Unit II - Lecture 7 Chemistry The Molecular Nature of Matter and Change Fifth Edition Martin S. Silberberg Copyright ! The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sample Problem 7.5 SOLUTION: PLAN: Applying the Uncertainty Principle PROBLEM: An electron moving near an atomic nucleus has a speed 6x10 6 ± 1%. What is the uncertainty in its position ("x)? The uncertainty ("x) is given as ±1% (0.01) of 6x10 6 m/s. Once we calculate this, plug it into the uncertainty equation. "u = (0.01)(6x10 6 m/s) = 6x10 4 m/s " x * m " u ! h 4# "x ! 4# (9.11x10 -31 kg)(6x10 4 m/s) 6.626x10 -34 kg*m 2 /s ! 1 x 10 -9 m The Schrödinger Equation H$ = E$ d 2 $ dy 2 d 2 $ dx 2 d 2 $ dz 2 + + 8# 2 m % h 2 (E-V(x,y,z)$(x,y,z) = 0 + how & changes in space mass of electron total quantized energy of the atomic system potential energy at x,y,z wave function Figure 7.16 Electron probability density in the ground-state H atom. Quantum Numbers and Atomic Orbitals An atomic orbital is specified by three quantum numbers. n the principal quantum number - a positive integer l the angular momentum quantum number - an integer from 0 to n-1 m l the magnetic moment quantum number - an integer from -l to +l Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals Name, Symbol (Property) Allowed Values Quantum Numbers Principal, n (size, energy) Angular momentum, l (shape) Magnetic, m l (orientation) Positive integer (1, 2, 3, ...) 0 to n-1 -l,…,0,…,+l 1 0 0 2 0 1 0 3 0 1 2 0 0 -1 +1 -1 0 +1 0 +1 +2 -1 -2
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Unit II - Lecture 7
ChemistryThe Molecular Nature of
Matter and Change
Fifth Edition
Martin S. Silberberg
Copyright ! The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Sample Problem 7.5
SOLUTION:
PLAN:
Applying the Uncertainty Principle
PROBLEM: An electron moving near an atomic nucleus has a speed 6x106 ± 1%.
What is the uncertainty in its position ("x)?
The uncertainty ("x) is given as ±1% (0.01) of 6x106 m/s. Once we
calculate this, plug it into the uncertainty equation.
"u = (0.01)(6x106 m/s) = 6x104 m/s
" x * m " u ! h
4#
"x !
4# (9.11x10-31 kg)(6x104 m/s)
6.626x10-34 kg*m2/s! 1 x 10-9 m
The Schrödinger Equation
H$ = E$
d2$
dy2
d2$
dx2
d2$
dz2+ +
8#2m%
h2(E-V(x,y,z)$(x,y,z) = 0+
how & changes in space
mass of electron
total quantized energy of the atomic system
potential energy at x,y,zwave function
Figure 7.16
Electron probability density in the ground-state H atom.
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