Focus 1: Atoms PDF

Focus 1: Atoms 1. Investigating Atoms a. The Nuclear Model of the Atom i. J.J. Thomson's cathode ray (high voltage is applied between 2 metal electrodes in an evacuated glass tube) 1. Noted that cathode rays are streams of negatively charged particles 2. Charged particles = electrons 3. Measured e/m...

Focus 1: Atoms 1. Investigating Atoms a. The Nuclear Model of the Atom i. J.J. Thomson's cathode ray (high voltage is applied between 2 metal electrodes in an evacuated glass tube) 1. Noted that cathode rays are streams of negatively charged particles 2. Charged particles = electrons 3. Measured e/me 4. Plum pudding model (electrons embedded in positively charged sphere) ii. Robert Millikan’s oil drop experiment 1. Discovered smallest increment of charge 2. e = 1.602 x 10-19 C (fundamental charge) 3. Proton = +e, electron = -e 4. Used Thomson’s measurement to find mass of electron (9.11 x 10 -31 kg) iii. Ernest Rutherford shot α particles at a very thin sheet of platinum foil 1. Found that most particles passed through onto the other side, but 1 in 20,000 were deflected 2. Nuclear model = nucleus is a tiny, dot-like structure that sits in the center of the atom and is surrounded by a large volume of mostly empty space, where the electrons are located (electron cloud) 3. Protons have a positive charge, neutrons are neutral 4. Atomic number (Z) is the number of protons and determines the type of element 5. If an atom is electrically neutral: +Ze = -Ze b. Electromagnetic Radiation i. Spectroscopy 1. The analysis of light emitted or absorbed by substances 2. Electromagnetic radiation consists of oscillating (time-varying) electric and magnetic fields that travel through empty space at about 3 x 10 8 ms-1 (light is a form of this) 3. c = 3 x 108 ms-1 (speed of light) 4. As light passes an electron, its electric field pushes the electron in one direction and then in the opposite direction over and over again (cyclical) a. Frequency (𝜈) = the number of cycles per second (Hz: s-1) b. Amplitude = height of the wave above the centerline c. Intensity = square of amplitude, brightness d. Wavelength (λ) = peak-to-peak distance 5. Short wavelength corresponds to high-frequency, long wavelength corresponds to low-frequency a. λ x 𝜈 = c (wavelength x frequency = speed of light) 6. Electromagnetic spectrum a. Visible light = wavelengths between 400 nm and 700 nm, detectable by the human eyes (color) b. Ultraviolet radiation = wavelength less than 400 nm, higher frequency c. Infrared radiation = wavelength greater than 800 nm, lower frequency c. Atomic Spectra i. ii. When electric current is passed through hydrogen gas... 1. Strong electric field strips electrons off H2 forming plasma of H+ ions and electrons which conduct current 2. Electrons attach to H+ ions forming energetically excited hydrogen atoms 3. These atoms discard their excess energy by giving off electromagnetic radiation, then recombine to form H2 again Spectral lines = the radiation founded to consist of a number of distinct components when the light emitted from an excited element is passed through a prism 1. Johann Balmer = first person to identify a pattern in the lines of the visible region of the hydrogen spectrum a. λ ∝ 2 2 4 where n = 3, 4, … 2. Johannes Rydberg a. 1/λ ∝ 3. 𝜈 = c/λ = R{ 1 2 1 2 12 - 1 - 1 2 22 where n = 3, 4, … } where n1 = 1, 2, … and n2 = n1 + 1, n2 + 2, … a. R (Rydberg constant) = 3.29 x 1015 Hz b. Balmer series = lines with n1 = 2 (visible light) c. Lyman Series = lines with n2 = 1 (UV light) 4. Energy levels a. Lines arises from transition (change of state between 2 energy levels) b. Difference of energy levels is carried away by electromagnetic radiation emitted iii. Absorption spectrum = series of dark lines on an otherwise continuous spectrum 1. Have same frequencies as lines in emissions spectrum 2. Suggests atoms can absorb radiation only of those same frequencies 2. Quantum Theory a. Radiation, Quanta, and Photons i. Electromagnetic Radiation 1. Incandescence = at high temperatures, objects begin to glow a. Color of light changes from red to orange, yellow and then white (qualitative observation) b. Intensity of radiation at each wavelength at different temperatures (quantitative observation) 2. Black body = hot object that does not favor one wavelength over another in the sense of absorption and emission a. Black-body radiation = radiation emitted at different wavelengths by a heated body, for a series of temperatures b. Stefan-Boltzmann law: total intensity = constant x T4 