The Quantum Mechanical Model of the Atom

Why Study Quantum? Quantum mechanics is a mathematical theory that can be used to predict chemical properties. Since nowadays we have better computational methods and faster computers, we can calculate properties for systems that previously were too difficult to isolate experimentally. With this advanced technology, the renewed interest in and continued study of quantum chemistry may result in a quantitative understanding of transition states, prediction of alternate reaction pathways, and hopefully even prediction of new chemical concepts! The Quantum Mechanical Model of the Atom Most definitions of quantum theory and quantum mechanics offer the same description for both. These definitions essentially describe quantum theory as a theory in which both energy and matter have characteristics of waves under some conditions and characteristics of particles under other conditions. Quantum theory suggests that energy comes in discrete packages called quanta (or, in the case of electromagnetic radiation, photons). Quantum theory has some mathematical development, often referred to as quantum mechanics, that offers explanations for the behavior of electrons inside the electron clouds of atoms. The wave-particle duality of electrons within the electron cloud limits our ability to measure both the energy and the position of an electron simultaneously. The more accurately we measure either the energy or the position of an electron, the less we know about the other. The fact that we cannot accurately know both the position and the momentum of an electron at the same time causes an inability to predict a trajectory for an electron. Consequently, electron behavior is described differently than the behavior of normal sized particles. We use statistical descriptions that show, not the electron path, but the region where it is most likely to be found. Since it is the electron in the electron cloud of an atom that determines its chemical behavior, the quantum mechanics description of electron configuration is necessary to understanding chemistry. Reading Comprehension Review A 1. What does “wave-particle duality” mean? 2. Why can’t we describe an electron’s behavior by discussing its path as it orbits around the nucleus? Schröndinger Equation to Describe Electron Movement The most common way to describe electrons in atoms according to quantum mechanics is to solve the Schrodinger equation for the energy states of the electrons within the electron cloud. When the electron is in these states, its energy is well-defined but its position is not. The position is described by a probability distribution map called an orbital. Schrodinger found that the values that solve the equation to show the stable energy levels for an electron in an electron cloud produce a wave function that described a standing wave. These solutions to Schrödinger’s equation involve four special numbers called quantum numbers. (Three of the numbers, , , and , come from Schrödinger’s equation, and the fourth one comes from an extension of the theory). These four numbers completely describe the energy of an electron. Each electron has exactly four quantum numbers, and no two electrons have the same four numbers. The statement that no two electrons can have the same four quantum numbers is known as the Pauli exclusion principal. The principal quantum number, , is a positive integer that indicates the main energy level of an electron within an atom. According to quantum mechanics, every principal energy level has one or more sub-levels within it. The number of sub-levels in a given energy level is equal to the number assigned to that energy level. That is, principal energy level 1 will have 1 sublevel, principal energy level 2 will have two sub-levels, principal energy level 3 will have three sub-levels, and so on. Table 1 Number of Sub-Levels and Electrons by Principal Quantum Number Principal Number of Total Number Quantum Number Sub-Levels of Electrons 1 1 2 2 2 8 3 3 18 4 4 32 In any energy level, the maximum number of electrons possible is 2n2. Therefore, the maximum number of electrons that can occupy the first energy level is 2 (2 x 12). For energy level 2, the maximum number of electrons is 8 (2 x 22), and for the 3rd energy level, the maximum number of electrons is 18 (2 x 32). Table 1 lists the number of sub-levels and electrons for the first four principal quantum numbers. Reading Comprehension Review B 1. What is the Pauli Exclusion Principle? Why is it important? 2. What does the principal quantum number (n) tell us? What is the principal quantum number of Sulfur? 3. How many sublevels could be in energy level 5? How many electrons could occupy the 5 th energy level? The largest known atom contains slightly more than 100 electrons. Quantum mechanics sets no limit as to how many energy levels exist, but no more than 7 principal energy levels are needed to describe all the electrons of all the known atoms. Each energy level can have as many sub-levels as the principal quantum number, as discussed above, and each sub-level is identified by a letter. Beginning with the lowest energy sub-level, the sub-levels are identified by the letters s, p, d, f, g, h, i, and so on. Every energy level will have an s sub-level, but only energy levels 2 and above will have p sub-levels. Similarly, d sub-levels occur in energy level 3 and above, and f sub-levels occur in energy level 4 and above. Energy level 5 could have a fifth sub-energy level named g, but all the known atoms can have their electrons described without ever using the g sub-level. Therefore, we often say there are only four sub-energy levels, although theoretically there can be more than four sub-levels. The principal energy levels and sub-levels are shown in the following diagram. The principal energy levels and sub-levels that we use to describe electrons are in red. The energy levels and sub-levels in black are theoretically present but are never used for known atoms. Reading Comprehension Review C 1. Why do we say we only have 4 sub-energy levels? Why don’t we need to use g, h and i? 2. What would make us need the g sub-energy level in the future? Quantum mechanics also tells us how many orbitals are in each sub-level. In Bohr’s model, an orbit was a circular path that the electron followed around the nucleus. In quantum mechanics, an orbital is defined as an area in the electron cloud where the probability of finding the electron is high. The number of orbitals in an energy level is equal to the square of the principal quantum number. Hence, energy level 1 will have 1 orbital (1 2 ), energy level 2 will have 4 orbitals (2 2 ), energy level 3 will have 9 orbitals (3 2 ), and energy level 4 will have 16 orbitals (4 2 ). The sub-level has only one orbital. Each of the sub-levels has three orbitals. The sub-levels have five orbitals, and the sub-levels have seven orbitals. As a result, the single orbital in energy level 1 is the orbital. The four orbitals in energy level 2 are a single orbital and three orbitals. The nine orbitals in energy level 3 are a single orbital, three orbitals, and five orbitals. The sixteen orbitals in energy level 4 would be the ________________________________________________________________ (complete the table below). Table 2 shows the relationship between (the principal quantum number), the number of orbitals, and the maximum number of electrons in a principal energy level. Theoretically, the number of orbitals and number of electrons continue to increase for higher values of . However, no atom actually has more than 32 electrons in any of its principal levels. Principal Energy Number of Orbitals Present Level (n) Total Number Maximum Number of Orbitals (n 2 ) of Electrons (2n 2 ) 1 1 2 2 4 8 3 9 18 4 16 Each orbital will also have a probability pattern that is determined by interpreting Schrödinger's equation. The 3-dimensional probability pattern for the single orbital in the sub-level is a sphere. The probability patterns for the three orbitals in the sub-levels are shown below – independently and all three together. The probability patterns for the five orbitals are more complicated, and the seven orbitals are even more complicated. You should keep in mind that no matter how complicated the probability pattern is, each shape represents a single orbital, and the entire probability pattern is the result of the various positions that either one or two electrons can take. Reading Comprehension Review D