Kevin D. McMahon Reseda Science Magnet The Quantum-Mechanical Atom and the Properties of Elements I. THE QUANTUM—MECHANICAL ATOM A. Electromagnetic Radiation Radio waves, microwaves, visible light, ultraviolet, X-rays, and gamma rays are all examples of electromagnetic radiation. Electromagnetic radiation has three primary characteristics: wavelength, frequency, and speed. Wavelength (symbolized by the Greek letter lambda, "") is the distance between two consecutive peaks or troughs in a wave. Question #1: Which form of electromagnetic radiation has the shortest wavelength? 1 The frequency (symbolized by the Greek letter nu, "") is defined as the number of waves (cycles) per second that pass a given point in space. The speed of all electromagnetic radiation is the speed of light (symbolized by "c") which is equal to 2.9979 X 108 meters per second. There is an inverse relationship between wavelength and frequency: • waves with the smallest wavelength have the highest frequency, • waves with the largest wavelengths have the lowest frequency. The mathematical expression: = c where is in meters is in cycles/seconds or Hertz (Hz) describes the relationship between the primary characteristics of electromagnetic radiation. Question #2: Calculate the frequency of green light if it has a wavelength of 5 X 10-7m. Question #3: An FM radio station broadcasts at a frequency of 105.1 MHz. What is the wavelength of the radio waves? Hint! To use the formula = c, the frequency must be in Hz. 1MHz = 1,000,000 Hz. Quantum Theory At the end of the 19th century classical physics held that matter and energy were distinct: • matter was thought to consist of particles while energy was in the form of a wave (ie: electromagnetic energy), • particles were things that had mass and whose position in space could be specified, • waves were described as massless and delocalized; that is, their position could not be specified. Classical physics also held that matter should be able to absorb or emit any quantity of energy. Absorbing energy was like a ball being pushed up a ramp. Losing energy was like a ball rolling down a ramp. Any amount of energy gain or loss is possible. In 1901, the German physicist, Max Planck, explained that the 2 radiation by solid bodies heated to incandescence could not be explained by classical physics. To account for his observations Planck proposed that energy could be gained or lost only in whole-number multiples of "h." "h" is called Planck's constant. It was determined by experimentation to be 6.626 X 10-34 Js (a joule, J, is a unit of energy; 1J = 0.24 calories) The energy gained or lost is given by the expression: E = nh where "n" is an integer (1,2,3...), h is Planck's contant, and is the frequency of electromagnetic radiation absorbed or emitted. This relationship implied that energy was "quantized," that is, not continuous. This was very different than classical physics. Instead of the ball rolling up and down the ramp as in classical physics, the ball would jump up or fall down certain allowed or quantized steps. This relationship implied that energy was "quantized," that is, not continuous: • Energy could only occur in units of h. • Each "packet" of energy is called a "quantum." Albert Einstein proposed that electromagnetic radiation is quantized and that it was packaged in "particles" he called "photons." The energy of each photon is given by: E photon = h = hc/ Question #4: The laser in a compact disc player uses light with a wavelength of 7.8 X 102 nm. Calculate the energy of a single photon of this light. Einstein also demonstrated that a photon of electromagnetic radiation had an associated mass which could be determined by: m = h/c. Einstein had previously shown in his special theory of relativity that: E = mc2 . This expression can be re-written as: m = E/c2 . In addition, we know that the energy of a photon is given by: E = hc/ , therefore, we can substitute E in our first equation with hc/ and obtain: m = hc/ = h/c c2 Question #5: Calculate the mass of the photon of light emitted by the laser from the previous question. 3 Remember, the light had a wavelength of 7.80 X 102 nm. Your answer should be expressed in kg. Light, which had been previously thought of as being purely wave-like, was now being described by Quantum Theory as having the particulate properties of matter. This would ultimately revolutionize our understanding of the atom. According to classical physics the light emitted by the sun possess only wave characteristics. But as we know, the white light emitted by the sun is made up of numerous colors. The light emitted by the sun consist of a continuous spectrum of colored light of different wavelengths. According to Quantum Theory the light emitted by the sun consists of a stream of tiny energy packets called photons. Each photon has an energy equal to h and depending on its frequency () it will have its own unique color. Louis de Broglie (1892- 1987) suggested if energy exhibited particulate behavior then matter should exhibit wave characteristics. Using Einstein's equation: m = h/lc and considering a particle with a velocity (v) de Broglie obtained: m = h/v de Broglie rearranged the expression to obtain: = h/mv de Broglie equation demonstrated that a particle with a mass (m) will have a wavelength (l) when traveling at velocity (v). • Notice that as mass increases the wavelength decreases. • de Broglie's equation has been shown to be true through experimentation. Question #6: Calculate the de Broglie wavelength for an electron with a velocity of 15% of the speed of light. The mass of an electron is equal to 9.1 X 10-31kg Hint! To calculate 15% of the speed of light... (.15) (3 X 108 m/s) Question #7: Calculate the de Broglie wavelength for the fastest measured baseball (5.2 oz) with a velocity of 100.8 mph). Hint! Okay, but you should know this... 4 5.2 oz x 1 lb x 1 kg = 16 oz 2.2 lb 100.8 miles x 1 km x hr .62 miles 1000m x 1 km 1 hr x 60 min 1 min = 60 s The work of Planck, Einstein, and de Broglie led to the postulation of the "Wave-Particle Duality of Matter" which stated that matter exhibits both particulate and wave characteristics. Large pieces of matter, such as baseballs, exhibit predominately particulate properties whereas tiny pieces of matter, such as photons, exhibit predominately wave properties. Pieces of matter with intermediate mass, such as electrons, show both particulate and wave properties. Appreciating the dual characteristics of the electron is fundamental to understanding the structure of the atom and the properties of the elements. Quantum Model When hydrogen gas receives a high-energy spark it absorbs the energy. Hydrogen gas then releases the absorbed energy as light. When this light is passed through a prism or diffraction grating the hydrogen emission spectrum has distinct lines rather than being a continuous spectrum. The emission spectra of hydrogen shows that hydrogen atoms will absorb and emit only certain wavelengths of energy. • This is in accordance with Planck's quantum theory. • The energy absorbed and emitted can be determined using the equation: E = hc/ ( is equal to the wavelength of light emitted). In 1913, a Danish physicist named Niels Bohr used the emission spectra of hydrogen to develop his "Quantum Model of the Atom." Bohr suggested that the electron in a hydrogen atom moves around the nucleus only in certain allowed circular orbits. An electron in the orbit closest to the nucleus is in its lowest energy state or its "ground state." The electron can absorb a particular photon of energy and jump to another orbit. When the electron falls down to a lower orbit it will emit a 5 photon of light. Bohr's mathematical computation for the energy of various orbits corresponded perfectly with the energies associated with the emission spectra of hydrogen. Each line in hydrogens emission spectra responds to a photon of light emitted when an electron drops from one orbit to a lower orbit. The most important equation to come from the Bohr model is the expression for the energy levels available to the electron in the hydrogen atom: n is an integer (the larger the value of n, the larger the orbit), z is the nuclear charge which for the hydgrogen atom is 1. The negative sign in the equation means that the energy of the electron bound to the nucleus is lower than it would be if the electron were not bound to the nucleus. The equation can be used to calculate the change in energy of an electron and the wavelength of light emitted or absorbed when the electron changes orbit. E = Efinal - Einitial Question #8: Calculate the energy of an electron in a hydrogen atom when n = 6. Question #9: Calculate the energy emitted when an n= 6 electron falls down to the n =1 (ground state) level. Hint. Calculate E then use this value in the formula below: E = Efinal - (- 6.05 X 10-20 J) Questions #10: 6 Calculate the energy required to excite the hydrogen electron from level n = 1 to level n = 2. Hint! Calculate the value for E and then use this and the value you calculated previously for E in the formula: E = Efinal - Einitial Question #11: Calculate the wavelength of light that must be absorbed to produce this change. Hint: Use the formula = hc/E and use the value of E that you calculated in the previous problem. Remember, c = 3 X 108 m/s. When Bohr's Quantum Model was applied to atoms other than hydrogen it did not work. • The Quantum Model is fundamentally incorrect in that the electrons do not move around the nucleus in circular orbits. • The Quantum Model was important because it helped pave the way for other theories. Bohr was awarded the Nobel Prize in Physics in 1922. The Wave Nature of Electrons Erwin Schrodinger, an Austrian physicist, applied de Broglie's