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Sample Problem 7.5 Unit II - Lecture 7 PROBLEM: PLAN: Chemistry Applying the Uncertainty Principle An electron moving near an atomic nucleus has a speed 6x106 ± 1%. What is the uncertainty in its position ("x)? The uncertainty ("x) is given as ±1% (0.01) of 6x106 m/s. Once we calculate this, plug it into the uncertainty equation. SOLUTION: The Molecular Nature of Matter and Change "u = (0.01)(6x106 m/s) = 6x104 m/s h "x * m "u ! 4# Fifth Edition "x ! 6.626x10-34 kg*m2/s ! 1 x 10-9 m 4# (9.11x10-31 kg)(6x104 m/s) Martin S. Silberberg Copyright ! The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Figure 7.16 The Schrödinger Equation H$ = E$ wave function d2$ dx2 + d2$ dy2 mass of electron + d2$ dz2 how & changes in space + 8#2m% h2 potential energy at x,y,z (E-V(x,y,z)$(x,y,z) = 0 total quantized energy of the atomic system Electron probability density in the ground-state H atom. Quantum Numbers and Atomic Orbitals Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals Name, Symbol Allowed Values (Property) Quantum Numbers 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 Principal, n Positive integer (size, energy) (1, 2, 3, ...) Angular momentum, l (shape) 0 to n-1 1 2 0 0 0 0 3 1 0 1 2 ml the magnetic moment quantum number - an integer from -l to +l Magnetic, ml -l,…,0,…,+l (orientation) 0 -1 0 +1 -1 0 -2 -1 +1 0 +1 +2 Sample Problem 7.6 PROBLEM: PLAN: Determining Quantum Numbers for an Energy Level What values of the angular momentum (l) and magnetic (ml) quantum numbers are allowed for a principal quantum number (n) of 3? How many orbitals are allowed for n = 3? Sample Problem 7.7 Determining Sublevel Names and Orbital Quantum Numbers 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 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 SOLUTION: 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. What is wrong with each of the following quantum numbers designations and/or sublevel names? n l ml (a) 1 1 0 1p 4 3 +1 4d (c) 3 1 -2 3p (a) n = 1 only l = 0. Name 1s. (b) l = 3 is an f sublevel. Name 4f. (c) l The 2p orbitals. # 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 Figure 7.17 l = 1 can only have ml of -1, 0, +1. Figure 7.19 Figure 7.18 sublevel name possible ml values Name (b) SOLUTION: n Identifying Incorrect Quantum Numbers Sample Problem 7.8 PROBLEM: Combine the n value and l designation to name the sublevel. Knowing l, we can find ml and the number of orbitals. PLAN: The 3d orbitals. -3, -2, -1, 0, 1, 2, 3 7 Figure 7.20 Figure 7.19 continued Figure 7.21 The energy levels in the H atom. One of the seven possible 4f orbitals.