional standing waves, the nodes are lines on the surface (for three-dimensional standing waves, the nodes are two-dimensional surfaces within the three-dimensional volume).
Bohr Model
Bohr's expression for the quantized energies: En(subscript)=−k/n^2,n=1,2,3,... •k is a constant comprising fundamental constants such as the electron mass and charge and Planck's constant.
Scientists in the 19th century with light
Early in the nineteenth century, Thomas Young demonstrated that light passing through narrow, closely spaced slits produced interference patterns that could not be explained in terms of Newtonian particles but could be easily explained in terms of waves. Later in the nineteenth century, after James Clerk Maxwell developed his theory of electromagnetic radiation and showed that light was the visible part of a vast spectrum of electromagnetic waves, the particle view of light became thoroughly discredited. By the end of the nineteenth century, scientists viewed the physical universe as roughly comprising two separate domains: matter composed of particles moving according to Newton's laws of motion, and electromagnetic radiation consisting of waves governed by Maxwell's equations. Today, these domains are referred to as classical mechanics and classical electrodynamics (or classical electromagnetism).
Bohr Model
Electrons move to higher energy as light is absorbed and they move to lower energy as light is emitted
The Schrödinger Equation
In the most general form, the Schrödinger equation can be written as: •Hψ = Eψ • Ĥ is the Hamiltonian operator, a set of mathematical operations representing the total energy of the quantum particle. • ψ is the wavefunction of the particle. • E is the total energy of the particle.
Isaac Newton and his work with light
In the seventeenth century, Isaac Newton performed experiments with lenses and prisms and was able to demonstrate that white light consists of the individual colors of the rainbow combined together. Newton explained his optics findings in terms of a "corpuscular" view of light, in which light was composed of streams of extremely tiny particles travelling at high speeds according to Newton's laws of motion.
Bohr Model
Inserting the expression for the orbit energies into the equation for DE gives 1λ=k/hc(1/n1^2-1/n2^2) which is identical to the Rydberg equation in which R∞=k/hc.
Wavelengths in real life
Many valuable technologies operate in the radio (3 kHz-300 GHz) frequency region of the electromagnetic spectrum. At the low frequency (low energy, long wavelength) end of this region are AM (amplitude modulation) radio signals (540-2830 kHz) that can travel long distances. FM (frequency modulation) radio signals are used at higher frequencies (87.5-108.0 MHz). In AM radio, the information is transmitted by varying the amplitude of the wave (Figure 6.5). In FM radio, by contrast, the amplitude is constant and the instantaneous frequency varies Other technologies also operate in the radio-wave portion of the electromagnetic spectrum. For example, 4G cellular telephone signals are approximately 880 MHz, while Global Positioning System (GPS) signals operate at 1.228 and 1.575 GHz, local area wireless technology (Wi-Fi) networks operate at 2.4 to 5 GHz, and highway toll sensors operate at 5.8 GHz. The frequencies associated with these applications are convenient because such waves tend not to be absorbed much by common building materials
Other scientists in the 17th century
Others in the seventeenth century, such as Christiaan Huygens, had shown that optical phenomena such as reflection and refraction could be equally well explained in terms of light as waves travelling at high speed through a medium called "luminiferous aether" that was thought to permeate all space
Predicting Electron Configurations of Ions
Predicting Electron Configurations of Ions •What is the electron configuration of: (a) Na+ (b) P3- (c) Al2+ (d) Fe2+ (e) Sm3+ •First, write out the electron configuration for the parent atom, and then add or remove the appropriate number of electrons from the correct orbital(s).