concept that the electron exhibited wave behavior to develop the Wave-Mechanical Model of the Atom. The electron has significant wave characteristics because of its size. Because of this wave characteristic the electron's position could not be determined with certainty. Werner Heisenberg demonstrated mathematically that there was a fundamental limitation to just how precisely we can know both the position and momentum of a particle at a given time. • This principle is known as Heisenberg's Uncertainty Principle. • This uncertainty is insignificant when dealing with substantial mass, but becomes very significant when dealing with particles such as electrons. There was a philosophical implication with Heisenberg's Uncertainty Principle. It meant that there was a fundamental limitation to how precisely we could know something, especially like the location of an electron. Given this, Schrodinger recognized that it wouldn't be possible to know the exact location of the electron. But perhaps he could determine the "probability" of locating an electron. Schrodinger's analysis allowed him to predict the probability of finding the electron at some distance from the hydrogen nucleus. His calculations showed that the probability of finding an electron increased moving away from the nucleus to a maximum of 0.529Å . The probability of finding an electron diminished as the distance from the nucleus increased. 7 When the probabilities of finding electrons are plotted with respect to their distance from the nucleus one obtains a nearly sphrical distribution of electrons around the nucleus with the greatest concentration approximately 0.529 Å away from the nucleus. There is no way of knowing where the electron is within the sphere because of uncertainty. Because of its wave nature the electron effectively occupies the entire volume of the sphere. The sphere has no distinct limit since the probability of finding an electron at some further distance from the nucleus never goes to zero. Therefore, the size of the sphere is arbitrarily defined as the radius of the sphere that encloses 90% of the total electron probability. The sphere of electron probability surrounding the nucleus is called the "1s" orbital. The 1s orbital corresponds to the innermost orbit of the Bohr Model. • An "orbital" is not an orbit which has a defined circular path with a specified radius. • The path or motion of the electron in an orbital is not known. • The distance from the nucleus of an electron in an orbital is specified by the radial probability. This model of the atom is called the Wave-Mechanical Model. • de Broglie, Schrodinger, and Heisenberg are credited with the development of this model. • The Wave-Mechanical Model not only explains the behavior of hydrogen atoms, but all other known atoms. Quantum Numbers When the Schrodinger equation is solved for hydrogen, it is determined that there are many orbitals that satisfy it. Each orbital is characterized by "quantum numbers" which describe various properties of these orbitals. The Principle Quantum Number (n): The principle quantum number (n) is related to the size and energy of the orbital. n can have integral values of 1, 2, 3 .... n is sometimes referred to as an energy "level" or "shell" and corresponds to the Bohr energy level. As n increases... • the orbital becomes larger, • the electron spends more time further from the nucleus, • the electron has higher energy. 8 The Azimuthal Quantum Number (l): l relates to the shape of the atomic orbital. l can have integral values of 0 to n-1. The value of l for a particular orbital is commonly assigned a letter: • when l = 0, the orbital is called "s." • when l = 1, the orbital is called "p." • when l = 2, the orbital is called "d." • when l = 3, the orbital is called "f." This system of naming orbitals arises from early spectral studies. l also indicates the number of orbitals for each value of n: • when n=1, there is only 1 orbital, ie: the s orbital, • when n=2, l can be 0 or 1; thus there are two orbitals—s and p, • when n= 3, l can be 0, 1, or 2; thus there are three orbitals- s, p, and d. • when n=4, l can be 0, 1, 2, or; thus there are four orbitals- s, p, d, and f. Each orbital is designated by the number for the principle quantum number and the letter designation for l. For example: 2p orbital Here the 2 indicates that n = 2, and the p indicates that l = 1. Sometimes l is referred to as a "sublevel" or "subshell." The Magnetic Quantum Number (ml): ml relates to the orientation of the orbital in space relative to the other orbitals in the atom. ml can have integral values between l and -1, including 0. When l = 0 = s, there is only 1 value of ml, that is, 0. The sublevel s has only one orbital which has a spherical shape. The 1s and 2s orbitals are separated by a "node," a region of zero probability of finding an electron. Here we see the 