Radial and angular nodes and their significance

1. Radial and angular nodes and their significance.
Mr. Mithil Fal Desai, Ph. D.
Assistant Professor in Chemistry
Shree Mallikarjun and Shri Chetan Manju Desai College
Canacona, Goa

2. Wave function
𝛙(r, 𝛉, 𝛟) = R(r), Y(𝛉, 𝛟)
Angular
wave
function
Radial
wave
function

3. Orbital
Quantum
number
Radial component of wave function ‘R(r)’
Radial component of wave
function ‘Y(𝛉, 𝛟)’
1s
n = 1, l =
0
2(
𝑧
𝑎°
)3/2 𝑒−𝑍𝑟/𝑛𝑎° (
1
4𝜋
)
1
2
2p
n = 2, l =
1
1
24
1
2
𝑧
𝑎°
3
2 2𝑍r
n𝑎°
𝑒−𝑍𝑟/𝑛𝑎° (
3
4𝜋
)
1
2 𝑐𝑜𝑠𝜃
Radial and angular component of wave function
r = distance from nucleus, e = base of natural logarithm, Z = atomic number,𝑎°= Bohr radius
(52.9 pm) and n = principal quantum number

4. R(r)= 2(
𝑧
𝑎°
)3/2
𝑒−𝑍𝑟/𝑛𝑎°
Radial wave function of 1s orbital

5. 𝛙 =
𝟏
(𝜋𝑎°
3
)
𝟏
𝟐
𝐞−𝐫/𝐚°
Wave function of S orbital and boundary surface
x
y
z

6. R(r) =
𝟏
𝟖
𝟏
𝟐
𝒛
𝒂°
𝟑
𝟐
𝟐 −
𝟐𝒁𝒓
𝒏𝒂°
𝒆−𝐙𝐫/𝐧𝒂°
Radial wave function of 2s electrons
Node Radial wave function is zero

7. Radial wave function of 2p orbital
𝑅 𝑟 =
1
24
1
2
𝑧
𝑎°
3
2 2𝑍r
n𝑎°
𝑒−𝑍𝑟/𝑛𝑎°

8. 𝛙 =
𝟏
𝟒(2𝜋)
𝟏
𝟐
𝒛
𝐚°
𝟓
𝟐
𝐫 𝐜𝐨𝐬𝛉 𝐞−𝐙𝐫/𝟐𝐚°
Wave function of p orbital
+
x
y
z
-

9. Boundary surface of p orbital
x
y
z
-
+
Angular wave function
zero along xy plane
Angular node