Video Lesson

Lesson Objectives

At the end of this lesson, students will be able to:

• State Heisenberg’s uncertainty principle
• Describe the significance of electron probability distribution.
• Explain the quantum numbers n, l, m, ms
• Write all possible sets of quantum numbers of electrons in an atom.
• Describe the shapes of orbitals designated by s, p and d.

Brainstorming Question

• If particles have wavelike motion, why can’t we observe its motion in the macroscopic world?
• If electrons possess particle nature it should be possible to locate
  electrons.
• How can an electron be located?
• Is there any wave associated with a moving elephant?

Key Terms and Concepts

1.6 The Quantum Mechanical Model of the Atom

The Bohr model of an electron orbiting around the nucleus, looking like a planet Around The Sun, doesn’t explain properties of atoms.

The planetary view of one charged particle orbiting another particle of opposite charge does not match some of the best-known laws of classical physics. Scientists have developed quantum mechanics, which presents a different view of how electrons are arranged about the nucleus in the atom. This View Depends on two central concepts the wave behavior of matter and the uncertainty principle. The combination of these two ideas leads to a mathematical description of electronic structure .

1.6.1 The Heisenberg Uncertainty Principle

• The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics formulated by Werner Heisenberg in 1927. It states that there is a limit to how precisely we can simultaneously know certain pairs of physical properties of a particle. The most commonly discussed pair is position and momentum.
• The Heisenberg Uncertainty Principle states that it is impossible to simultaneously and precisely measure both the position (x) and momentum (p) of a particle with perfect accuracy. More formally, the principle is expressed as:

• where; Δx- is the uncertainty in position,
• Δp- is the uncertainty in momentum,
• h- Planck constant, approximately 6.63×10⁻³⁴

This inequality implies that if you measure the position of a particle very precisely (Δx is small), then the uncertainty in its momentum (Δp) will be large, and vice versa.

• Erwin Schrödinger developed a complex mathematical equation that relates the wave properties associated with electrons to their energies.

• The results of a quantum mechanical treatment show that the electron may be visualized as being in rapid motion with in a region of space around the nucleus. The region in which an electron is most likely found is called an orbital.
• The electron may be located anywhere within an orbital at any instant of time, but it spends most of its time in certain high probability region like an “electron cloud” whose density varies with in the orbital.

1.6.2 Quantum Numbers

• In quantum mechanics, each electron in an atom is described by four quantum numbers.
• Three of the quantum numbers 𝑛, 𝓁 𝑎𝑛𝑑 𝑚𝑙 specify the wave function that gives the probability of finding the electron at various points in space.
• The Fourth quantum number (ms) refers to a magnetic property of electrons called spin.

The allowed values and general meanings of each of the four quantum numbers are discussed as follows:

The principal quantum number (n)

• describes the main energy level, or shell, an electron occupies. It maybe any positive integer, n=1,2,3,4,etc.
• It describes the Size And Energy Of the shell in which the orbital resides and it is analogous to the energy levels in Bohr’s model.

The Angular momentum quantum number (ℓ)

• The Angular Momentum Quantum Number(ℓ) designates the shape of the Atomic orbitals.
• With in a Shell different Sublevels or Subshells are Possible, each with a characteristic Shape .
• It takes values from 0 to n-1. Example , in n =4, ℓ has values of 0,1,2,and 3
• Orbitals of the same n, but different ℓ are said to belong to different subshells.
• The Angular Momentum Quantum Number(ℓ) is also called azimuthal quantum number.
• Each value of ℓ corresponds to an orbital label and an orbital shape.

The Magnetic Quantum Number(mℓ)

Determines the orientation of the orbital in space.

• Takes integral values from −𝓁 to +𝓁.
   For 𝓁 = 0 (s-sub shell), the allowed quantum number is 0 there is only one orbital in the s-sub shell.
   For 𝓁 = 1 (P-sub shell), ml= -1,0,+1, shows three different orbital’s in p-sub shell (Px, Py, Pz)
• The number of possible mℓ values or orbitals for a given ℓ value is 2ℓ+1

The electron spin quantum number(ms)

• Specifies the direction of spin of an electron about its own axis.

• The spin can be either counter clockwise or clockwise and is designated by either 𝑚𝑠 = +1/2 𝑜𝑟 𝑚𝑠 = −1/2.

n	𝓁	orbital designation	m𝓁	number of orbital
1	0	1s	0	1
2	0	2s	0	1
	1	2p	-1,0,+1	3
3	0	3s	0	1
	1	3p	-1,0,+1	3
	2	3d	-2-1, 0,+1,+2	5
4	0	4s	0	1
	1	4p	-1, 0,+1,	3
	2	4d	-2-1, 0,+1,+2	5
	3	4f	-3-2-1, 0,+1,+2,+3	7

Examples

1. What values of  are permissible when n=3?

Solution:-   ℓ ranges from 0 to n-1, i.e. ℓ=0,1,2.

The number of possible values of , ℓ = n i.e. ℓ= 3 values.

2. Give the sublevel notation for each of the following sets of quantum numbers

               a. n=3,   l=2                                             c) n=4,   l=1

               b. n=3,   l=0                                             d) n=4,   l=3

Solution:-

               a.   3d because when l =2, then the sub shell is d.

                b. 3s because when l =0, then the sub shell is s.

                c. 4p because when l =1, then the sub shell is p.

                d. 4f because when l =3, then the sub shell is f.

3. How many orbital’s are possible when n=3?

Solution: The number of orbital’s is given by n² .  Thus, (3)²=9 orbital’s are possible when n=3

4. Determine the number of electrons in the following quantum numbers.

a. n=4,l=2

b. n=3,l=1,ml=-1

c. n=3,l=0,ml=0

d. n=5,l=3,m_l=-2,m_s=+1/2

Solution:-

a.   When n=4 & l=2 it represents 4d, there are 10 electrons in d- orbital.             

b. When n=3 and l=1 and m_l=-1, it represents  3Px, there are only two electrons in 3Px.         

 c. When n=3,l=1,ml=-1 It represents 3s, and there are only 2e- in 3s.  

d. When n=5,l=3,ml=0 it represents 5f, but it specifically denotes a single electron. i.e. 5f¹

5. Indicate whether each of the following is a permissible set of quantum numbers. If the set is not permissible, state why it is not.

                a. n=3,   l=1 , m_l=+2                         

                b. n=4,   l=3 , m_l=-3                          

                c. n=3,   l=2, m_l=-2

                d. n=0,   l=0,m_l=0

                e. n=3,   l=3,m_l=-3

Solution:-

                a. not permissible because m_l cannot be +2 when l=1.

                b. Permissible and labeled as 4f.

                c. Permissible and labeled as 3d.

                d. Not permissible because n cannot be zero.

                e. Not permissible because ℓ cannot be 3 when n=3.

1.6.3 Shapes of Atomic Orbitals

a. S- orbital

• S- orbital has a spherical shape

• As n increase, the electron is more likely to be located further away from the nucleus

b. P. orbitals

• The p orbitals of any principal quantum number are arranged along three mutually perpendicular axes, x, y, and z, so that the region of the highest electron density are in dumb bell-shaped boundary surface (Figure ). The three p orbitals are

• The three p – orbitals have two lobes of each p orbital lie along a line with the nucleus at their center.

c. d-Orbitals

• There are five d-orbital’s in each of the principal quantum numbers above the second (n=2).

• The orientations of d-orbital are much more complicated than p-orbital.