1s orbital nestled within the 2s orbital. When l = 1 = p, there are three values of ml: -1, 0, 1. The sublevel p consist 9 of three orbitals corresponding to ml = -1, 0, 1. There are no p orbitals when n = 1. The shape of these orbitals are often referred to as looking like dumb bells. When l = 2 = d, there are five values of ml: -2, -1, 0, 1, 2. The sublevel d has five orbitals. There are no d orbitals when n = 1 and n = 2. When l = 3= f, there are seven values of ml: -3, -2, -1, 0, 1, 2, 3. The sublevel f has seven orbitals. There are no f orbitals when n = 1, 2, or 3. Electrons can move from one orbital to another. In the lowest energy state, the ground state, the electron resides in the 1s orbital. If energy is absorbed by the atom, the electron can be transferred to a higher energy orbital producing an "excited state." • If the electron drops to a lower energy level the atom emits energy. • Like the Bohr atom this can account for the spectral lines of hydrogen. But unlike the Bohr model the Wave-Mechanical model can account for the emission spectra of all other atoms. Electron Spin and the Pauli Exclusion Principle: Samuel Goudsmit and George Uhlenbeck (University of Leyden in the Netherlands) found that a fourth quantum number was necessary to account for additional characteristics of emission spectra. The electron spin quantum number, ms, can have only two values: +1/2 and- 1/2. ms = +1/2 or -1/2 is interpreted that the electron can spin in one of two opposite directions. Since a spinning electric charge produces a magnetic field, the two opposing spins produce oppositely directed magnetic fields. The Austrian physicist, Wolfgang Pauli, proposed his Exclusion Principle which stated that in a given atom no two electrons can have the same set of four quantum numbers. Since electrons in the same orbital have the same values of n, l, and ml they must have different values of ms. Since only two values of ms are allowed, an orbital can only have two electrons, and they must have opposite spins. Question #12: 10 The dots represent the probable electron distribution for the 1s orbital. Which percent that corresponds to the outer limit of the orbital? Question #13: The quantum number that corresponds to the Bohr orbital is... a. n b. l c. ml d. ms Question #14: When l = 2, the orbital is... a. d orbitals b. f orbitals c. s orbitals d. p orbitals Question #15: When n = 3, l can be ... a. 1 b. 0, 1 c. 0,1,2 d. 0,1,2,3 and the corresponding orbitals are... a. s b. s, p c. s, p, d d. s, p, d, f Question #16: Which orbital has the lowest energy? a. 4d b. 3d c. 4s d. 4p Question #17: When l = 2= d, values of d included: a. -2, -1, 0, 1, 2 b. 0, 1, 2 c. -1, 0, 1 d. 0 Question #18: Which of the following are possible sets of quantum numbers for an electron? a. n = 1, l = 0, ml = 1, ms = +1/2 b. n = 9, l = 7, ml = -6, ms = -1/2 c. n = 2, l = 1, ml = 0, ms = 0 d. n = 1, l = 1, ml = 1, ms = +1/2 Quantum Numbers and the Periodic Table The Periodic Table can be understood using our know- ledge of the Wave-Mechanical Model of the Atom. The numbers running down the left hand side of the table correspond to the principle quantum number "n." The Periodic Table is also divided into clearly identifiable sections. These sections correspond to the Azimuthal Quantum number (l) or the orbitals s, p, d, and f. 11 As we go across the Periodic Table electrons fill orbitals in accordance with the rules set forth by the Wave Mechanical Model and the Pauli Exclusion Principle. Hydrogen has 1 electron which occupies the 1s orbital, which can be depicted as: The superscript "1" represents the number of electrons occupying the 1s orbital. The arrow represents an electron spinning in a particular direction. Helium completes the 1s orbital: Note the arrows are in opposite directions indicating opposite spin. The n = 1 energy level is now filled. Electrons must now begin to fill the n = 2 energy level and orbitals: 12 Carbon has one more electron than Boron. Should the next electron be added to the p orbital already containing an electron or a completely empty p orbital? The German physicist, F.H. Hund proposed a rule stating that atoms seek to be at their lowest energy state. • This is achieved when an atom has a maximum number of unpaired electrons. • All unpaired electrons have parallel spins. Therefore... Carbon's electron configuration is... Continuing across the row or period 2 The elements of period 3 (n = 3) add electrons to the orbitals in the same manner as just described for period 2. The valence electrons are the electrons in the outermost principle quantum level of the atom. For example, Silicon has the electron configuration: Si: 1s2 2s2 2p6 3s2 3p2 Silicon has 4 valence electrons: 3s2 3p2 Valence electrons are the most important in chemistry since they are involved in chemical reactions. The inner electrons (non valence) are