Question

In a one electron system, the probability of finding the electron within a shell of thickness or at a radius of r from the nucleus is given by the radial distribution function P() PR). An electron in a 1s hydrogen orbital has the radial wavefunction R(r) given by: R(r)-21" ne rn, where ao is the Bohr radius (52.9 pm) Calculate the probability of finding the electron in a sphere of radius 2.4ao centered at the nucleus. Number 95

Answer 1

The probability of finding the electron is, 2r 4 e a dr P 2r 4 (2.4 e a dr 3 0. 2.4a, 2r 2r 4 |dr -2 2r -2 a 2.4a 2r -ar'e 2r 4 a +afre dr 3 2.4a, 2r -ar'e +a rfe a dr-fe 2r 2r 4 a (1)đr 2.4a 2r 2r 2r -ar'e a 4 +a r -2 3 -2 a ||

2.4a, 2r 2r 2r 4-ar'e a a e +a r -2 -2 a 2.4a 2r 2r 2r a² 4 + ar- (-2) 4 2( 2.4ao) 2(2.4a) 2(2.4a) -a, (2.4a,) e (2.4a,)e =D4 -1 %3D 2 4 = 4(-0.0237–0.00986+0.248) = 0.8577 = 85.77% ||