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Quantum Mechanics • Erwin Schrödinger developed a mathematical treatment into which both the wave and particle nature of matter could be incorporated. • It is known as quantum mechanics. The Quantum Mechanical Model • Energy is quantized - It comes in chunks. • A quantum is the amount of energy needed to move from one energy level to another. • Since the energy of an atom is never “in between” there must be a quantum leap in energy. • In 1926, Erwin Schrodinger derived an equation that described the energy and position of the electrons in an atom. Schrodinger’s Wave Equation d V 8 m dx h 2 2 2 2 E Equation for the probability of a single electron being found along a single axis (x-axis) Erwin Schrodinger Quantum Mechanics • The wave equation is designated with a lower case Greek psi (). • The square of the wave equation, 2, gives a probability density map of where an electron has a certain statistical likelihood of being at any given instant in time. Quantum Numbers • Solving the wave equation gives a set of wave functions, or orbitals, and their corresponding energies. • Each orbital describes a spatial distribution of electron density. • An orbital is described by a set of three quantum numbers. Principal Quantum Number, n (Shell) • The principal quantum number, n, describes the energy level on which the orbital resides. • The values of n are integers ≥ 0. distance of e- from the nucleus n=1 n = 1, 2, 3, 4, …. n=2 n=3 Azimuthal Quantum Number, l • This quantum number defines the shape of the orbital. • Allowed values of l are integers ranging from 0 to n − 1. • We use letter designations to communicate the different values of l and, therefore, the shapes and types of orbitals. for a given value of n, l = 0, 1, 2, 3, … n-1 n = 1, l = 0 n = 2, l = 0 or 1 n = 3, l = 0, 1, or 2 Value of l 0 1 2 3 Type of orbital s p d f l=1 l=0 l=2 p orbital (principal) s orbital (Sharp) d orbital (diffuse) l=3 f orbital (fundamental) Magnetic Quantum Number, ml • Describes the three-dimensional orientation of the orbital. • Values are integers ranging from -l to l: −l ≤ ml ≤ l. • Therefore, on any given energy level, there can be up to 1s orbital, 3p orbitals, 5d orbitals, 7f orbitals, etc. if l = 1 (p orbital), ml = -1, 0, or 1 if l = 2 (d orbital), ml = -2, -1, 0, 1, or 2 orientation of the orbital in space Magnetic Quantum Number, ml • Orbitals with the same value of n form a shell. • Different orbital types within a shell are subshells. Quantum Numbers and Atomic Orbitals An atomic orbital is specified by three quantum numbers. n the principal quantum number - a positive integer l the angular momentum quantum number - an integer from 0 to n-1 ml the magnetic moment quantum number - an integer from -l to +l Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals Name, Symbol (Property) Allowed Values Quantum Numbers Principal, n (size, energy) Positive integer (1, 2, 3, ...) 1 Angular momentum, l (shape) 0 to n-1 0 0 0 0 Magnetic, ml (orientation) -l,…,0,…,+l 2 3 1 0 1 2 0 -1 0 +1 -1 0 +1 -2 -1 0 +1 +2 Sample Problem Determining Quantum Numbers for an Energy Level What values of the angular momentum (l) and magnetic (ml) PROBLEM: quantum numbers are allowed for a principal quantum number (n) of 3? How many orbitals are allowed for n = 3? PLAN: Follow the rules for allowable quantum numbers found in the text. l values can be integers from 0 to n-1; ml can be integers from -l through 0 to + l. SOLUTION: For n = 3, l = 0, 1, 2 For l = 0 ml = 0 For l = 1 ml = -1, 0, or +1 For l = 2 ml = -2, -1, 0, +1, or +2 There are 9 ml values and therefore 9 orbitals with n = 3.0 Determining Sublevel Names and Orbital Quantum Numbers Sample Problem PROBLEM: Give the name, magnetic quantum numbers, and number of orbitals for each sublevel with the following quantum numbers: (a) n = 3, l = 2 (b) n = 2, l = 0 (c) n = 5, l = 1 (d) n = 4, l = 3 PLAN: Combine the n value and l designation to name the sublevel. Knowing l, we can find ml and the number of orbitals. SOLUTION: n l sublevel name possible ml values # of orbitals (a) 3 2 3d -2, -1, 0, 1, 2 5 (b) 2 0 2s 0 1 (c) 5 1 5p -1, 0, 1 3 (d) 4 3 4f -3, -2, -1, 0, 1, 2, 3 7