Question

1. Consider the wavefunction of the 2s orbital of the hydrogen atom: -Dexp (-) where do is the Bohr's radius (0.52918 nm). (25) = 42 (a) (15pt) Determine the expectation value of the potential < > of the 2s orbital in ev. (b) (10pt) Determine the expectation value of the kinetic energy of the 2s orbital in eV. (c) (5pt) Determine the location of the radial node (if there is any) in nm. (d) (5pt) Determine the location of the angular node (if there is any). (e) (10pt) Determine the most probable distance of 2s in nm.

Answer 1

Detailed solution of the above problem are as follows

diven that Honce ce2 40€ T t1 = 2 X . - h290 wave function of the 25 orbital of the hydrogen atom: 4 (25) stogon (2-% Jexp(-a) Bohar's Radius a=0.52918 hm. (9) The expectation value of the potential <cente) = me <> men <$) - no where h=2 because wave function. iA a orbital 25 Cnss = 1.44 ev.nmix I 4x0.52916 mm. = 0.68029 evi Kerst) = 0.60 629 evil (6) The Expectation value of kinetic Energy of the 25 orbital. using Virial. theorem. <k. f) = n <P.E .P. f = ez ft. 45€ so n=-1 =-1&(-0.68029) <R.E.) = 0.3 40148 evl (c) Location of Radial node At the radical node wave function will be zero 30 4 (25) = 1 1 2 4 28/12 (2-2) exp ( 1 ) = 0 (2-3 ) = 0 1 1 1 932 exp(-2) to 8=290 8=2x0.52918 hm. 18 = 1.0588 6 hom

(6) Location of the angular node Gimce wave function is of the orbital 2s. So h=2 & d=0. Due to l=0 there is no ongular hode Honce - angular hodes=0 (e) most probable dentonce for the most probable distance. dr 1 Rne 122] =0 2 a Catar) ² (2-8 3 explored 82] =0 ak [14 12 + y - 49 ) exp(-3) = 0 18r+ 483 2 - 12 8 2 ) exp (-) + (-2) e 190 (4+ 8 + 413 (-1er + 8 82 + B - 8 ) = 0 9. → = 5-2.3an m. or 8 = 5.23x 0.52910 [8=2-7676 him.