Quantum Theory and the Electronic Structure of Atoms PDF

Quantum Theory and the Electronic Structure of Atoms / 12 said alabbar Chapter two QUANTUM THEORY AND THE ELECTRONIC STRUCTURE OF ATOMS Quantum Mechanics and Quantum Numbers Electron Configura...

Quantum Theory and the Electronic Structure of Atoms / 12 said alabbar Chapter two QUANTUM THEORY AND THE ELECTRONIC STRUCTURE OF ATOMS Quantum Mechanics and Quantum Numbers Electron Configurations and the Aufbau Principle QUANTUM MECHANICS AND QUANTUM NUMBERS STUDY OBJECTIVES 1. Describe the use of the terms probability, electron density, and orbital in the wave mechanical model of the hydrogen atom. 2. Describe the significance of the three quantum numbers that correspond to an orbital in an atom. 3. Describe the shapes of s, p, and d orbitals. 4. State the Pauli exclusion principle. The dual nature of the electron is further described by the Heisenberg uncertainty principle which states that it is impossible to know simultaneously both the momentum and the position of an electron with certainty. The principle can be interpreted as follows: If we want to know the energy of an electron in an atom with only a small uncertainty, then we will have a large uncertainty in knowing the position of the electron in some region of the atom. The result of this principle is that we can only know the probabilty of finding the electron in a certain region of the atom. Quantum Mechanics. In 1926 Erwin Schrödinger incorporated all the wave and particle properties into a form of mechanics called wave mechanics or quantum mechanics. He formulated the now famous Schrödinger wave equation and obtained a set of mathematical functions called wave functions, ψ. For any set of coordinates within the atom, ψ has a numerical value. The square of the wave function ψ2 is related to the probability that an electron will be found in a small region (volume) in space. Where ψ2 is large, the probability of finding the electron is high. Where ψ2 is small, the probability of finding the electron is low. The term electron density (Figure 7.22 in the text) is used to represent the probability concept. The term electron density refers to the probability of finding the electron in a certain region of the atom. The electron density must not be interpreted to mean that one electron occupies the whole of the space covered by the probability distribution at one time. Rather, the electron is considered a point charge, and the electron density refers to the probability of finding the electron in a specific volume of the atom. As a result of the probability concept, we no longer speak of distinct Bohr orbits. In the wave mechanical model of an atom, the Bohr orbit is replaced with an orbital or atomic orbital. Orbitals are regions in an atom within which electrons have a high probability, usually 90 percent, of being found. Each atomic orbital has a characteristic shape and energy. Atomic Orbitals. An orbital is a graphical representation of the electron probability around the nucleus. The density of the probability cloud at each region in space (see Figure 7.22 text) represents the probability of finding the electron there. Orbitals are often drawn using equal-probability contours. If we go far enough from the nucleus in a particular direction, we come to a point at which 90 percent of the time the electron, when it is in that direction, will be inside that point. Connecting all such points produces an equal-probability boundary surface such that 90 percent of the time the electron will be inside that surface no matter what direction from the nucleus we consider. This graphical representation gives the shape of the orbital. The different types of orbitals can be recognized by their shapes. While there are more than three types of orbitals only the characteristics of s, p, and d orbitals are given in Table 7.1. The shapes of s and p orbitals are shown in Figure 7.18 and 7.19 of the text. Back Forward Main Menu TOC Study Guide TOC Textbook Website MHHE Website  13 / Quantum Theory and the Electronic Structure of Atoms Table 7.1 Symbols and Shapes of Orbitals __________________________________________________________________ Numberof orbitals Shape of 90 percent Orbital Designation per shell boundary surface __________________________________________________________________ s ns 1 spherical p np 3 dumbbell (2 lobes) d nd 5 4 lobes __________________________________________________________________ Quantum Numbers. Both the energy and the probability distribution are described by a set of numbers called quantum numbers. The principal quantum number n determines the energy of a hydrogen atom according to the formula En = –R H (1/n 2 ). This number can have any integral value, n = 1, 2, 3,…. The larger the value of n, the greater the energy, and the farther the electron is from the nucleus on the average. When n = 1, the electron in the