EXAM 1 CHEM
Introduction
- Absolutely small (or quantum) electrons behave differently than large (or macroscopic) objects. - Absolutely small particles like electrons can be in 2 different states at the same time. - In the quantum world, a particle can be in a state of emitting a particle and not emitting a particle simultaneously; in the macroscopic world, only one can occur at a time. - However, when we measure a quantum particle, the act of measurement pushes the particle into one state or the other. - Quantum-mechanical model: a model that explains the strange behavior of electrons, describing electrons as they exist within atoms and how those electrons determine the chemical and physical properties of elements.
The uncertainty principle
- Any experiment designed to observe an electron as it travels through slits results in the detection of an electron "particle" traveling through a single slit and no interference pattern. An unobserved electron can occupy 2 different states, but the act of observation forces the electron into one state or the other. Thus, observing an electron as it goes through both slits causes it to go through only one slit. - A laser beam can be used to determine which slit an electron will pass through. It makes the interface pattern absent and causes the electrons to act like particles instead of waves. - We can never see both the interference pattern and simultaneously determine which hole the electron goes through. In quantum mechanics, the observation of an event affects its outcome. - We cannot simultaneously observe the wave nature and particle nature of the electron. Thus, the 2 are complementary properties. Complementary properties exclude one another: the more we know about one, the less we known about the other. - An electron's velocity is related to its wave nature; an electron's position is related to its particle nature (particles have well-defined positions; waves do not). Since we can't measure wave nature and particle nature, then we can't simultaneously measure an electron's position and velocity: Δx * mΔv ≥ (h/4π). This is Heisenberg's uncertainty principle. - Heisenberg's uncertainty principle: the more accurately you know the position of an electron (the smaller the Δx), the less accurately you know its velocity (bigger Δv) and vice versa. - The product of the uncertainties in both the position and speed of a particle is inversely proportional to its mass: Δx * Δv ≥ (h/4π)(1/m).
Atomic spectroscopy and the Bohr model
- Atomic spectroscopy: the study of electromagnetic radiation absorbed and emitted by atoms. When an atoms absorbs energy (in the form of heat, light, or electricity) it often re-emits that energy as light. Atoms of each element emit light of a characteristic color; the color visible is determined by its wavelength. Light contains several distinct wavelengths that can be separated by passing it through a prism. - Emission spectrum: a series of bright lines that is always the same for a particular element; consists of the same bright lines at the same characteristic wavelengths; can be used to identify an element. - White light has a continuous spectrum (no sudden interruptions in the intensity of the light as a function of wavelength) because it consists of light of all wavelengths. Light from a specific element is not continuous and has bright liens at specific wavelengths with complete darkness in between. - Equation for predicting the wavelength of the hydrogen emission spectrum: 1/λ = R(1/m^2 - 1/n^2), where R is the Rydberg constant (1.097 x 10^7 m^-1). This equation doesn't explain why atomic spectra are discrete, atoms are stable, or why the equation works. - In Bohr's model, electrons travel around the nucleus in circular orbits, with orbits only existing at specific, fixed distances from the nucleus (or have fixed energies). The energy of each Bohr orbit is fixed, or quantized. Bohr called these orbits stationary states (which we now know are a manifestation of the wave nature of the electron). Bohr proposed that no radiation is emitted by an electron orbiting the nucleus in a stationary state (contrary to what classic electromagnetic theory believed), and only when an electron transitions from one stationary state to another is radiation emitted or absorbed. Therefore, a spectral line in the emission spectrum is released when an electron falls from one stable orbit (or stationary state) to another of lower energy. - The electron is never observed between states; it's observed only in one state or another. The energy emitted when an electron makes a transition from one stationary state to another is the energy difference between the two stationary states. Transitions between stationary states that are closer together produce light of lower energy (longer wavelength). Bohr's model was ultimately replaced by a more complete quantum-mechanical model. - Absorption: an electron transitions from a lower energy level to a higher one. Emission: an electron transitions from a higher energy level to a lower one.