i. Constant = 5.67 x 10-8 Wm-2K-4 3. Used classical physics to explains laws of black body radiation a. Ultraviolet catastrophe = classical physics predicted that any hot body should emit intense UV radiation b. Solved by Max Planck who proposed the exchange of energy between matter and radiation occurs in quanta, or packets of energy i. E = h𝜈 ii. h (planck’s constant) = 6.6261 x 10-31 Js ii. Photoelectric Effect 1. The ejection of electrons from a metal when its surface is exposed to UV radiation 2. Observations a. No electrons are ejected unless the radiation has a frequency above a certain threshold value that is characteristic of the metal b. Electrons are ejected immediately, however low the intensity of the radiation c. The kinetic energy of the ejected electrons increases linearly with the frequency of the incident radiation 3. Einstein proposed that electromagnetic radiation consists of particles (photons) a. E = h𝜈 is a measure of the energy of each individual photon 4. ɸ (work function) = energy required to remove an electron from the metal a. If energy of photon is less than the energy required to remove an electron from the metal, then an electron will not be ejected b. If h𝜈 > ɸ, then an electron is ejected with a kinetic energy equal to ½ m ev2 c. E = h𝜈 - ɸ d. ½ mev2 = h𝜈 - ɸ (kinetic energy of an ejected electron = energy supplied by a photon - the energy required to eject an electron/work function) 5. Einstein’s Theory a. An electron can be ejected from a metal only if the energy from the photon is equal to the work function b. Provided a photon has enough energy, a collision results in the immediate ejection of an electron c. The KE of the electron from the metal increases linearly with the frequency of the incident radiation 6. Bohr frequency condition: h𝜈 = Eupper - Elower 7. Supports the idea that electrons behave like particles b. The Wave-Particle Duality of Matter i. Diffraction 1. The pattern of high and low intensities generated by an object in the path of a ray of light 2. Diffraction pattern = results when the peaks and troughs of waves traveling along one path interfere with the peaks and troughs traveling along another path a. Constructive interference = peaks coincide (enhancement) b. Destructive interference = peak and trough coincide (diminished) 3. Shows wave-like property of electromagnetic radiation ii. Wave-particle duality 1. In the wave model, the intensity of the radiation is proportional to the square of the amplitude of the wave 2. In the particle model, intensity is proportional to the number of photons present at each instant iii. λ = h/mv 1. de Broglie relation: λ = h/p c. The Uncertainty Principle i. Problems with Classical Mechanics 1. Trajectory in classical mechanics, particle has a definite path 2. One cannot specify the precise position of a particle if it behaves like a wave 3. Complementarity = the impossibility of knowing the precise position if the linear momentum is known precisely a. If one property is known, the other cannot be known simultaneously ii. Heisenberg 1. Expressed complementarity quantitatively 2. If the location of a particle is known to within an uncertainty Δx, then the linear momentum, p, parallel to the x-axis can be known simultaneously only to within an uncertainty Δp, where… a. Δp ྾ Δx ≥ ½ ħ b. ħ = h/2π 3. Wavefunctions and Energy Levels a. The Wavefunction and Its Interpretation i. Schrödinger replaced trajectory of a particle by a wavefunction, Ѱ 1. Mathematical function with values that vary with position 2. Born interpretation = the probability of finding the particle in a region is proportional to the value Ѱ2 in that region a. Ѱ2 = probability density (probability that the particle will be found in a small region divided by the volume of the region) 3. Node = where Ѱ passes through zero, particle has zero probability density wherever the wavefunction has nodes ii. Schrӧdinger Equation = used to calculate wavefunction for any particle 1. For a particle of mass m moving in one direction in a region where the potential energy is V(x), the equation is… a. ħ2 2 Ѱ 2 iii. + V(x)Ѱ = EѰ (kinetic energy + potential energy = total energy) 2Ѱ b. c. 2 2 ħ2 = measure of how sharply the wavefunction is curved = hamiltonian operator, written as HѰ 2 d. EѰ = HѰ 2. Used to calculate the wavefunction and corresponding energy Particle in a Box 1. A single particle, of mass m, confined in a one-dimensional box between 2 rigid walls a distance L apart 2 2. Ѱn(x) = ( )½ sin( ) where n = 1, 2, 3, … a. Integer n labels the