Amplitude
The amplitude corresponds to the magnitude of the wave's displacement and so, in Figure 6.2, this corresponds to one-half the height between the peaks and troughs. The amplitude is related to the intensity of the wave, which for light is the brightness, and for sound is the loudness
Electromagnetic radiation in a vacuum
The product of a wave's wavelength (λ) and its frequency (ν), λν, is the speed of the wave. Thus, for electromagnetic radiation in a vacuum, speed is equal to the fundamental constant, c: c=2.998×10^8ms^−1=λν
Visible light role in chemistry
Visible light and other forms of electromagnetic radiation play important roles in chemistry, since they can be used to infer the energies of electrons within atoms and molecules. Much of modern technology is based on electromagnetic radiation. For example, radio waves from a mobile phone, X-rays used by dentists, the energy used to cook food in your microwave, the radiant heat from red-hot objects, and the light from your television screen are forms of electromagnetic radiation that all exhibit wavelike behavior
Relationship between wavelength and frequency
Wavelength and frequency are inversely proportional: As the wavelength increases, the frequency decreases. Each of the various colors of visible light has specific frequencies and wavelengths associated with them, and you can see that visible light makes up only a small portion of the electromagnetic spectrum. Because the technologies developed to work in various parts of the electromagnetic spectrum are different, for reasons of convenience and historical legacies, different units are typically used for different parts of the spectrum. For example, radio waves are usually specified as frequencies (typically in units of MHz), while the visible region is usually specified in wavelengths (typically in units of nm or angstroms)
Waves and matter
Waves need not be restricted to travel through matter. As Maxwell showed, electromagnetic waves consist of an electric field oscillating in step with a perpendicular magnetic field, both of which are perpendicular to the direction of travel. These waves can travel through a vacuum at a constant speed of 2.998 × 10^8 m/s, the speed of light (denoted by c)
wavelength
denoted by λ the wavelength is the distance between two consecutive peaks or troughs in a wave (measured in meters in the SI system) Electromagnetic waves have wavelengths that fall within an enormous range-wavelengths of kilometers (103 m) to picometers (10−12 m) have been observed
frequency
denoted by ν The frequency is the number of wave cycles that pass a specified point in space in a specified amount of time (in the SI system, this is measured in seconds). A cycle corresponds to one complete wavelength. The unit for frequency, expressed as cycles per second [s^−1], is the hertz (Hz). Common multiples of this unit are megahertz, (1 MHz = 1 × 10^6 Hz) and gigahertz (1 GHz = 1 × 10^9 Hz)
magnetic quantum number (name)
symbol: ml (subscript) allowed values: - l ≤ ml ≤ l physical meaning: orientation of the orbital
spin quantum number (name)
symbol: ms (subscript) allowed values: 1/2,−1/2 physical meaning: direction of the intrinsic quantum "spinning of electron"
Principle quantum number (name)
symbol: n allowed values: 1,2,3,4,... physical meaning: shell, the general region for the value of energy for an electron in the orbital
angular momentum quantum number (name)
symbol: ℓ allowed values: 0 ≤ l ≤ n - 1 physical meaning: subshell, the shape of the orbital
De Broglie Wavelength
λ= h/mv = h/p •h = Planck's constant •m = particle mass •v = particle velocity •p = particle momentum •l = de Broglie wavelength •The de Broglie wavelength is a characteristic of particles, not electromagnetic radiation.
Photoelectric Effect
• According to classical wave theory, a wave's energy depends on its intensity (which depends on its amplitude), not its frequency. •Albert Einstein was able to resolve the paradox by incorporating Planck's quantization findings into the particle view of light (Nobel prize). •Light striking the metal surface should not be viewed as a wave, but instead as a stream of particles (later called photons).
Atomic Model Prior to Bohr
• Atoms consisted of tiny dense nuclei surrounded by lighter and even tinier electrons continually moving about the nucleus. •This classical mechanical description of the atom is incomplete, since an electron moving in an elliptical orbit would be accelerating. •According to classical electromagnetism, the electron should continuously emit electromagnetic radiation. •This loss in orbital energy should result in the electron's orbit getting continually smaller until it spirals into the nucleus. •Niels Bohr attempted to resolve the atomic paradox by ignoring classical electromagnetism's prediction that the orbiting electron in hydrogen would continuously emit light.
The Quantum-Mechanical Model of an Atom
•A few years later, Max Born: •Electrons are still particles, and so the waves represented by ψ are not physical waves but, instead, are complex probability amplitudes. •The square of the magnitude of a wavefunction ∣ψ∣^2 describes the probability of the quantum particle being present near a certain location in space. •Wavefunctions can be used to determine the distribution of the electron's density with respect to the nucleus in an atom.
Abbreviated Electron Configurations
•Abbreviated electron configurations emphasize the similarity of the configurations of elements in the same group. • Group 1: •Li: [He]2s1 •Na: [Ne]3s1 •K: [Ar]4s1 • Group 2: •Be: [He]2s2 •Mg: [Ne]3s2 •Ca: [Ar]4s2
Electron Configurations of Ions
•An anion (negatively charged ion) forms when one or more electrons are added to a parent atom. •The electrons are added in the order predicted by the Aufbau principle.
