2.3: Question 2.E.40 PASS - quantum numbers, orbital characteristics

Exercise \(\PageIndex{2.E.40}\)

Consider the orbitals shown here in outline…

1. What is the maximum number of electrons contained in an orbital of type (x)? Of type (y)? Of type (z)?

2. How many orbitals of type (x) are found in a shell with n = 2? How many of type (y)? How many of type (z)?

3. Write a set of quantum numbers for an electron in an orbital of type (x) in a shell with n = 4. Of an orbital of type (y) in a shell with n = 2. Of an orbital of type (z) in a shell with n = 3.

4. What is the smallest possible n value for an orbital of type (x)? Of type (y)? Of type (z)?

5. What are the possible l and ml values for an orbital of type (x)? Of type (y)? Of type (z)?

Answer

1. type (x) = 2 electrons; type (y) = 2 electrons; type (z) = 2 electrons

2. type (x) = 1 orbital; type (y) = 3 orbitals; type (z) = 0 orbitals

3. type (x) = [4, 0, 0, ½]; type (y) = [2, 1, 0, -½]; type (z) = [3, 2, -1, ½]

4. type (x): n = 1; type (y): n = 2; type (z): n = 3

5. type (x): l = 0, ml = 0; type (y): l = 1, ml = -1, 0, 1; type (z): l = 2; ml = -2, -1, 0, 1, 2

See LibreText 2.4 Quantum Mechanics and the Atom

Strategy Map

Step	Hint
This problem covers concepts from several topics in section 2: Quantum Theory and Electronic Structure	Refer to: For a visual representation of the orbital types and their suborbital positions see figure 2.5.7. Libre Text 2.6, Electronic Structure of Atoms (Electron Configurations) For a visual representation of the hierarchy of subshell filling/energy levels see figure 2.6.3.
1. Identify what orbital types are represented in the figure.	In CHEM1500 four types of orbitals are introduced: s, p, d, and f. Recall this mnemonic the letter can help describe the shape s (sphere - 1 lobe) p (propeller - 2 lobes) d (daisy - 4 lobes; or donut - two lobes and a ring shape) f (funky - 7 lobes)
2. Consider each requirement of this multi-part problem and work through the answer step by step to not miss anything	Be familiar with the fundamental concepts of atomic orbital theory and the key characteristics of each orbital type. Consider Hund's Rule, Aufbau's Principal, and the Pauli Exclusion Principal to guide your thinking.