Indeterminacy
- Classical physics described particles as moving in a trajectory (path) determined by a particle's velocity (the speed and direction of its travel), its position, and the forces acting on it. In classical physics, both position and velocity are needed to determine trajectory. - Newton's laws of motion are deterministic i.e. the present determines the future. - Since we can't know the position and velocity of an electron simultaneously, we can't know its trajectory. In quantum mechanics, trajectories are replaced with probability distribution maps, which are statistical maps that show where an electron is likely to be found under a given set of conditions. In these maps, darker shading indicates great probability of electrons being located in that spot. - An electron, even under identical conditions, will be found in a different place every time; this is called indeterminacy. Unlike a baseball for example, whose future path is determined by its position and velocity, the future path of an electron is indeterminate and can only be described statistically. - Quantum-mechanical electron orbitals are essentially probability distribution maps for electrons as they exist within atoms.
Atomic spectroscopy explained
- Each wavelength in the emission spectrum of an atom corresponds to an electron transition between quantum-mechanical orbitals. - When an atom absorbs energy, an electron in a lower energy orbital is excited, or promoted to higher energy orbital. However, this excited electron is unstable and the electron quickly falls back (relaxes) to a lower energy orbital; as it does so, it releases of photon of light with an amount of energy exactly equal to the energy difference between the 2 energy levels. The difference in energy between two levels, n-intitial and n-final, is given by ΔE = Efinal - Einitial. Substituting En into the expression for ΔE, we get: ΔE = -2.18 x 10^-18 J x ((1/nf^2) - (1/ni^2)). - An atom transitioning from a higher state to a lower state will have a negative ΔE because it emits energy; an atom transitioning from a lower state to a higher state will have a positive ΔE because it absorbs energy. - ΔEatom = -Ephoton - This energy determine the frequency and wavelength of the photon since E = hc/λ. The wavelength of a photon can be calculated as: λ = hc/E. - Transitions between orbitals that are farther apart in energy produce light that is higher in energy and therefore shorter in wavelength, than transitions between orbitals that are close together. E.g. n= 5 to n = 2 has 434 nm light; n = 4 to n = 2 has 486 nm light; n =3 to n = 2 has 656 light. - Energy differences get closer together with increasing n, so the greater energy difference between n= 3 and n = 2 (in comparison to n= 4 and n = 3) results in a photon emitted with greater energy and therefore shorter wavelength.
Interference and diffraction
- Interference: waves cancel each other out or build each other up spending upon interaction. - Constructive interference: if two waves of equal amplitude are in phase when they interact (they align with overlapping crests), then a wave with twice the amplitude results. Equal path lengths. - Destructive interference: if two waves are completely out of phase when they interact (they align so that the crest from one source overlaps with the trough from the other source), then the waves cancel. Path lengths differ by λ/2. - Diffraction: when a wave encounters an obstacle or slit that's comparable in size to its wavelength, it bends (or diffracts) around it and spreads out after passing through. Particles do not diffract; they simply pass through the opening. - Interference pattern: the diffraction of light coupled with interference, with each slit acting as a new wave sources, creating a series of bright and dark lines. Out of phase waves make dark spots, while in phase waves make bright spots.
Light
- Light has wave-particle duality, similar to electrons. Light is electromagnetic radiation. - Electromagnetic radiation: a type of energy embodied in oscillating electric (a region of space where an electrically charged particle experiences a force) and magnetic (a region of space where a magnetic particle experiences a force) fields. A wave of oscillating electric and magnetic fields that are perpendicular to one another. - Light moves at a speed of 3 x 10^8 m/s. - Amplitude: the vertical height of a wave's crest (or depth of a trough) from the node; determines light intensity i.e. brightness, higher amplitude = greater brightness. - Wavelength (λ): the distance between adjacent crests (or any 2 analogous points) of a wave; measured in meters or nanometers; determines color and energy of wave, shorter wavelength = more energy. - Frequency (ν): the number of cycles (or wave crests) that pass through a stationary point in a give period of time; measured in s^-1 or Hz; directly proportional to the speed at which the wave is traveling, the faster the wave, the greater the frequency; inversely proportional to wavelength. - Electromagnetic spectrum from lowest energy (i.e. longest wavelength, lowest frequency) to highest energy (i.e. shortest wavelength, highest frequency): radio waves, microwaves, infrared waves, visible light (400 nm to 750 nm), ultraviolet waves, x-ray, gamma rays.