The Pauli Exclusion Principle
•An electron in an atom is completely described by four quantum numbers: n, l, ml, and ms. •Wolfgang Pauli, an Austrian physicist, formulated a general principle. •The Pauli Exclusion Principle: No two electrons in the same atom can have exactly the same set of all four quantum numbers.
Blackbody Radiation and the Ultraviolet Catastrophe
•Around 1900, Max Planck derived a theoretical expression for blackbody radiation that fit the experimental observations exactly (within experimental error). •Planck developed his theoretical treatment on the premise that the atoms composing the oven vibrated. •The atoms vibrated at increasing frequencies (or decreasing wavelengths), as the temperature increased.
Electron Configurations
•Beginning with the transition metal scandium (atomic number 21), additional electrons are added successively to the 3d subshell. • d subshells are filled to capacity with 10 electrons. •For l = 2 [d orbitals], there are 2l + 1 = 5 values of ml. •Five d orbitals have a combined capacity of 10 electrons. • After the 3d subshell is filled, electrons are next added to the 4p subshell.
De Broglie Wavelength
•Bohr had postulated the electron as being a particle orbiting the nucleus in quantized orbits. •De Broglie argued that Bohr's assumption of quantization can be explained if the electron is considered a circular standing wave. •Only an integer number of wavelengths can fit exactly within the orbit.
Bohr Model
•Bohr incorporated the following into the classical mechanics description of the atom. •Planck's ideas of quantization. •Einstein's finding that light consists of photons whose energy is proportional to their frequency. •Bohr assumed that the electron orbiting the nucleus would not normally emit any radiation. •Rather the electron emits or absorbs a photon if it moved to a different orbit.
Limitations of the Bohr Model
•Bohr was unable to extend his theory to the next simplest atom, He, which only has two electrons. •It is does not account for electron-electron interactions in atoms with more than one electron. •Bohr's model was severely flawed, since it was still based on the classical mechanics notion of precise orbits.
Moving Forward
•Bohr won a Nobel Prize in Physics for his contributions. •It became clear to most physicists at that time that the classical theories that worked so well in the macroscopic world were fundamentally flawed and could not be extended down into the microscopic domain of atoms and molecules. •A proper model of quantum mechanics was later developed to supersede classical mechanics.
Wave Nature of Electrons
•C. J. Davisson and L. H. Germer, demonstrated experimentally that electrons can exhibit wavelike behavior. • They showed that electrons travelling through a regular atomic pattern in a crystal produced an interference pattern. • The interference pattern is strikingly similar to the interference patterns for light.
Line Spectra
•Each emission line consists of a single wavelength of light. •This implies that the light emitted by a gas consists of a set of discrete energies. •The origin of discrete spectra in atoms and molecules was extremely puzzling to scientists. •According to classical electromagnetic theory, only continuous spectra should be observed.
The Pauli Exclusion Principle
•Electrons can share the same orbital (the same set of quantum numbers n, l, and ml). •But only if they have different ms numbers. •Since the spin quantum number can only have two values (± ½), no more than two electrons can occupy the same orbital. •If two electrons are located in the same orbital, they must have opposite spins.
Understanding Quantum Theory of Electrons in Atoms
•Electrons in atoms can exist only in discrete energy levels but not between them. • The energy of an electron in an atom is quantized. • The energy can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels.
Photoelectric Effect (E=hv)
•Electrons were ejected when hit by photons having sufficient energy (a frequency greater than the threshold). •The greater the frequency, the greater the kinetic energy imparted to the escaping electrons by the collisions. •Somehow, at a deep fundamental level still not fully understood, light is both wavelike and particle-like. This is known as wave-particle duality. -Photons with low frequencies do not have enough energy to cause electrons to be ejected via the photoelectric effect. For any frequency of light above the threshold frequency, the kinetic energy of ejected electron will increase linearly with the energy of the incoming photon.
The Quantum-Mechanical Model of an Atom
•Erwin Schrödinger extended de Broglie's work by incorporating the de Broglie relation into a wave equation. •Today this equation is know as the Schrödinger equation. •Schrödinger properly thought of the electron in terms of a three-dimensional stationary wave, or wavefunction, represented by the Greek letter psi, ψ.