wavefunction, called quantum number b. The Quantization of Energy i. Boundary Conditions 1. The particle has zero potential energy inside the box, and infinite outside 2. Walls trap particle in box 3. E = 2 ℎ2 8 2 where n = 1, 2, 3, … ii. iii. a. Energy of the particle is quantized (restricted to a series of discrete values) b. These discrete values are called energy levels According to quantum mechanics, only certain wavelengths fit into the box and each wavelength corresponds to a different energy, energy is quantized An electron in an atom has a wavefunction that must satisfy certain constraints in three dimensions, only some solutions of the Schrӧdinger equation and their corresponding energies are accepted 1. As L or m increases, the separation between neighboring energy levels decreases 2. Particle in a container cannot have zero energy, because lowest value of n is 1 a. Zero-point energy = lowest possible energy, E1= 3. En + 1 - En = (2n + 1) ℎ 2 ℎ2 8 2 2 2 8 4. The Hydrogen Atom a. Energy Levels i. Certain constraints result in the quantization of energy and the existence of discrete energy levels 1. V(r) = ( ) 4 ( ) 2 =- 4 a. (charge of electron x charge of proton)/(4𝜋𝜀𝑜x distance of electron from the proton) ℎ 2. En = 3. En = - 2 with R = 2ℎ 2 4 8ℎ 3 2 where n = 1, 2, 3, … where n = 1, 2, 3, … 4. Electron has a lower energy in the atom than when it is far from the nucleus ii. Principle quantum number = n 1. Ground state = the state of lowest energy when n = 1, for hydrogen: -hR 2. Ionization = when the electron leaves the atom, occurs when E = 0 and n approaches infinity 3. Ionization energy = the minimum energy needed to achieve ionization starting from the ground state b. Atomic Orbitals i. Atomic orbitals = the wavefunctions of electrons in an atom 1. The square of the wavefunction describes the probability density of an electron at each point ii. Spherical polar coordinates 1. Each point is labeled r, θ, and ɸ a. r = distance from nucleus b. θ = the angle from the positive z-axis, “latitude” c. ɸ = the angle about the z-axis, “longitude” 2. Ѱ(r, θ, ɸ) = R(r) x Y(θ, ɸ) a. R(r) = radial wavefunction, expresses how the wavefunction varies on moving away from the nucleus b. Y(θ, ɸ) = angular wavefunction, expresses how the wavefunction varies as the angles θ and ɸ change 3. Wavefunction corresponding to the ground state of the hydrogen atom a. Ѱ(r, θ, ɸ) = ( 1 3 )½ e-r/ao i. ao = Bohr radius (52.9 pm) ii. Spherically symmetric (independent of the angles θ and ɸ) iii. Probability density is highest closest to nucleus (r = 0) c. Quantum Numbers, Shells, and Subshells i. 3 quantum numbers needed to label each wavefunction for 3-dimensional Schrӧdinger’s 1. n is related to the size and energy of the orbital a. All atomic orbitals with the same principle quantum number have the same energy and are said to belong to the same shell 2. l is related to the shape a. Orbital angular momentum quantum number b. l = 0, 1, 2, …, n - 1 c. The orbitals of a shell with principal quantum number n fall into n subshells, groups of orbitals that have the same value of l d. Value of l 0 1 2 3 Orbital type s orbital p orbital d orbital f orbital e. l tells us the orbital angular momentum of the electron f. Orbital angular momentum = { l (l + 1) }1/2 ħ i. Electrons in the s orbital are distributed spherically around the nucleus ii. Electrons in the p, d, and f orbitals circulate around the nucleus iii. Interesting characteristic of atoms with 1 electron: all orbitals of a given shell have the same energy, regardless of their orbital angular momentum (degenerate) 3. ml is related to its orientation in space a. Magnetic quantum number b. Distinguishes the individual orbitals within a subshell c. ml = l, l - 1, l - 2, …, -l i. There are 2l + 1 orbitals in a subshell of quantum number l 1. Example: for p-orbital l = 1, so there are 3 p-orbitals in a given shell d. Specifies the orientation of the orbital motion of the electron i. Orbital angular momentum around an arbitrary axis is equal to m lħ ii. -ħ vs +ħ implies clockwise and counterclockwise d. The Shapes of Orbitals i. Each possible combination of the 3 quantum numbers specifies an individual orbital 1. Example: an electron in the ground state of hydrogen has n = 1, l = 0, m = 0 ii. Each shell has 2 s-orbital 1. The s-orbital in the shell with quantum number n is called an ns-orbital 2. Probability density of an electron in 1s orbital: Ѱ(r, θ, ɸ) = ( iii. Radial distribution function 1 3 )½ e-r/ao 1. Used to calculate the total probability