Quantum Numbers and Orbitals: PECTAA Class 11 Chemistry, Punjab Board

Bohr gave one number for an electron. Schrödinger's model needs three, plus a fourth for spin. Learn what each one tells you and the exam questions become counting: shells, subshells, orbitals, electrons.

In the PECTAA Chemistry 11 book, Chapter 2 (Atomic Structure) section 2.4 explains why. The Bohr model was one-dimensional. It used one quantum number, n, for the size and energy of the orbit. Schrödinger's model lets the electron occupy three-dimensional space, so it needs three coordinates, or three quantum numbers. This note is for FSc Part 1 (1st year, Class 11) students in Lahore, Faisalabad, Sargodha, Rawalpindi, Multan, Gujranwala and the other Punjab boards. Gujrat district comes under the Gujranwala board. PECTAA writes the books. Your own board sets your paper.

What to know

Principal quantum number (n). It takes positive whole values 1, 2, 3, 4 and so on, called shells K, L, M, N. It describes the size and energy of the orbital. The larger n is, the farther the electron is from the nucleus on average, the higher its energy, and the less tightly it is bound. The maximum number of electrons in a shell is 2n². So K, L, M and N hold 2, 8, 18 and 32.

Azimuthal quantum number (ℓ). It describes the shape of the orbital. For each n it takes whole values from 0 to (n − 1).

• n = 1: ℓ = 0 only.
• n = 2: ℓ = 0 and 1.
• n = 3: ℓ = 0, 1 and 2.
• n = 4: ℓ = 0, 1, 2 and 3.

The values 0, 1, 2 and 3 are called s, p, d and f. The book says the letters stand for sharp, principal, diffused and fundamental, words used for spectral lines. The number of subshells in a shell equals the shell number. A subshell holds 2(2ℓ + 1) electrons.

Magnetic quantum number (m). It describes the orientation of an orbital in space. For a given ℓ there are (2ℓ + 1) whole values of m, from −ℓ through 0 to +ℓ. The number of values of m is the number of orbitals in the subshell.

• s: one orbital (m = 0)
• p: three orbitals (m = −1, 0, +1)
• d: five orbitals (m = +2, +1, 0, −1, −2)
• f: seven orbitals (m = +3 down to −3)

Orbitals of one subshell have the same energy. The book calls them degenerate orbitals. A magnetic field separates them, and that gives m its name.

Spin quantum number (s). An electron spins about its own axis, and a spinning charge makes a magnetic field. There are two spins, clockwise and anticlockwise. The values of s are +½ and −½. Three quantum numbers describe an orbital. The fourth tells apart the two electrons that can occupy it.

Key points

• Orbital. The book defines an atomic orbital as the three-dimensional region around the nucleus where the probability of finding the electron is maximum.
• s orbital. Spherical. Electron density is uniform in all directions, and a higher n gives a larger s orbital.
• p orbitals. Two lobes along an axis. They are named px, py and pz.
• d orbitals. Five shapes: dxy, dxz and dyz lie in the xy, xz and yz planes. The lobes of dx²−y² lie along the x and y axes. The dz² orbital has two lobes along the z-axis and a "doughnut" in the xy plane.
• f orbitals. Seven orientations, with very complicated shapes.
• Book "Did you know". In the nth shell there are n subshells, n² orbitals and at most 2n² electrons.

Worked examples

Example 1. List the subshells of the M shell.

1. M means n = 3.
2. ℓ = 0, 1, 2.
3. The subshells are 3s, 3p and 3d. There are 3, the same as the shell number.

Example 2. How many orbitals and how many electrons can the M shell hold?

1. Orbitals: 1 (s) + 3 (p) + 5 (d) = 9, which is n² = 9.
2. Electrons: 2 × 9 = 18, which is 2n² = 18.

Example 3 (style of book Quick Check 2.3). If n = 3 and ℓ = 2, which shell and subshell is the electron in?

1. n = 3 is the M shell.
2. ℓ = 2 is the d subshell.
3. The electron is in the 3d subshell of the M shell.

Example 4 (style of book Quick Check 2.3). If n = 2 and ℓ = 1, how many orientations in space are possible?

1. The number of values of m is 2ℓ + 1 = 2(1) + 1 = 3.
2. So there are 3 orientations: m = −1, 0 and +1. These are the 2px, 2py and 2pz orbitals.

Common mistakes in the exam

1. Writing ℓ up to n. The top value is (n − 1).
2. Mixing orbitals and electrons. A d subshell has 5 orbitals and 10 electrons.
3. Confusing 2n² with 2(2ℓ + 1). 2n² is for a whole shell. 2(2ℓ + 1) is for one subshell.
4. Listing the wrong range of m. It runs from −ℓ to +ℓ, including 0.
5. Saying a shell has n orbitals. It has n² orbitals and n subshells.
6. Forgetting the fourth number. The spin quantum number separates the two electrons in one orbital.

Exam technique

• Write n first, then ℓ, then m, then s. The order follows the book.
• For counting questions, build a small table of n, ℓ, subshells and orbitals.
• Use the exact words: "orbital" is a region of space, "shell" is set by n and "subshell" is set by ℓ.
• Check your board's notice for which sections are in your paper.

Practice questions

1. What does the principal quantum number tell you about an orbital? Answer: Its size and energy.
2. How many subshells are in the N shell, and what are they? Answer: Four: 4s, 4p, 4d and 4f.
3. Write the values of m for ℓ = 2. Answer: +2, +1, 0, −1, −2. That is five values, so five d orbitals.
4. What is the maximum number of electrons in the L shell? Answer: 2n² = 2 × 4 = 8.
5. Name the shape of the s orbital and the p orbital. Answer: The s orbital is spherical. The p orbital is polar (dumb-bell), with two lobes.

Ready to practise this? Continue in 1st Year Chemistry.