De Broglie Wavelength
•For a circular orbit of radius r, the circumference is 2πr, and so de Broglie's condition is: 2πr = nλ, n=1,2,3,...
Electron Configurations
•For the two periods of inner transition metals, lanthanum (La) through lutetium (Lu) and actinium (Ac) through lawrencium (Lr). •Electrons are added to an f subshell. •For l = 3 [f orbitals], there are 2l + 1 = 7 values of ml. •Seven f orbitals have a combined capacity of 14 electrons.
Orbital Relative Energies
•However, in atoms with more than one electron, this degeneracy is eliminated by electron-electron interactions. •Orbitals that belong to different subshells have different energies. •Orbitals within the same subshell are still degenerate and have the same energy.
Electron Configurations and Orbital Diagrams
•Hund's rule: the lowest-energy configuration for an atom with electrons within a set of degenerate orbitals is that having the maximum number of unpaired electrons.
Magnetic Quantum Number
•If an orbital has an angular momentum (ℓ ≠ 0), then this orbital can point in different directions. •The magnetic quantum number, ml, specifies the orientation of the orbital in space.
Louis de Broglie
•If electromagnetic radiation can have particle-like character, can electrons and other submicroscopic particles exhibit wavelike character? •In 1925 Louis de Broglie extended the wave-particle duality of light that Einstein used to resolve the photoelectric-effect paradox to material particles.
Line Spectra
•In 1885, Johann Balmer was able to derive an empirical equation. •The equation related the four visible wavelengths of light emitted by hydrogen atoms to whole integers 1/λ=k(1/4 - 1/n^2), n=3,4,5,6 •Other discrete lines for the hydrogen atom were found in the UV and IR regions
Orbital Energies and Atomic Structure
•In any atom with two or more electrons, the repulsion between the electrons makes energies of subshells with different values of l differ. • The energy of the orbitals increases within a shell in the order s < p < d < f. • Electrons in atoms tend to fill low-energy orbitals first.
Line Spectra
•In contrast to continuous spectra, light can also occur as discrete or line spectra having very narrow line widths interspersed throughout the spectral regions. •Exciting a gas at low partial pressure using an electrical current, or heating it, will produce line spectra. •Each element displays its own characteristic set of lines. -Neon signs operate by exciting a gas at low partial pressure using an electrical current. This sign show the elaborate artistic effects that can be achieved.
•What is the energy (in joules) and the wavelength (in meters) of the line in the spectrum of hydrogen that represents the movement of an electron from Bohr orbit with n = 4 to the orbit with n = 6? In what part of the electromagnetic spectrum do we find this radiation?
•In this case, the electron starts out with n = 4, so n1 = 4. It comes to rest in the n = 6 orbit, so n2 = 6. The difference in energy between the two states is given by this expression where k = 2.179 × 10-18 J: ΔE=k(1/n1^2-1/n2^2) ΔE=2.179*10^-19 J (1/4^2 - 1/6^2) = 7.566 *10^-20 J •This energy difference is positive, indicating a photon enters the system (is absorbed) to excite the electron from the n = 4 orbit up to the n = 6 orbit. The wavelength of a photon with this energy is found by the expression: ΔE=hc/λ Rearrangement gives: λ=hc/ΔE = (6.626 *10 ^-34 J*s)(2.998*10^8 m*s^-1) / 7.566 * 10^-20 J = 2.626 * 10^-6 m •This wavelength is found in the infrared portion of the electromagnetic spectrum
Electron Configurations and the Periodic Table
•Inner transition elements are metallic elements in which the last electron added occupies an f orbital. •The valence electrons (those added after the last noble gas configuration) in these elements include the ns, (n - 2)f, and if present the (n - 1)d subshells. •Example: Promethium (Pm): [Ar]6s24f5 •Pm has seven valence electrons (6s2 and 4f5).
Angular Momentum Quantum Number
•Instead of using numbers, the angular momentum quantum number is often designated by letters. •For ℓ = 0: s orbital •For ℓ = 1: p orbital •For ℓ = 2: d orbital •For ℓ = 3: f orbital
Interference Patterns of waves
•Interference patterns arise when light passes through narrow slits closely spaced about a wavelength apart. •The dark regions correspond to regions where the peaks for the wave from one slit happen to coincide with the troughs for the wave from the other slit (destructive interference). •The brightest regions correspond to the regions where the peaks for the two waves (or their two troughs) happen to coincide (constructive interference). •Such interference patterns cannot be explained by particles moving according to the laws of classical mechanics.