called "core electrons." The core electrons of silicon are: 1s2 2s2 2p6 Because the core electrons have the same configuration as the Group 8A element of the previous period, electron configuration are often abbreviated using the symbol of the previous Group 8A element followed by the valence electrons. For example, silicon's electron configuration could be depicted as Si: [Ne] 3s2 3p2 As we move to the fourth period of the Periodic Table we fill the 4s orbital: 13 K: [Ar] 4s1 Ca: [Ar] 4s2 The next element, Scandium, begins a series of elements known as the "transition metals." The transition metals add electrons by filling the d orbitals. In filling the transition elements of period 4 (n=4) the 3d orbitals are filled. The 3d orbitals are filled before the 4p orbitals because they are lower in energy. Use the Periodic Table to observe the electron configuration of the period 4 transition metals. Note the anomalous configurations of Cr and Cu. The elements of period 5 add electrons to the orbitals in very much the same manner as described for period 4. Again, you may use the Periodic Table to review the electron configuration of these elements. The element Barium has the electron configuration: Ba: [Xe] 6s2 is followed by Lanthanum, which has the expected electron configuration: La: [Xe] 6s2 5d1 The element following lanthanum is Cerium. This element belongs to a group of elements called "lanthanides." The lanthanides consist of 14 elements in which the 4f orbitals are being filled. The 4f and the 5d energy sublevels are so close that sometimes electrons will fill the 5d orbital before the 4f. Electrons fill elements of period 7 in a manner similar to that of period 6. The actinides, like the lanthanides are elements that are having their f orbitals filled. The Lanthanides and Actinides are sometimes referred to as the f-transition elements. The Wave-Mechanical Model of the Atom and the electron configurations we obtain by its application provides us with the necessary framework to understand why elements possess certain properties and why these properties are periodically repeated through the Periodic Table. Question #19: The electron configuration for silicon is : [Ne] 3s2 3p2 The electrons fill the orbitals as follows... a. b. c. 14 Question #20: Which of the following is a possible electron configuration? a. [Kr] 5s2 4d1 b. [Kr] 5s2 5d1 c. [Kr] 5s2 5p1 The element described above is a... a. transition element b. lanthanide c. actinide Question #21: Which element has a ground state electron configuration in which it has three unpaired p electrons? a. Boron b. Argon c. Nitrogen d. Scandium Question 22: Identify the element that has an excited electron configuration of 1s2 2s2 2p5 3s1 . a. Neon b. Sodium c. Carbon d. Fluorine THE PROPERTIES OF ELEMENTS Periodicity Mendeleev discovered that when elements were arranged by increasing atomic mass the properties (such as valence and reactivity) of the elements would periodically repeat. He called his arrangement of elements the Periodic Table of the Elements. Modern Periodic Table arranges elements by increasing atomic number. Even with this fundamental change the elements are still observed to be grouped in families that share the same or similar physical and chemical properties. The Wave-Mechanical Model of the Atom allows us to understand the periodicity of the elements and why families of elements exist. The questions that follow will help you to explore the relationship between the Wave-Mechanical Model and periodicity. Question #23: Which statement best explains the concept of periodicity? a. Members of the same period or row complete their s orbitals before filling their p orbitals. b. Members of the same group or column have similar electron configurations. c. Members of the transition elements have partially full or completely full d orbitals. d. Elements further down a group have higher principle quantum numbers. 15 Question #24: Which of the following groups of elements would have similar chemical properties. a. boron, carbon, nitrogen b. manganese, iron, silver c. nitrogen, phosphorus, antimony d. gadolinium, europium, holium Atomic Radius The size of an atom cannot be precisely defined because the size of its orbitals cannot be determined exactly. Atomic radii must be determined by measuring the distance between atoms in a compound. The distance between the two nuclei in Br has been determined to be 2.28Å . The atomic radius of Br is assumed to be 1/2 this distance or 1.14Å . Atomic radii have been determined for most of the elements of the Periodic Table Question #25: Which statement best describes the trend of atomic radii in the Periodic Table? a. The atomic radius increases going across a period and down a group. b. The atomic radius decreases going across a period and increases going down a group. c. The atomic radius increases going across a