of finding the electron at distance r from the nucleus at all possible orientations 2. P(r) = r2R2(r) 3. For s-orbitals... a. Ѱ = RY = R/(2π)½, so R2 = 4πѰ2 b. P(r) = 4πr2Ѱ2(r) 4. Important to distinguish the radial distribution function from the wavefunctions and its square a. The wavefunction tells us, through Ѱ(r, θ, ɸ), the probability of finding the electron in the small volume δV at a particular location specified by r, θ, and ɸ b. The radial distribution function tells us, through P(r)δ(r), the probability of finding the electron between r and r + δ summed over all values of θ and ɸ 5. Even though Ѱ2 is at a maximum at r = 0 (nucleus), P(r)δ(r) is 0 therefore there is no probability of finding an electron at this position 6. The highest probability of finding an electron occurs at some point r, and this radius is the size of that particular atom iv. Boundary Surface 1. Smooth surface that encloses most of the cloud 2. S orbital a. Boundary surface is spherical b. The probability density inside the boundary wave is not uniform 3. P orbital a. Boundary surface has 2 lobes (labeled as “+” and “-”) b. 2 lobes of the p orbital are separated by the nodal plane (cuts through the nucleus, where Ѱ = 0) c. P-electron = electron in a p-orbital d. Electrons are not found at nodes (Ѱ = 0 there) e. 3 p-orbitals in each subshell 4. d and f orbitals a. 5 d orbitals in each subshell b. 7 f orbitals in each subshell v. Nodes 1. An orbital with quantum number n and l has… a. l angular nodes b. n - l - 1 radial nodes 2. Total number of orbitals in a shell with principal quantum number n is n 2 e. Electron Spin i. Samuel Goudsmit and George Uhlenbeck first suggested that electrons behave like spinning spheres (property called spin) ii. According to Quantum Mechanics 1. Electrons have 2 spin states: ↑/α (up/alpha) and ↓/β (down/beta) 2. Distinguished by 4th quantum number, the spin magnetic quantum number (m s) a. Only 2 values i. +½ = ↑/α (up/alpha) ii. -½ = ↓/β (down/beta) f. The Electronic Structure of Hydrogen i. In ground state, the electron is in the lowest energy level (n = 1), where there is only 1 orbital (1s), electron is said to occupy a 1s-orbital 1. Described by the following quantum numbers a. n = 1 b. l = 0 c. ml = 0 d. ms = +½ or -½ ii. When atom acquires enough energy for its electron to reach n = 2 1. Can occupy any of the 4 orbitals 2. 1 2s-orbital and 3 2p-orbitals 3. Eventually enough energy is absorbed that the electron can escape the pull of the nucleus and leave the atom 5. Many-Electron Atoms a. Orbital Energies i. Many-electron atom = a neutral atom with more than one electron ii. More complicated factors 1. Nucleus is more highly charged, attracting electrons 2. Electrons repel one another 3. Electrons are shielded from the full attraction to the nucleus by other electrons 4. Effective nuclear charge (Zeffe) experienced by electron is always less than the actual nuclear charge a. E = - iii. 2ℎ 2 5. An s-electron of any shell can be found very close to the nucleus and is said to penetrate through the inner shells 6. s electrons are bound to the nucleus more tightly than p electrons, and therefore have a lower (more negative) energy Helium Atom 1. Potential energy is proportional to the attraction of electron 1 to the nucleus x the attraction of electron 2 to the nucleus x repulsion between the 2 electrons 2. V ∝ (- 2 2 1 )(- 2 2 2 )(+ 2 ) 12 3. r1 = distance of electron 1 to the nucleus, r2 = distance of electron 2 to the nucleus, r12 = distance between the 2 electrons 4. (-) represents attraction, while (+) represents repulsion iv. In a many-electron atom, because of the effects of penetration and shielding, the order of orbital energies in a given shell is s < p < d < f b. The Building-Up Principle i. Electron configuration = a list of all of an atom’s occupied orbitals, with the numbers of the electrons that occupy each one ii. Pauli exclusion principle = no more than 2 electrons may occupy each given orbital. When 2 electrons do occupy one orbital, their spins must be paired 1. Paired spins = one is ↑ and the other is ↓ 2. No 2 electrons in an atom can have the same set of 4 quantum numbers iii. iv. Closed shell = a shell containing the maximum number of electrons allowed by the exclusion principle 1. Atoms have an inner core consisting of electrons in filled orbitals surrounded by the valence electrons (electrons in outermost shell) 2. Generally only valence electrons can be lost in chemical reactions Building up principle: to predict the ground-state configuration