Show work for each please

4. Consider the hydrogen atom consisting of one proton and one electron. The best distance unit for the hydrogen atom is the bohr radius x -raind cans of 5.29=10""m, 0.529 Angstroms (1 Angstrom - 10 meters) or 52.9 y =rain sing picometyers. The radius & below is in these units. That is r=1 means that r= 0.529 Angstroms, or roughly half an angstrom. The average radius for a Is electron in hydrogen is 1.5 bohr radii or 0,794 Angstroms or 79.4 picameters (1 picometer = 10#2 meters). In spherical coordinates the wave function for the 2s orbital of a hydrogen atom is given by: The wave function for the 2pr orbital of a hydrogen atom is given by: if cos D 412n The wave function for the 3d , orbital of a hydrogen atom is given by: -ple " (Joos' 0-1) The orbital of hydrogen is normalized if | | | Y ar -1. Where d is the appropriate volume element. The average value. (f). of a quantity /(r.0, c] is calculated as the integral () - [ [ [ v prav. Where off' is the appropriate volume element (a) (4 pts) Write the full integral needed to show that the 2p, orbital is normalized. To calculate the average radius (ie. () = (r)) of an electron in an orbital we only have to consider the nonnalized radial wave function and integrate over . For the 2s, 2p. and 3d orbitals, the radial wave flanctions are: R()=-(2-re R (r ) = -red Rule]= 2v2 V24 81VIS (b) (4 pis) Perform the necessary integrations to show that the 2p, orbital is normalized. Name: (c) (4 pts) Use the normalized radial wave function on the bottom of Page-6 to show that the average radius of an electron in a 2s orbital of hydrogen is 6 bohr radii. (d) (2 pts) Given values of 1.5 for the Is orbital, 5 for the 2p orbital, 10.5 for the 3d orbital and 13.5 for the 35 orbital, what is the significance of this value for 2s