Guided Solution

Guided Solution	Hint
This problem requires comprehension of the following concepts: 'quantum mechanics and the atom', 'shapes of atomic orbitals', and 'electronic structure of atoms' Consider the orbitals shown here in outline…	Refer to: Libre Text 2.4, Quantum Mechanics and The Atom Libre Text 2.5, The Shape of Atomic Orbitals Libre Text 2.6, Electronic Structure of Atoms (Electron Configurations)
Recall the quantum numbers: what are they and what do they represent? what are the guidelines/rules/principals that inform each quantum number?	Quantum numbers: [n, l, ml, ms] n (principal quantum number; represents shell energy level) must be a whole number integer (i.e., n = 1, 2, 3, ... n) bigger number = greater energy = greater distance from nucleus = larger size l (azimuthal / orbital angular momentum quantum number, represents orbital shape) must be zero or a positive whole number integer, restricted by n (i.e., l = 0, 1, 2, ... n-1; l≠n) when l=0, type s (sphere shaped: 1 lobe) when l=1, type p (propeller shaped: 2 lobes) when l=2, type d (daisy shaped: 4 lobes; or donut shaped: two lobes and a ring shape) when l=3, type f (funky shaped: 7 lobes) ml (magnetic quantum number, represents the orientation of the orbital within a subshell) must be a positive or negative whole number integer, or zero; restricted by l (i.e., ml = -l, ... -2, -1, 0, 1, 2, ... +l) for s type: l=0; ml must be 0 for p type; l=1; ml can be -1, 0, or +1 for d type; l=2; ml can be -2, -1, 0, +1, or +2 ms (spin quantum number, represents the direction of spin of the electron) can be +½ OR -½, not restricted by any other quantum number
Question Part 1 What is the maximum number of electrons contained in an orbital of type (x)? Of type (y)? Of type (z)? Answer Part 1 type (x) = 2 electrons type (y) = 2 electrons type (z) = 2 electrons	Think about how many electrons can you put into any orbital? Think about electron pairing. Regardless of orbital type, each orbital can contain at most 2 electrons; one electron spins in a direction (ms=+½), and the other electron spins in the opposite direction (ms=-½).
Question Part 2 How many orbitals of type (x) are found in a shell with n = 2? How many of type (y)? How many of type (z)? First, identify each orbital type based on the shape in the provided figure:	Here we must consider what quantum numbers are allowed for each orbital type when n=2 type (x) - 1 lobe, sphere shaped (s); l=0 type (y) - 2 lobes, propeller shaped (p); l=1 type (z) - 4 lobes, daisy shaped (d); l=2
Consider the relationship between quantum numbers n and l when n=2; is each orbital type (i.e., x, y, and z) allowed?	Recall: l must be zero or a positive whole number integer, restricted by n (i.e., l = 0, 1, 2, ... n-1) type (x) - when n=2; l=0 (allowed) type (y) - when n=2; l=1 (allowed) type (z) - when n=2; l≠2 (NOT allowed) Summary: a type (z) or 'd-type' orbital does not exist in the n=2 shell
Consider the relationship between quantum numbers l and ml How many orientations of each allowed orbital exist when n=2?	Recall: for a given l value, the allowed values for ml are the total number of orbitals of that type that can exist type (x) - when n=2; l=0; ml = 0 one allowed ml value therefore one 's' orbital when n=2 type (y) - when n=2; l=1; ml = -1, 0, +1 three allowed ml values therefore three 'p' orbitals when n=2
Answer Part 2 Type (x) = 1 orbital Type (y) = 3 orbitals Type (z) = 0 orbitals	This makes sense because: an s-orbital (sphere shaped electron density cloud) is an identical sphere regardless of its orientation in space... resulting in only 1 possible s-orbital at each energy level a p-orbital (propellor shaped electron density cloud) can exist in three different axes in 3-dimensional space; it can lay along the x-axis, the y-axis, and the z-axis... resulting in 3 possible p-orbitals (px, py, and pz) at each energy level
Question Part 3 Write a set of quantum numbers for an electron in an orbital of type (x) in a shell with n = 4. Of an orbital of type (y) in a shell with n = 2. Of an orbital of type (z) in a shell with n = 3. Consider what each quantum number represents? Summarize the quantum numbers you have been provided in the question.	Recall there are four distinct quantum numbers [n, l, ml ,ms] The question provides the principal quantum number (n) and the shape of the orbital defines the angular momentum quantum number (l): Type (x): [n=4, l=0] Type (y): [n=2, l=1] Type (z): [n=3, l=2]
What quantum numbers must be defined to answer the question? Are there any restrictions that must be considered? Is there more than one correct answer?	Still need ml and ms ml must be a positive or negative whole number integer, or zero; restricted by l (i.e., ml = -l, ... -2, -1, 0, 1, 2, ... +l) ms can be +½ OR -½, not restricted by any other quantum number Yes, more than one allowed quantum number sets for each orbital type as defined in the question
Answer Part 3 type (x) = [4, 0, 0, ½] type (y) = [2, 1, 0, -½] type (z) = [3, 2, -1, ½]	type (x) = [n=4, l=0, ml=0, ms=+½] ml must equal 0 because l=0 ms = -½ would also be allowed 2 correct answers This makes sense because the 4s orbital can hold 2 electrons type (y) = [n=2, l=1, ml=0, ms=-½] ml = -1 and ml = +1 would also be allowed ms = +½ would also be allowed 6 correct answers This makes sense because the 2p orbitals can hold 6 electrons 2 electrons in each px, py, and pz type (z) = [n=3, l=2, ml=-1, ms=+½] ml = -2, 0, +1, and +2 would also be allowed ms = -½ would also be allowed 10 correct answers This makes sense because the 3d orbitals can hold 10 electrons 2 electrons in each dxy, dyz, dxz, dx^2–y ^2 and dz^2
Question Part 4 What is the smallest possible n value for an orbital of type (x)? Of type (y)? Of type (z)? Consider the relationship between quantum numbers n and l.	Recall: l must be zero or a positive whole number integer, the value of l is restricted by n (i.e., l = 0, 1, 2, ... n-1; l≠n) Therefore the lowest value of n = (l+1)
Recall the value of l for each type of orbital	type (x) - 1 lobe, sphere shaped (s); l=0 type (y) - 2 lobes, propeller shaped (p); l=1 type (z) - 4 lobes, daisy shaped (d); l=2
Answer Part 4 Type (x): n = 1 Type (y): n = 2 Type (z): n = 3	Type (x): lowest n =(l + 1)=(0 + 1)= 1 Type (y): lowest n =(l + 1)=(1 + 1)= 2 Type (z): lowest n =(l + 1)=(2 + 1)= 3
Question Part 5 What are the possible l and ml values for an orbital of type (x)? Of type (y)? Of type (z)? Consider the relationship between ml and l quantum numbers.	ml must be a positive or negative whole number integer, or zero; ml is restricted by l (i.e., ml = -l, ... -2, -1, 0, 1, 2, ... +l)
Recall the value of l for each type of orbital	type (x) - 1 lobe, sphere shaped (s); l=0 type (y) - 2 lobes, propeller shaped (p); l=1 type (z) - 4 lobes, daisy shaped (d); l=2
Answer Part 5 Type (x): l=0; ml must be 0 Type (y): l=1; ml can be -1, 0, or +1 Type (z): l=2; ml can be -2, -1, 0, +1, or +2	Type (x) = s orbital: l=0; ml must be 0 Type (y) = p orbital; l=1; ml can be -1, 0, or +1 Type (z) = d orbital; l=2; ml can be -2, -1, 0, +1, or +2

(question source adapted question 6.3.11 from 6.E: Electronic Structure and Periodic Properties (Exercises): https://chem.libretexts.org/Bookshelves/General_Chemistry/Chemistry_1e_(OpenSTAX)/06%3A_Electronic_Structure_and_Periodic_Properties_of_Elements/6.E%3A_Electronic_Structure_and_Periodic_Properties_(Exercises) , shared under a CC BY 4.0 license, authored, remixed, and/or curated by OpenStax, original source question 41 https://openstax.org/books/chemistry-2e/pages/6-exercises , Access for free at https://openstax.org/books/chemistry/pages/1-introduction)