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The expectation value, (r), for a hydrogen atom in the 3d₂2 orbital can be written 4 [° dr r³ (1) * (r) = 8 (3⁹.5) ₂³ e -2r/3a0 where the integrals over 0 and have already been evaluated and included in this expression. (a) Starting with the integral shown above, define x = r/ao and use this to simplify this integral by expressing it as an integration over x. Be careful and don't forget about the volume element and the integration limits. 5 16πt (b) From part (a), you should now have an integral over x and this variable essentially represents the radial distance of the electron in units of að. Determine (r) for this state. You may find the tabulated integral given above (before problem 3) helpful. (c) The angular part of the 3d₂² orbital is given by the spherical harmonic, Y₂⁰ (0, 6): Y₂(0, 4) = Using the standard limits of these variables (0 ≤ 0≤ ñ; 0 ≤ 0 ≤ 2π), determine the angles at which this function has nodes. (3 cos² 0 - 1)

The expectation value, (r), for a hydrogen atom in the 3d₂2 orbital can be written 4 [° dr r³ (1) * (r) = 8 (3⁹.5) ₂³ e -2r/3a0 where the integrals over 0 and have already been evaluated and included in this expression. (a) Starting with the integral shown above, define x = r/ao and use this to simplify this integral by expressing it as an integration over x. Be careful and don't forget about the volume element and the integration limits. 5 16πt (b) From part (a), you should now have an integral over x and this variable essentially represents the radial distance of the electron in units of að. Determine (r) for this state. You may find the tabulated integral given above (before problem 3) helpful. (c) The angular part of the 3d₂² orbital is given by the spherical harmonic, Y₂⁰ (0, 6): Y₂(0, 4) = Using the standard limits of these variables (0 ≤ 0≤ ñ; 0 ≤ 0 ≤ 2π), determine the angles at which this function has nodes. (3 cos² 0 - 1)

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purpose, the integral [ ²x² e-Pxdr = 3² 0 will be helpful after you have made a suitable change of coordinate. n! Bn+1
Transcribed Image Text:purpose, the integral [ ²x² e-Pxdr = 3² 0 will be helpful after you have made a suitable change of coordinate. n! Bn+1
The expectation value, (r), for a hydrogen atom in the 3d₂2 orbital can be written ∞ 4 r = [² dr r² (2) * e-2r/3a0 (r) = = 8 (3⁹.5) a ³ where the integrals over 0 and have already been evaluated and included in this expression. (a) Starting with the integral shown above, define x = r/ao and use this to simplify this integral by expressing it as an integration over x. Be careful and don't forget about the volume element and the integration limits. (b) From part (a), you should now have an integral over x and this variable essentially represents the radial distance of the electron in units of do. Determine (r) for this state. You may find the tabulated integral given above (before problem 3) helpful. (c) The angular part of the 3d₂2 orbital is given by the spherical harmonic, Y₂ (0,4): 5 -√√ 16π Y₂ (0,0) = (3 cos² 0 - 1) Using the standard limits of these variables (0 ≤ 0≤ л; 0≤ ≤ 2π), determine the angles at which this function has nodes. (d) Describe the nodal surfaces for the orbital, as defined by the values of the angles where Y₂ (0,4) is zero. In other words, identify the locations and types of the angular nodal surfaces in the Cartesian axis system.
Transcribed Image Text:The expectation value, (r), for a hydrogen atom in the 3d₂2 orbital can be written ∞ 4 r = [² dr r² (2) * e-2r/3a0 (r) = = 8 (3⁹.5) a ³ where the integrals over 0 and have already been evaluated and included in this expression. (a) Starting with the integral shown above, define x = r/ao and use this to simplify this integral by expressing it as an integration over x. Be careful and don't forget about the volume element and the integration limits. (b) From part (a), you should now have an integral over x and this variable essentially represents the radial distance of the electron in units of do. Determine (r) for this state. You may find the tabulated integral given above (before problem 3) helpful. (c) The angular part of the 3d₂2 orbital is given by the spherical harmonic, Y₂ (0,4): 5 -√√ 16π Y₂ (0,0) = (3 cos² 0 - 1) Using the standard limits of these variables (0 ≤ 0≤ л; 0≤ ≤ 2π), determine the angles at which this function has nodes. (d) Describe the nodal surfaces for the orbital, as defined by the values of the angles where Y₂ (0,4) is zero. In other words, identify the locations and types of the angular nodal surfaces in the Cartesian axis system.
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