of a neutral atom of an element with atomic number Z with its Z electrons 1. Add Z electrons to the orbitals with no more than 2 electrons in any one orbital *Pauli Exclusion Principle 2. If more than 1 orbital in a subshell is available, add electrons with parallel spins (⇈) to different orbitals of the same subshell rather than pairing 2 electrons in v. one of the orbitals *Hund’s Rule Excited state = atoms with electrons in energy states higher than predicted by the building up principle Valence shell = occupied shell with the largest value of n vi. 6. Periodicity a. The General Structure of the Periodic Table i. Periodic law = Mendeleev’s observation that elements fell into families with similar properties when arranged in order of increasing atomic mass ii. Table divided into blocks, named for the last subshell that is occupied according to the building-up principle iii. s and p blocks form main groups of periodic table iv. Periods (horizontal rows) are numbered according to the principal quantum number of the valence shell b. Atomic Radius i. Atomic radius = half the distance betweens the centers of neighboring atoms 1. If element is metal: atomic radius is half the distance between the centers of neighboring atoms in a solid sample 2. If element is nonmetal or metalloid: atomic radius is half the distance between the nuclei of atoms joined by a chemical bond (covalent radius) 3. If element is a noble gas: van der Waals radius is used, which is half the distance between the centers of neighboring atoms in a sample of the solidified gas ii. Atomic radius decreases from left to right across a period and increases down a group 1. More protons = higher effective nuclear charge, drawing electrons in 2. Higher quantum number = more occupied successive shells c. Ionic Radius i. Ionic radius = the shared distance between neighboring ions in an ionic solid ii. Cations are smaller than their parent atoms, while anions are larger iii. Atoms and ions with the same number of electrons are called isoelectronic 1. Radi between isoelectronic atoms differ based on effective nuclear charge iv. Ionic radii generally increase down a group and decrease from left to right across a period d. Ionization Energy i. The minimum energy needed to remove electrons from an atom in the gas phase 1. J (g) ⟶J+ (g) + e- (g) I = E(J+) - E(J) e. f. g. h. 2. E(J) = energy of the species 3. Ionization energies are reported as kJ mol-1 or eV (electron volts) a. Electron volts = the change in the energy of an electron when it moves through a potential difference of 1 volt 4. Elements with high ionization energies are unlikely to form cations and are unlikely to conduct electricity ii. First ionization energy, I1 = minimum energy needed to remove an electron from a neutral atom in the gas phase iii. Second ionization energy, I2 = minimum energy needed to remove an electron from a singly charged gas-phase cation iv. Trend 1. First ionization energies typically decrease down a group a. Outermost electrons occupy shells further from the nucleus, therefore are less tightly bound 2. First ionization energies generally increase across a period a. Effective nuclear charge increases across a period 3. Successive ionizations are always higher (I3 > I2 > I1) 4. Metals are found toward the lower left of the periodic table because these elements have low ionization energies and can readily lose their electrons Electron Affinity i. Electron affinity (Eea) = energy released when an electron attaches to an gas-phase atom 1. Positive Eea = energy is released 2. Negative Eea = energy must be supplied to attach electron ii. Electron affinities are highest toward the right of the periodic table (in groups 16 and 17) The Inert-Pair Effect i. The tendency to form ions two units lower in charge than expected from the group number ii. Due to relative energies of the valence p- and s-electrons iii. Most pronounced for heavy elements in the p-block Diagonal Relationships i. Diagonal relationships = the similarity in properties among diagonal neighbors in the mains groups of the periodic table ii. Helpful for making predictions about the properties of an element The General Properties of the Elements i. S-block elements are reactive metals that form basic oxides ii. P-block elements tend to gain electrons to complete closed shells; they range from metals through metalloids and nonmetals iii. D-block elements are metals with properties between those of s-block and p-block; many d-block elements form cations in more than one oxidation state