Line Spectra
•Johannes Rydberg generalized Balmer's work. •He developed an empirical formula that predicted all of hydrogen's emission lines 1/λ=R∞(1/n1^2-1/n2^2) •n1 and n2 are integers •n1 < n2 •R∞ is the Rydberg constant (1.097 × 107 m−1)
Electron Configurations and the Periodic Table
•Main group elements or representative elements are those in which the last electron added enters an s or a p orbital in the outermost shell. •The valence electrons for main group elements are those with the highest n level. •Example: Gallium (Ga): [Ar]4s23d104p1 •Ga has three valence electrons (4s2 and 4p1). •The completely filled 3d orbitals count as core, not valence electrons.
Standing Waves
•Not all waves are travelling waves. •Standing waves (also known as stationary waves) remain constrained within some region of space. •Standing waves play an important role in our understanding of the electronic structure of atoms and molecules. •A vibrating string that is held fixed at its two end points is an example of a one-dimensional standing wave.
Bohr Model
•One of the fundamental laws of physics is that matter is most stable when it has the lowest possible energy. •When the electron is in this lowest energy orbit (n = 1), the atom is said to be in its ground state. •If the atom receives energy from an outside source, it is possible for the electron to move to an orbit with a higher n value (excited state), which has a higher energy.
Standing Waves
•Only those waves having an integer number, n, of half-wavelengths between the end points can form. • A system with fixed end points restricts the number and type of possible waveforms. • This is an example of b. •Only discrete values from a more general set of continuous values are observed. •Harmonic waves (those waves displaying more than one-half wavelength) all have one or more points between the two end points that are not in motion. • These special points are nodes.
Electron Configurations and Orbital Diagrams
•Orbital diagrams are pictorial representations of the electron configuration, showing the individual orbitals and the pairing arrangement of electrons. •An upward arrow represents and electron with ms = + ½. • A downward arrow represents and electron with ms = - ½.
Development of Quantum Theory
•Particles and waves are very different phenomena in the macroscopic realm. •By the 1920s it became increasingly clear that very small pieces of matter follow a different set of rules from those we observe for large objects. •The unquestionable separation of waves and particles was no longer the case for the microscopic world.
Planck's constant
•Planck had to assume that the vibrating atoms required quantized energies, which he was unable to explain E=nhv n=1,2,3,... •The quantity h is a constant now known as Planck's constant h=6.626 x 10^-34 J*s
Electron Configurations of Ions
•Recall, ions are formed when atoms gain or lose electrons. •A cation (positively charged ion) forms when one or more electrons are removed from an atom. •For main group elements, the electrons that were added last are the first electrons removed. •For transition metals and inner transition metals, the highest ns electrons are lost first, and then the (n - 1)d or (n - 2)f electrons are removed.
Quantum Mechanics
•Schrödinger's work, as well as that of Heisenberg and many other scientists following in their footsteps, is generally referred to as quantum mechanics.
Line Spectra
•Scientists in the late nineteenth century struggled with understanding the light emitted from atoms and molecules. •When solids, liquids, or condensed gases are heated sufficiently, they radiate some of the excess energy as light. •Photons produced in this manner have a range of energies, and thereby produce a continuous spectrum in which an unbroken series of wavelengths is present.
Blackbody Radiation and the Ultraviolet Catastrophe
•Sunlight consists of a range of broadly distributed wavelengths that form a continuous spectrum. • A blackbody is a convenient, ideal emitter that approximates the behavior of many materials when heated. • A good approximation of a blackbody that can be used to observe blackbody radiation: •A metal oven that can be heated to very high temperatures.
The Bohr Model
•The Bohr model does introduce several important features of all models used to describe the distribution of electrons in an atom. •The energies of electrons (energy levels) in an atom are quantized, described by quantum numbers: integer numbers having only specific allowed values and used to characterize the arrangement of electrons in an atom. •An electron's energy increases with increasing distance from the nucleus. •The discrete energies (lines) in the spectra of the elements result from quantized electronic energies.