The wave nature of matter
- Louis de Broglie proposed the quantum-mechanical theory that replaced Bohr's model. Electrons, which are particles with a known mass, also have a wave nature. This wave nature is seen most clearly in diffraction: if a electron beam is aimed at two closely spaced slits, the interference patter is similar to that observed for light and other waves. - The interference pattern is caused by single electrons interfering with themselves NOT pairs of electrons interfering with each other. A low level of electrons creates the same interference pattern, thus the wave nature of the electron is an inherent property of individual electrons. The unobserved electron passing through the slit is both a wave and particle at the same time, causing it to interfere with itself. - A beam of electrons creates an interference pattern when passed through 2 slits. A beam of particles produces 2 smaller beams of particles when passed through 2 slits. - The electron's wave nature manifests itself in 3 important ways: the de Broglie wavelength, the uncertainty principle, and indeterminacy.
Phases of orbitals & shape of atoms
- Orbitals are 3-D waves. - Phase: the sign of the amplitude of a wave, which can be positive or negative; determines how wave interferes with another wave; the phase of quantum-mechanical orbitals is important to bonding. - Atoms are usually drawn as spheres because most atoms contain many electrons occupying a number of different orbitals. Therefore, the shape of an atom is obtained by superimposing all of its orbitals, making a roughly spherical shape.
The particle nature of light
- Photoelectric effect: the observation that many metals emit electrons when light shines upon them. This effect was thought to be caused by the transfer of energy from the light to an electron in the metal, resulting in the dislodgment of the electron. - According to the classical description of light, only the amplitude of light affects the emission electrons, not the wavelength. Thus, the rate at which electrons leave the metal due to the photoelectric effect increases with increasing intensity of light. However, this theory was not supported by experiments. Low-intensity, high-frequency light was shown to produce electrons without the lag time predicted by the photoelectric effect. - The light used to dislodge electrons in the photoelectric effect has a threshold frequency, below which no electrons are emitted from the metal, no matter how long the light shines on the metal. The increasing intensity of light does NOT change the threshold frequency i.e. low-frequency (long-wavelength) light does not eject electrons from a metal regardless of its intensity or its duration. However, high-frequency (short-wavelenght) light does eject electrons, even if its intensity is low. - Einstein's explanation for the photoelectric effect: light energy comes in packets; the amount of energy (E) in a light packet is dependent on frequency (ν) according to the equation: E = hν. - A packet of light is called a photon or quantum of light. The energy of a photon can be expressed as: E = (hc)/λ. - Classical electromagnetic theory viewed light as a wave with a continuously variable intensity. Einstein saw light as lumpy, with a shower of photons. - The emission of electrons from a metal depends on whether or not a single photon has enough energy (hν) to dislodge a single electron. For an electron bound to a metal with binding energy Φ, the threshold frequency is reached when the energy of the photon is equal to Φ: hv = Φ. Φ is the binding energy of emitted electron. - Low frequency light does NOT eject electrons because no single photon has the minimum energy needed to dislodge the electron. Increasing the intensity has no effect on electron emission. Increasing frequency, which increases the energy of each photon, does. - The excess energy of the photon (beyond what is needed to dislodge the electron) is transferred to the electron in the form of kinetic energy: KE = hν - Φ. Short wavelengths of light (highest energy per photon) produce photoelectrons with the greatest kinetic energy. Long wavelengths, which have the lowest energy per photon, produce no observable photoelectrons. - Wave-particle duality: light appears to behave like a wave, and at other times like a particle.