period and decreases going down a group. Ionization Energy Ionization energy is the energy required to remove an electron from an atom: X(g) + ionization energy --> X +(g) + ewhere the atom is assumed to be in the ground state. More than one electron can be removed from an atom. Consider the ionization of aluminum: kJ/mole Al (g) ---> Al+ (g) + e- I= 580 Al+ (g) ---> Al+2 (g) + e- I= 1815 Al+2 (g) ---> Al+3 (g) + e- I= 2740 Al+3 (g) ---> Al+4 (g) + e- I=11,600 The first ionization energy (I1 ) is significantly lower than the second ionization energy 16 (I2) because of the increased charge of the nucleus and resultant attraction to the electron. The difference in ionization energies between I1 and I2 can also be explained because the first electron to be ionized was a 3p whereas the I2 electron is in the 3s orbital. Much energy is required to remove the 4th e- because it's a core electron. Question #26: What trend do you observe in the Periodic Table with respect to ionization energy? a. Ionization energy increases going across a period and decreases going down a group. b. Ionization energy decreases going across a period and increases going down a group. c. Ionization energy increases going across a period and going down a group. Question #27: There are a couple noticeable exceptions to the trend you observed in the previous question. One of these exceptions suggests that... a. atoms seek to fill partially full orbitals first. b. electrons fill orbitals to maximize electron-electron repulsion. c. it is more stable to have all of your p orbitals partially full. Electron Affinity Electron affinity is the energy change associated with the addition of an electron to a gaseous atom: X(g) + e- ---> X- (g) The negative values (example, F has an electron affinity of -328 kJ/mole) indicates that energy is released when the electron is added. The more negative the electron affinity the more stable the electron configuration achieved when an electron is added. The Group 7 elements have the most negative electron affinity. This is due to the fact that when Group 7 elements (ns2 p5 ) add an electron they complete their "p" orbitals (ns2 p6) which is a stable electron configuration. Question #28: Ionization energy tells us how willing an atom is to part with an electron. Electron affinity suggests how much an atom would like to add an electron. Given this which of the following would you expect to be true? a. Atoms tend to want electrons less going across and up the periodic table. b. Atoms tend to want electrons more going across and down the periodic table. c. Atoms tend to want electrons more going across and up the periodic table. Stable Electron Configuration You will observe by examining the Periodic Table that all of the Group 8 elements have 17 their outer shell full of electrons. This is known as a "stable electron configuration." Elements try to achieve a stable electron configuration if they do not already have one. They can do this by reacting with other elements who are also trying to achieve stability. Achieving stability can be accomplished by losing, gaining, or sharing electrons. In doing this unstable elements combine or bond with other unstable elements through chemical reactions to form compounds. In compounds each element has achieved a stable electrons configuration. The Group 8 elements already have a stable electron configuration and hence have little incentive to react with other elements. Consequently, these elements are known as "inert." They are also referred to as "Noble" because of their lack of association with "common" reacting elements. Question #29: The alkali metals (Group IA) are very reactive. Given their electron configuration, ionization energy, and electron affinity which of the following statements best describes how the alkali metals would achieve a stable electron configuration. a. Alkali metals attract additional electrons to fill up their s and p orbitals. b. Alkali metals are in fact already stable and do not need to alter their electron configurations. c. Alkali metals give up their one valence electron. Question #30: The halogens (Group VIIA) elements are also very reactive. Given their electron configuration, ionization energy, and electron affinity which of the statements best describes how the halogens might try to achieve a stable electron configuration. a. A halogen will attract an electron to fill its p orbital thereby achieving a stable electron configuration. b. Halogens lose electrons to become stable. c. Halogens form "metal bonds" a share many electrons between them. 18 Question # 31: Identify the regions of s, p, d, and f orbital fillings. Question #32: Write electron configurations for each element identified by the letters a through h. Question #33: Use arrows to indicate increasing atomic radii, increasing electron affinity, and decreasing electronegativity. 19