Radial distribution function and most probable distance of 1s and 2s electron

Radial distribution function and most probable distance of 1s and 2s electron

1. Radial and angular parts of the hydrogenic wave functions

2. -----------1   Schrodinger equation with Cartesian coordinates

3. Cartesian coordinates and polar coordinates θ φ r (x, y, z) or (r, θ , φ ) x axis z axis y axis x = cos y = r z = r cos m =  

4. -----------2   Schrodinger equation with polar coordinates We have,

5. -----------2   (r, = R(r) , Θ ( Φ )   Radial and angular component of wave Angular component Radial component

6. Radial component ‘R(r)’ of wave function ’ gives the distribution of electron as a function of radius ‘ r ’(distance from the nucleus) Radial wave function = R(r)   Radial component of wave function

7. Radial wave function depends on principle quantum number ‘ ’ and azimuthal quantum number ‘ l ’ and have a common function     Where, r = distance from nucleus e = base of natural logarithm Z = atomic number = Bohr radius (52.9 pm) = principal quantum number   Distribution curves for the Radial wave function

8. Orbital Quantum number Radial component of wave function ‘ R(r) ’ 1s n = 1, l = 0 2 2s n = 2, l = 0 Orbital Quantum number Radial component of wave function ‘ R(r) ’ 1s n = 1, l = 0 2s n = 2, l = 0 ‘ [ R(r)] 2 ’ gives the probability of finding electron. Distribution curves for the Radial wave function

9. r in pm R(r) [R(r)] 2 4π r 2 [R(r)] 2 0.005198122 2.70205E-05 52.5 0.001926797 3.71255E-06 0.128 52 2758 52.9 0.001912282 3.65682E-06 0.128 53 0144 53.5 0.001890715 3.5748E-06 0.128 51 3735 224.8 7.41823E-05 5.50301E-09 0.003492872 R(r) = 2   Most probable distance of 1s electron

10. Radial wave function Radial distribution function Most probable distance of 1s electron

11. R(r) =   Most probable distance of 2s electrons

12. Radial distribution function Radial wave function Most probable distance of 2s electrons

13. Probability of finding 1s electron is closer to nucleus than 2s electron. There are no radial nodes in 1s orbital. There is one radial node in 2s orbital. End Note

14. You can read more always Lee, J. D., (2018), Concise Inorganic Chemistry , Fifth Edition, New Delhi, Wiley India. Huheey , J. E., Keiter , E. A., Keiter , R. L., Medhi , O. K., (2019) Inorganic Chemistry: Principle of structure and reactivity , Fourth Edition, Noida, Pearson Education India. Atkins, P., Overton T., Rourke, J., Weller, M., Armstrong F., (2006 ) Shriver and Atkin’s Inorganic Chemistry , Fourth Edition, Great Britain, Oxford University Press.