Quantum numbers
- Principal Quantum Number (n): an integer that determines the overall size and energy of an orbital. Its possible values are n = 1, 2, 3... and so on. For a hydrogen atom, the energy of an electron in an orbital with quantum number n is given by the equation: En = -2.18 x 10^-18 J (1/n^2). The energy is negative because the electron's energy is lowered by its interaction wit the nucleus. 2.18 x 10^-18 J is the Rydberg constant for hydrogen (Rh). Orbitals with higher values of n have greater (less negative) energies. Also, as n increases, the spacing between the energy levels becomes smaller. - Angular Momentum Quantum Number (l): an integer that determines the shape of an orbital. The possible values of l are 0, 1, 2...(n - 1). For a given value of n, l can be any integer (including 0) up to n - 1. Each value of l is assigned a letter: l = 0 (s), l = 1 (p), l =2 (d), and l = 3 (f). - Magnetic Quantum Number (ml): an integer that specifies the orientation of the orbital. The possible values of ml are the integer values (including 0) ranging from -l to +l. E.g. the possible values of ml for l = 2 are: = -2, -1, 0, +1, and +2. - Spin Quantum Number (ms): specifies the orientation of the spin of the electron. Electron spin is a fundamental property of an electron (like its negative charge), and one electron does' have more or less spin than another, they all have the same amount. The orientation of an electron's spin is quantized, with only 2 possibilities: spin up (ms = +1/2) and spin down (ms = -1/2). - Orbitals with the same value of n are said to be in the same principal level (or principal shell). Orbitals with the same value of n and l are said to be in the same sublevel (or subtle). The principal level is specified by n, the sublevel is specified by l (letter value). - The number of sublevels in any level is equal to n, e.g. n = 1 has 1 sublevel, n = 2 has 2 sublevels and so on. - The number of orbital in any sublevel is equal to 2l + 1. E.g. the s sublevel (l = 0) has one orbital, the p sublevel (l = 1) has 3 orbitals, the d sublevel (l = 2) has 5 orbitals, and so on. - The number of orbitals in a level is equal to n^2. E.g. the n = 1 level has one orbital, the n = 2 level has 4 orbitals, the n = 3 level has 9 orbitals, and so on. See page 317. - N = 4 has 4s (1 orbital), 4p (3 orbitals), 4d (5 orbitals), and 4f (7 orbitals) for a total of 4^2 = 16 orbitals.
De Broglie wavelength
- The faster an electron is moving, the higher its kinetic energy and shorter its wavelength. The wavelength of an electron has a mass(m) moving at velocity (v) is given by the de Broglie relation: λ = h/(mv), where h is Planck's constant. - The velocity of a moving electron is related to its wavelength; knowing one is equivalent to knowing the other.