Angular Momentum Quantum Number
•The angular momentum quantum number (ℓ) is an integer that defines the shape of the orbital. •ℓ takes on the values, ℓ = 0, 1, 2, ..., n - 1. •For an orbital with n = 1; ℓ = 0 •For an orbital with n = 2; ℓ = 0 and 1 •Orbitals with the same value of ℓ form a subshell.
Electron Configuration
•The arrangement of electrons in the orbitals of an atom is called the electron configuration of the atom. •An electron configuration consists of symbols that contain three pieces of information: 1)The principal quantum shell, n. 2)The letter that designates the orbital type (l). 3)A superscript number that designates the number of electrons in that particular subshell.
Valence and Core Electrons
•The electrons occupying the orbital(s) in the outermost shell (highest value of n) are called valence electrons. • The electrons occupying the inner shell orbitals are called core electrons. • The core electrons represent noble gas electron configurations. • Electron configurations can be expressed in an abbreviated format by writing the noble gas that matches the core electron configuration, along with the valence electrons. A core-abbreviated electron configuration (right) replaces the core electrons with the noble gas symbol whose configuration matches the core electron configuration of the other element.
The Bohr Model
•The emission spectra for hydrogen was eventually explained by Niels Bohr in 1913. •Bohr's work convinced scientists to abandon classical physics and spurred the development of modern quantum mechanics.
Bohr Model
•The energy absorbed or emitted would reflect differences in the orbital energies according to this equation: ∣ΔE∣=∣Ef−Ei∣=hν=hcλ •Ei and Ef are the initial and final orbital energies. • Bohr assumed only discrete values for the energy difference (DE) were possible.
Principle Quantum Number
•The energy levels are labeled with an n value, where n = 1, 2, 3... • Generally speaking, the energy of an electron in an atom is greater for larger values of n. • n is referred to as the principal quantum number or shell number. • The principal quantum number defines the location of the energy level and is similar in concept to the Bohr model. -Different shells are numbered by principle quantum numbers
Electronic Structure of Atoms (Electron Configurations)
•The energy of atomic orbitals increases as the principal quantum number, n, increases. •As the principal quantum number, n, increases, the size of the orbital increases and the electrons spend more time farther from the nucleus. •Thus, the attraction to the nucleus is weaker and the energy associated with the orbital is higher (less stabilized).
Photoelectric Effect
•The energy of the photons depended on their frequency. • According to Planck's formula: E=hv E=hc/λ V=c/λ
Spin Quantum Number
•The first three quantum numbers, n, l, and ml(subscript), define the region of space where an electron is most likely to be located. • The electron spin is a different kind of property. • Electron spin describes an intrinsic electron "rotation" or "spinning". • An electron can only "spin" in one of two quantized states.
Blackbody Radiation and the Ultraviolet Catastrophe
•The maxima in the blackbody curves, λmax, shift to shorter wavelengths as the temperature increases. • Theoretical expressions as functions of temperature fit the observed experimental blackbody curves well at longer wavelengths. • But there were significant discrepancies at shorter wavelengths. • This became known as the "ultraviolet catastrophe".
Photoelectric Effect
•The next paradox in the classical theory to be resolved concerned the photoelectric effect. •It had been observed that electrons could be ejected from the surface of a metal. •The light had to have a frequency greater than some threshold frequency. •Surprisingly, the kinetic energy of the ejected electrons did not depend on the brightness of the light, but increased with increasing frequency of the light.
Radial Nodes
•The number of radial nodes in an orbital is n - ℓ - 1. •For a 1s orbital (n = 1, ℓ = 0) •The number of nodes = 1 - 0 - 1 = 0 •For a 2s orbital (n = 2, ℓ = 0) •The number of nodes = 2 - 0 - 1 = 1 •For a 3s orbital (n = 3, ℓ = 0) •The number of nodes = 3 - 0 - 1 = 2
Electron Configurations and the Periodic Table
•The periodic table arranges atoms so that elements with the same chemical and physical properties are in the same group. •Elements in the same group have similar valence electron configurations. •Valence electrons play the most important role in chemical reactions. •This describes why elements in the same group have similar chemical reactivity.
Atomic Orbitals
•The principal quantum number is one of three quantum numbers used to characterize an orbital. •An atomic orbital is a general region in an atom that an electron is most probable to reside. •An atomic orbital is distinct from an orbit. •The quantum mechanical model specifies the probability of finding an electron in a three-dimensional space around the nucleus.
