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

1. Radial and angular parts of the
hydrogenic wave functions

2. 𝐝𝟐𝛙
𝐝𝐱𝟐
+
𝐝𝟐𝛙
𝐝𝐲𝟐
+
𝐝𝟐𝛙
𝐝𝐳𝟐
+
𝟖𝛑𝟐𝐦
𝐡𝟐
(𝐄 − 𝐕) = 𝟎
-----------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
x = r sin θ cos ϕ
y = r sin θ sin ϕ
z = r cos θ
m = μ

4. 𝟏
𝐫𝟐
𝐝
𝐝𝐫
(𝐫𝟐
𝐝𝛙
𝐝𝐫
) +
𝟏
𝐫𝟐 𝐬𝐢𝐧 𝛉
𝐝
𝐝𝛉
(𝐬𝐢𝐧 𝛉
𝐝𝛙
𝐝𝛉
) +
𝟏
𝐫𝟐 𝐬𝐢𝐧𝟐 𝛉
𝐝𝟐
𝛙𝟐
𝐝𝛟𝟐
+
𝟖𝛑𝟐
𝛍
𝐡𝟐
(𝐄 − 𝐕) = 𝟎
-----------2
Schrodinger equation with polar
coordinates
We have,

5. 𝟏
𝐫𝟐
𝐝
𝐝𝐫
(𝐫𝟐
𝐝𝛙
𝐝𝐫
) +
𝟏
𝐫𝟐 𝐬𝐢𝐧 𝛉
𝐝
𝐝𝛉
(𝐬𝐢𝐧 𝛉
𝐝𝛙
𝐝𝛉
) +
𝟏
𝐫𝟐 𝐬𝐢𝐧𝟐 𝛉
𝐝𝟐
𝛙𝟐
𝐝𝛟𝟐
+
𝟖𝛑𝟐
𝛍
𝐡𝟐
(𝐄 − 𝐕) = 𝟎
-----------2
𝛙(r, 𝛉, 𝛟) = R(r), Θ(𝛉), Φ(𝛟)
Radial and angular component of wave
Angular
component
Radial
component

6. Radial component ‘R(r)’ of wave function ′𝛙’ 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 ‘𝒏’
and azimuthal quantum number ‘l’ and have a common function
𝐞−𝒁𝒓/𝒏𝒂°
Where,
r = distance from nucleus
e = base of natural logarithm
Z = atomic number
𝑎°= Bohr radius (52.9 pm)
𝑛 = 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(
𝑧
𝑎°
)3/2
𝑒−𝑍𝑟/𝑛𝑎°
2s n = 2, l = 0
1
8
1
2
𝑧
𝑎°
3
2
2 −
2𝑍r
n𝑎°
𝑒−𝑍𝑟/𝑛𝑎°
‘[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 4πr2 [R(r)]2
0 0.005198122 2.70205E-05 0
52.5 0.001926797 3.71255E-06 0.128522758
52.9 0.001912282 3.65682E-06 0.128530144
53.5 0.001890715 3.5748E-06 0.128513735
224.8 7.41823E-05 5.50301E-09 0.003492872
R(r) =2(
𝑧
𝑎°
)3/2
𝑒−𝑍𝑟/𝑛𝑎°
Most probable distance of 1s electron

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

11. R(r) =
𝟏
𝟖
𝟏
𝟐
𝒛
𝒂°
𝟑
𝟐
𝟐 −
𝟐𝒁𝒓
𝒏𝒂°
𝒆−𝐙𝐫/𝐧𝒂°
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.