Shapes of atomic orbitals
- The shape of an atomic orbital is primarily determined by l (the angular momentum quantum number). Each value of l is assigned a letter that corresponds to particular orbitals: l = 0 is the s orbital; l = 1 is the p orbital; l = 2 is the d orbital; and l = 3 is the f orbital. - An atomic orbital can be represented by a geometrical shape that encompasses the volume where the electron is likely to be found most frequently. - Radial distribution function: represents the total probability of finding the electron within a thin spherical shell at a distance r from the nucleus: total radial probability (at a given r) = (probability/unit volume) x volume of shell at r. Represents not the probability density at a point r, but total probability at a radius r; probability density has a maximum at the nucleus, while radial distribution has a value of zero at the nucleus. The shape of the radial distribution function is the result of multiplying: (1) the probability destiny function (ψ^2), the probability per unit volume, has a max at the nucleus, and decreases with increasing r; (2) the volume of the thin shell, which is zero at the nucleus and increases with increasing r. As r increases, the volume of the tin spherical shell increases. - Node: a point where the wave function (ψ), and therefore the probability density (ψ^2) and radial distribution, all go through zero; the probability of finding the electron at a node is zero. - S orbitals (l = 0): lowest energy orbital; is spherically symmetrical. - P orbitals (l = 1): has 3 p orbitals (ml = -1, 0, +1); have 2 lobes of electron density on either side of the nucleus and a node located at the nucleus; shaped like dumbbell; 3 p orbitals differ only in their orientation and are orthogonal (mutually perpendicular) to one another. 3p, 4p, 5p, etc. are all similar in shape to 2p orbitals but have additional nodes and are progressively larger in size. - D orbitals (l = 2): contains 5 d orbitals (-2, -1, 0, +1, +2); 4 of these orbitals have a cloverleaf shape, with 4 lobes of electron density around the nucleus and 2 perpendicular nodal planes; 5th orbital has 2 lobes and donut-shaped ring along axis. 4d, 5d, 6d etc. orbitals all similar in shape to the 3d orbitals, but have additional nodes and are progressively larger in size. - F orbitals (l = 3): each principal energy level has 7 orbitals (-3, -2, -1, 0, +1, +2, +3); have more lobes and nodes than d orbitals. - Nodal plane: where the electron probability density is 0.
Quantum mechanics
- We know that velocity is complementary to position, so since velocity is directly related to energy (KE = 0.5mv^2), position and energy are also complementary properties: the more we know about position, the less we know about energy. - Oftentimes we use probability distribution maps that allow us to specify the energy of an electron precisely, but not its location at a give instant. This means the electron's position is described in terms of an orbital, which is a probability distribution map showing where the electron is likely to be found. - Schrodinger equation: Ȟψ = Eψ, where Ȟ is the Hamiltonian operator, a set of mathematical operations that represent the total energy (kinetic and potential) of the electron within the wave function; E is the actual energy of the electron; and ψ is the wave function, a mathematical function that describes the wavelike nature of an electron. A plot of ψ^2 represents an orbital. - ψ^2 represents probability density, the probability (per unit volume) of finding the electron at a point in space. ψ^2 = probability density = (probability/unit volume). The magnitude of ψ^2 is proportional to the density of the dots. High dot density near the nucleus (center of the plot) indicates a higher probability density for the electron there. - Between measurements, an electron has no single location; only when it is measured does it become localized to one spot.
Solutions to the Schrodinger equation
- When the Schrodinger equation is solved, it yields many solutions (many possible wave functions). For the sake of simplicity, we will look at graphical representations of the orbitals that correspond to the wave functions. - Each orbital is specified by 2 interrelated quantum numbers: n, l, and ml. These quantum numbers all have integer values. A fourth quantum number, ms, specifies the orientation of the spin of the electron.
Formulas
- ν = c/λ where c = 3 x 10^8 m/s. ν(Hz), λ(m). - 1 nm = 10^-9 m - E = hν, where h = 6.626 x 10^-34 J*s = Planck's constant. E(J), h(J*s), ν (Hz). - E = (hc)/λ where c = 3 x 10^8 m/s, h = 6.626 x 10^-34 J*s. This is the equation for energy of a photon. - (Energy of pulse) / (Energy of photon) = number of photons. Energy of pulse = Given Joules of miliJoules of energy. - 1 J = 1 (kg x m^2) / (s^2) - Threshold frequency condition: hν = Φ. - Kinetic energy of a photon: KE = hν - Φ - de Broglie relation: λ = h/(mv), where h is Planck's constant (6.626 X 10^-34 J x s), m (kg), and v (m/s) - Heisenberg's uncertainty principle: Δx x mΔv ≥ (h/4π). Δx is the uncertainty in position, Δv is the uncertainty in velocity, and h is Planck's constant. - Schrodinger equation: Ȟψ = Eψ - Energy of a hydrogen atom's electron: En = -2.18 x 10^-18 J (1/n^2), where n is the principal quantum number. - ΔE = -2.18 x 10^-18 J x ((1/nf^2) - (1/ni^2))