hydrogen atom is said to be in the ground state. All other values of n correspond to excited states. When n = ∞ , the highest possible energy, the atom has lost the electron and exists as a H+ ion. The angular momentum quantum number l is related to the shape of the orbital. Within each principal energy level one or more orbital types will exist. Thus any one principle quantum level may have more than one value for l. For a given n, l has integral values from 0 to (n – 1); l = 1, 2, 3,…, (n – 1). Each value of l defines a group of orbitals composing a specific sublevel or subshell. The shape of atomic orbitals will be discussed in the next section. For now each orbital is specified by a letter designation. Orbitals for which l = 0 are also referred to as s orbitals. Electrons in s orbitals have zero units of angular momentum. The letters used to designate orbitals with different l values are l 0 1 2 3 4 … ________________________________________ orbital s p d f g … The magnetic quantum number m l describes the orientation of an orbital in space. The value of m l may have integral values from –l to +l, including zero. ml = –l, (–l + 1), …, 0, …, (l – 2), (l – 1), +l The number of ml values is very important because it designates the number of orbitals within a subshell. For example, when l = 2, then ml = –2, –1, 0, 1, 2. We see there are five ml values. d orbitals always occur in groups of five orbitals. These five d orbitals all have the same energy, but have five different orientations in space. They make up the d subshell. A subshell is a group of orbitals designated by a particular l value within a shell. Table 7.2 shows the number of orbitals in the most common subshells. Table 7.2 The Number of Orbitals in Various Subshells ___________________________________________________________ Type of l Number of Orbital Value ml Values Orbitals ___________________________________________________________ s 0 0 1 p 1 –1, 0, 1 3 d 2 –2, –1, 0, 1, 2 5 f 3 –3, –2, –1, 0, 1, 2, 3 7 ___________________________________________________________ The electron spin quantum number, ms, designates one of two possible spin directions for electrons. The two possible values of ms are +1/2 and –1/2. The Pauli exclusion principle states that no two Back Forward Main Menu TOC Study Guide TOC Textbook Website MHHE Website  Quantum Theory and the Electronic Structure of Atoms / 14 electrons in an atom can have all four quantum numbers the same. Accordingly, this principle limits the numbers of electrons each atomic orbital can hold to no more than two electrons, one electron will have m s = + 1/2 and the other ms = – 1/2. Let's take an electron in a particular orbital, say, a 3px orbital. The three quantum numbers associated with this orbital are n = 3, l = 1, and m l = –1. Any electrons in this orbital must have these identical n, l, and ml quantum numbers. The two sets of quantum numbers for electrons in this orbital differ in their ms values: electron 1: n = 3, l = 1, ml = –1, ms = + 1/2 electron 2: n = 3, l = 1, ml = –1, ms = – 1/2 The result of this qualitative treatment of wave mechanics gives several essential pieces of information about electrons in atoms: 1. Quantum numbers (n, l, m) and their relationships to each other. 2. Electron probability density and atomic orbitals. 3. The maximum number of electrons per orbital. 4. An electron shell is a collection of orbitals with the same value of n. 5. A subshell is a set of orbitals with the same values of n and l. _______________________________________________________________________________ EXAMPLE 7.5 Quantum Numbers If the principal quantum number of an electron is n = 2, what values could the following have? a. Its l quantum number b. Its m l quantum number c. Its m s quantum number Method of Solution a. Recall that the angular momentum quantum number l has values that depend on the value of the principal quantum number n; l = 0, 1, 2, …, (n – 1). Answer: When n = 2, l = 0, 1. b. The magnetic quantum number ml depends on the value of l where ml = –l , –l +1, …, 0,…, l + 1, l. Answer: When l = 1, ml = –1, 0, 1; and when l = 0, ml = 0. c. Answer: The ms quantum number for a single electron could be either +1/2 or – 1/2. _______________________________________________________________________________ EXAMPLE 7.6 Quantum Numbers and Orbitals List all the possible types of orbitals associated with the principal energy level n = 5. Method of Solution The type of orbital is given by the angular momentum quantum number. When n = 5; l = 0, 1, 2, 3, and 4. Answer: These correspond to subshells consising of s, p, d, and f orbitals. _______________________________________________________________________________ EXAMPLE 7.7 Quantum Numbers For the following sets of quantum numbers for electrons, indicate which quantum numbers — n, l, m l — could not occur and state why. a. 3, 2, 2 b. 2, 2, 2 c. 2, 0, –1 Back Forward