purpose, the integral[ ²x² e-Pxdr = 3²0will 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∞4r= [² dr r² (2) * e-2r/3a0(r) ==8(3⁹.5) a ³where the integrals over 0 and have already been evaluated and included in thisexpression.(a) Starting with the integral shown above, define x = r/ao and use this to simplify thisintegral by expressing it as an integration over x. Be careful and don't forget about thevolume element and the integration limits.(b) From part (a), you should now have an integral over x and this variable essentiallyrepresents 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-√√16πY₂ (0,0) =(3 cos² 0 - 1)Using the standard limits of these variables (0 ≤ 0≤ л; 0≤ ≤ 2π), determine the anglesat which this function has nodes.(d) Describe the nodal surfaces for the orbital, as defined by the values of the angles whereY₂ (0,4) is zero. In other words, identify the locations and types of the angular nodalsurfaces in the Cartesian axis system.

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Explain each step