Table of contents | |
Quantum Numbers | |
Orbitals | |
Shapes of Atomic Orbitals | |
Understanding Atomic Orbitals through Boundary Surface Diagrams |
In the solution to the Schrodinger equation for the hydrogen atom, three quantum numbers arise from the spatial geometry of the solution and a fourth arises from electron spin.
No two electrons can have an identical set of quantum numbers according to the Pauli exclusion principle, so the quantum numbers set limits on the number of electrons that can occupy a given state and therefore give insight into the building up of the periodic table of the elements.
The principal quantum number or total quantum number n arises from the solution of the radial part of the Schrodinger equation for the hydrogen atom. The bound state energies of the electron in the hydrogen atom are given by En
n = 1, ℓ = 0
n = 2, ℓ = 0, 1. ; i.e. 2 values of ℓ.
n = 3, ℓ = 0, 1, 2. i.e. 3 values of ℓ.
No. of values of `l' is equal to `n' principal quantum number
It identifies sub shell in an atom
The value of `l' gives the name of the sub-shell and the shape of the orbital.
ℓ | Notation | Name | Shape |
o | s | Sharp | Spherical |
1 | P | Principal | dumbbell-shaped |
2 | d | diffuse | double dumbbell |
3 | f | fundamental | complex |
These orbitals are designated by a set of 3 quantum number (n, l, m) which arise as a natural consequence in the solution of Schrodinger equation i.e. the values of 3 quantum numbers are restricted by the solution of Schrodinger equation.
Different Atomic Orbitals
The wave function (ψ) representing an electron in an atom is just a mathematical tool without any physical meaning. It varies for different orbitals, and while it's not something tangible, we can look at plots of these wave functions to understand certain aspects.
According to Max Born, a German physicist, the square of the wave function (ψ²) at a point tells us about the probability density of finding the electron there.
To visualize these probability density variations, you can look at charge cloud diagrams. Here, the density of dots in a region represents the electron's probability density in that area.
Charge Cloud Diagram of s-Orbitals
Boundary surface diagrams with constant probability density provide a helpful representation of orbital shapes. In this representation, a boundary or contour surface is drawn for an orbital where the probability density |ψ|² remains constant. While many such surfaces are possible, the chosen diagram encloses a region where the probability of finding the electron is high, typically around 90%.
You might wonder why we don't draw a boundary surface diagram enclosing a region with 100% probability of finding the electron. The answer lies in the fact that the probability density |ψ|² always has some value, no matter how small, at any finite distance from the nucleus. Therefore, it's not feasible to draw a fixed-size boundary surface diagram where the probability of finding the electron is 100%.
Sphere for s Orbitals:
Spheres for s-Orbitals
2p Orbitals with Lobes
2p Orbitals
d Orbitals and their Shapes:
d Orbitals and Their Shapes
- Besides the points where the probability density is zero along a line from the nucleus (radial nodes), the probability density functions for the np and nd orbitals are also zero at specific planes passing through the nucleus.
- For instance, in the case of the pz orbital, the xy-plane is a nodal plane. In the dxy orbital, two nodal planes pass through the nucleus and divide the xy plane that contains the z-axis. These planes are called angular nodes, and the number of angular nodes is determined by the quantum number 'l'. For p orbitals, there is one angular node, for d orbitals, there are two, and so on.
- The total number of nodes, which includes both radial and angular nodes, is given by the formula (n–1), where 'n' is the principal quantum number.
172 videos|306 docs|152 tests
|
1. What are quantum numbers and how do they relate to atomic orbitals? |
2. What are atomic orbitals and how are they related to the shapes of orbitals? |
3. How can we understand atomic orbitals through boundary surface diagrams? |
4. How does the principal quantum number (n) affect the energy level of an electron in an atomic orbital? |
5. What is the significance of the spin quantum number (ms) in atomic orbitals? |
|
Explore Courses for NEET exam
|