Main Menu TOC Study Guide TOC Textbook Website MHHE Website  15 / Quantum Theory and the Electronic Structure of Atoms Method of Solution a. When n = 3, l = 0, 1, 2. When l = 2; ml = –2, –1, 0, 1, 2. Answer: Thus set (a) can occur. b. When n = 2, l = 0, 1. Answer: The set in (b) cannot occur because l ≠ 2 when n = 2. This imeans that the n = 2 energy level cannot have d orbitals. c. When n = 2, l = 0, 1. When l = 0, ml = 0. Answer: The set in (c) cannot occur because ml ≠ –1 when l = 0. _______________________________________________________________________________ EXAMPLE 7.8 Quantum Numbers and Orbitals What values can ml take for a. a 3d orbital? b. a 2s orbital? Method of Solution The value of ml depends only on l and so first convert each orbital designation to the corresponding value of l a. For a d orbital l = 2; therefore, possible m l values are –2, –1, 0, 1, and 2. b. For an s orbital l = 0; therefore, ml = 0. _______________________________________________________________________________ EXERCISES 12. Which of the following sets of quantum numbers are not allowed for describing an electron in an orbital? n l ml ms a. 3 2 –3 1/2 b. 2 3 0 –1/2 c. 2 1 0 –1/2 13. Which choice is a possible set of quantum numbers for the last electron added to make up an atom of gallium (Ga) in its ground state? n l ml ms a. 4 2 0 –1/2 b. 4 1 0 1/2 c. 4 2 –2 –1/2 d. 3 1 +1 1/2 e. 3 0 0 –1/2 14. Which of the following are incorrect designations for an atomic orbital? a. 3f b. 4s c. 2d d. 4f 15. How many orbitals in an atom can have the following designations? a. 2s b. 3d c. 4p d. n = 3 16. For each of the following give the subshell designation, the ml values, and the number of possible orbitals. a. n = 3 l = 2 b. n = 4 l = 3 c. n = 5 l = 1 17. For each of the following subshells give the n and l values, and the number of possible orbitals. a. 2s b. 3p c. 4f Back Forward Main Menu TOC Study Guide TOC Textbook Website MHHE Website  Quantum Theory and the Electronic Structure of Atoms / 16 ELECTRON CONFIGURATIONS AND THE AUFBAU PRINCIPLE STUDY OBJECTIVES 1. Compare the order of orbital energies in the hydrogen atom to the order in a many-electron atom. 2. Use the Aufbau principle to write the electron configuration of an element. 3. Use Hund's rule to construct orbital diagrams for the electron configurations of elements. The Energies of Orbitals. The way electrons are distributed among the orbitals of an atom is its electron configuration and electronic structure. Each of the principal energy levels (called shells) of an atom can be divided into subshells, which are sets of orbitals. The s subshell has one s orbital, the p subshell has three p orbitals, the d subshell has five d orbitals, and so on. In addition each subshell has an electron capacity based on a maximum of two electrons per orbital. The orbitals available to a shell and the subshell capacities are summarized in Table 7.3. Table 7.3 Allowed Numbers of Orbitals per Energy Level ________________________________________________________________________ Principal Orbital Shape Orbital Number of Electron Capacity Quantum l Name Orbitals per Subshell Number per Subshell ________________________________________________________________________ 1 0 1s 1 2 2 0 2s 1 2 2 1 2p 3 6 3 0 3s 1 2 3 1 3p 3 6 3 2 3d 5 10 4 0 4s 1 2 4 1 4p 3 6 4 2 4d 5 10 4 3 4f 7 14 ________________________________________________________________________ In a hydrogen atom all of the orbitals in a particular energy level (same n value) have the same energy. In a hydrogen atom the electron resides in the 1s orbital because it has the lowest possible energy in that orbital. The ground state electron configuration of H is 1s1 , as shown below. The number of electrons per subshell 1 1s Principal electron shell (value of n) Type of orbital occupied Atoms of elements other than hyrogen have more than one electron and are called many-electron atoms. In these atoms each principal energy level is split into several energies. These energies correspond to the orbitals of one type having a slightly different energy than orbitals of another type. This splitting is shown in Figure 7.1. The energy of an orbital in a many-electron atom depends on both n and l. For a given value of n, the Back Forward Main Menu TOC Study Guide TOC Textbook Website MHHE Website  17 / Quantum Theory and the Electronic Structure of Atoms energy of orbitals increases as l increases. Therefore, for atoms other than hydrogen atoms, the 2p orbital has a higher energy than the 2s orbital. 4d 5s 4p 3d 4s 3p Energy 3s 2p 2s 1s Figure 7.1 The orbital energies in a many-electron atom. Within a shell the orbital energies increase as the l quantum number increases. The energy of an orbital depends on its n and l values. Here the 4s subshell is shown to be lower than the 3d subshell. The order of increasing energy for orbitals within a given principal energy level is s