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Unformatted text preview: Chem140B – Physical Chemistry II Lecture Homework #7 March 17, 2009 due March 24, 2009 Q9.6) Is it always true that the probability of finding the electron in the H atom is greater in the interval than in the interval r dr r r dr − < < + r dr r r dr d d d d θ θ θ θ θ φ φ φ φ φ − < < + − < < + − < < + ? The interval corresponds to a spherical shell, whereas the interval r dr r r dr − < < + r dr d r dr r d d d θ θ θ θ θ φ φ φ φ φ − < < + − < < + < + − < corresponds to a small portion of the same spherical shell. Because the probability in any interval is a non- negative number, the total probability of finding the electron in the H atom must be bigger in the entire spherical shell than in a small portion of the same spherical shell. Q9.7) What possible geometrical forms can the nodes in the angular function for p and d orbitals in the H atom have? What possible geometrical forms can the nodes in the radial function for s , p , and d orbitals in the H atom have? The angular nodes for the p orbitals are planes, and the angular nodes for the d orbitals can be planes or cones. All spatial nodes for the H atom orbitals are spherical surfaces, corresponding to radial nodes. Q9.8) If the probability density of finding the electron in the 1 s orbital in the H atom has its maximum value for r = 0, does this mean that the proton and electron are located at the same point in space? No, the probability is the product of the probability density and the volume over which the probability is being determined. Because the volume of the nucleus is very small, the probability of finding the electron in the volume corresponding to the nucleus is very small as shown in Example Problem 9.4. P9.6) How many radial and angular nodes are there in the following H orbitals set? a) ( 2 , , x p r ) ψ θ φ b) ( ) 2 s r ψ c) ( ) 3 , , xz d r ψ θ φ d) ( ) 2 2 3 , , x y d r ψ θ φ − The functions have n – l –1 radial nodes and l angular nodes. Therefore a) ( 2 , , x p r ) ψ θ φ has no radial nodes and one angular node. b) ( ) 2 s r ψ has one radial node and no angular nodes. c) ( 3 , , xz d r ) ψ θ φ has no radial nodes and 2 angular nodes. d) ( 2 2 3 , , x y d r ) ψ θ φ − has no radial nodes and 2 angular nodes. P9.11) Calculate the mean value of the radius r at which you would find the electron if the H atom wave function is ( ) 100 . r ψ r a d d r e r a r e dr r a r a = = d r z z z z − ∞ − ∞ 1 4 3 2 3 2 3 3 2 π φ θ θ π π sin Using the standard integral r e n n r n − ∞ + z = α α 1 !...
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