Orbital Shapes
•The s subshell has a spherical shape. •The p subshell has a dumbbell shape. •The d and f orbitals are more complex. •These shapes represent the three-dimensional regions where the electron is likely to be found.
Spin Quantum Number
•The spin quantum number, ms (subscript), describes the two possible states. •ms = +½ or -½ •Any electron, regardless of the atomic orbital it is located in, can only have one of those two values of the spin quantum number.
Magnetic Quantum Number
•The value of mℓ depends on the value of ℓ. •mℓ = -ℓ ... -1, 0, +1, ... + ℓ •There are 2ℓ + 1 orbitals with the same ℓ value. •One s-orbital for ℓ = 0 •Three p-orbitals for ℓ = 1 •Five d-orbitals for ℓ = 2 •Seven f-orbitals for ℓ = 3
Radial Nodes
•There are certain distances from the nucleus at which the probability density of finding an electron located in a particular orbital is zero. •The value of the wavefunction ψ is zero at this distance for this orbital. •Such a value of radius r is called a radial node.
Electron Configuration Exceptions
•There are some exceptions to the order of filling subshells. • Examples: Cu and Cr •There is stability associated with a half-filled or fully filled d subshell. •An electron shifts from the 4s into the 3d subshell to achieve this stability. •Cu: •Expect: [Ar]4s23d9 •Actual: [Ar]4s13d10 •Cr •Expect: [Ar]4s23d4 •Actual: [Ar]4s13d5
Wave-Particle Duality
•Thus, it appears that while electrons are small localized particles, their motion does not follow the equations of motion implied by classical mechanics. •Instead some type of wave equation governs a probability distribution for an electron's motion. Thus the wave-particle duality first observed with photons is actually a fundamental behavior intrinsic to all quantum particles
The Aufbau Principle
•To determine the electron configuration for an atom we add a number of electrons equal to its atomic number. • Beginning with hydrogen, and continuing across the periods of the periodic table, we add one electron to the subshell of lowest available energy. • This procedure is called the Aufbau principle, from the German word Aufbau ("to build up"). • When writing electron configurations and orbital diagrams keep in mind the Pauli Exclusion principle.
Electron Configurations and the Periodic Table
•Transition elements or transition metals are metallic elements in which the last electron added enters a d orbital. •The valence electrons (those added after the last noble gas configuration) in these elements include the ns and (n - 1)d electrons. •Example: Vanadium (V): [Ar]4s23d3 •V has five valence electrons (4s2 and 3d3).
Two dimensional standing waves
•Vibrational patterns on a flat surface are examples of two-dimensional standing waves. • The nodes can be made visible by sprinkling the drum surface with a powder that collects on the areas of the surface that have minimal displacement. • For one-dimensional standing waves, the nodes were points on the line. • For two-dimensional standing waves, the nodes are lines on the surface.
When we see light from a neon sign, we are observing radiation from excited neon atoms. If this radiation has a wavelength of 640 nm, what is the energy of the photon being emitted?
•We use the form of Planck's equation that includes the wavelength, λ, and convert units of nanometers to meters so that the units of λ and c are the same. E=hc/λ (6.626 * 10^-34 J*s)(2.998 * 10^8 m*s^-1) / (640 nm)(10^-9 m/1 nm)= E=3.10*10^-19 J
Heisenberg Uncertainty Principle
•Werner Heisenberg determined that there is a fundamental limit to how accurately one can measure both a particle's position and its momentum simultaneously. •The more accurately we measure the momentum of a particle, the less accurately we can determine its position at that time, and vice versa. •Heisenberg Uncertainty Principle: It is fundamentally impossible to determine simultaneously and exactly both the momentum and the position of a particle.
Orbital Relative Energies
•When referring to an orbital, usually both the n value and the letter designation for ℓ are reported. •Examples: 2s, 3p •In the case of a hydrogen atom, energies of all the orbitals with the same n are the same. •This is called a degeneracy, and the energy levels with the same principal quantum number, n, are called degenerate energy levels.
Bohr Model
•When the atom absorbs energy as a photon, the electron moves from an orbit with a lower n to a higher n. • When an electron falls from an orbit with a higher n to a lower n, the atoms emits energy as a photon. • Since DE can only be discrete values, the photon absorbed or emitted can only have a wavelength with a discrete value (not continuous). • This explained the line spectra for the hydrogen atom.
