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Table 2: Guided Solution

Guided Solution Ideas
This question is a theory problem that requires the understanding and application of concepts related to the shape of atomic orbitals, effective nuclear charge, and atomic radii. This question asks us to rank the given elements by their size. Refer to Atomic Radii (2). Show/Hide Don’t Forget! Recognize that as principal quantum number (n) increases (i.e., as you move down the periodic table), the size of the orbital increases. Refer to Section 2.5: The Shape of Atomic Orbitals (3). Show/Hide Don’t Forget! Recognize that as Zeff increases (i.e., as you move left to right across the periodic table), the size of the orbital decreases. Refer to: Section 3.2.1 Effective Nuclear Charge: Penetration and Shielding (1). Section 3.2.2: Variation in Covalent Radius (1).
Consider what characteristics may influence the distance between the nucleus and valence electron(s). Show/Hide Think About This! Principal quantum number (n): Shielding — consider how many complete shells of electrons are between the valence electrons and the nucleus. Effective Nuclear Charge (Zeff): Shielding — electrons in the same shell (i.e., across the same row) are not effective at shielding the attractive forces from the nucleus. As you move left to right across a row, the number of protons and effective nuclear charge (i.e., positive charge felt by the valence electron) increases. The electrons are held more tightly (i.e., closer) to the nucleus, and atomic radii decrease. Show/Hide Don’t Forget! (n) influences size more than (Zeff).
Summarize the trend of atomic radii across the periodic table. Show/Hide Don’t Forget! The atomic radius decreases from left to right and increases from top to bottom. The largest element is located at the bottom left of the periodic table. The smallest element is located at the top right of the periodic table.
Locate the elements on the periodic table Show/Hide Element Locations Ca (calcium) — atomic number 20, group 2, period 4 Cs (cesium) — atomic number 55, group 1, period 6 Mg (magnesium) — atomic number 12, group 2, period 3 Rb (rubidium) — atomic number 37, group 1, period 5
List the elements in order of increasing principal quantum number (n). Mg — period 3 Ca — period 4 Rb — period 5 Cs — period 6 Show/Hide Don’t Forget! As (n) increases, the size of the orbital increases. Recall: The valence electrons for all of the listed elements are in s-orbitals: Mg — 3s Ca — 4s Rb — 5s Cs — 6s Note that because all the elements are in different periods, we do not need to consider their group location (i.e., relative position from left to right). Refer to Section 3.2.2: Variation in Covalent Radius (1).

Table 3: Complete Solution

Complete Solution
Atomic radius increases as you move down rows from the top to the bottom of the periodic table. This increase is because the principal quantum number (n) increases, meaning the element requires more shells to house additional electrons around the nucleus. Elements at the bottom of the periodic table have a larger ‘period’ number.
Atomic radius decreases as you move from left to right across the columns of the periodic table. This decrease is because the effective nuclear charge (Zeff) increases, meaning that outer shell valence electrons feel a stronger attractive force from the protons that pull them closer to the nucleus. Elements towards the right of the periodic table have a larger ‘group’ number.
Because of the left-to-right trend, the atomic number (i.e., the number of electrons) is not the only factor to consider when ranking atomic size. First, consider the period — row: top to bottom (bottom is largest). If more than one element is in the same period, consider the group — column: left to right (left is smallest).
Locate each element on the periodic table to find their relative period and group numbers. Ca (calcium) — atomic number 20, group 2, period 4 Cs (cesium) — atomic number 55, group 1, period 6 Mg (magnesium) — atomic number 12, group 2, period 3 Rb (rubidium) — atomic number 37, group 1, period 5
Because each listed element is in a different row (i.e., period), they can be ranked for size using this information alone. Mg — period 3 Ca — period 4 Rb — period 5 Cs — period 6 Smallest to largest atomic radii: Mg, Ca, Rb, Cs

PASS Attribution

• LibreTexts PASS Chemistry Book CHEM 1510/1520 (4)
• Question 3.E.9 from LibreTexts PASS Chemistry Book CHEM 1500 (5) is used under a CC BY-SA 4.0 license.
  • Question 3.E.9 is question Q6.5.9 from LibreTexts Chemistry 1e (OpenSTAX) (6), which is under a CC BY 4.0 license.
  • Question Q6.5.9 is question 75 from OpenStax Chemistry 2e (7), which is under a CC BY 4.0 license. Access for free at https://openstax.org/books/chemistry-2e/pages/1-introduction.

Media Attributions

• Figure 1: Periodic Table of Elements Black and White [adapted from Dmarcus100] by Thompson Rivers University Open Learning (8) is used under a CC BY-SA 4.0 license.
• Figure 2: [Modified] Periodic Table Of Elements Black and White [adapted from Dmarcus100] by Thompson Rivers University Open Learning (8) is used and modified under a CC BY-SA 4.0 license.

References

1. OpenStax. 3.2: Periodic Variations in Element Properties. In CHEM 1500: Chemical Bonding and Organic Chemistry. LibreTexts, 2023. https://chem.libretexts.org/Courses/Thompson_Rivers_University/CHEM1500%3A_Chemical_Bonding_and_Organic_Chemistry/03%3A_Periodic_Relationships_Among_the_Elements/3.02%3A_Periodic_Variations_in_Element_Properties#Effective_Nuclear_Charge:_Penetration_and_Shielding.

2. LibreTexts. Atomic Radii. In Supplemental Modules (Physical and Theoretical Chemistry). LibreTexts, 2023. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Physical_Properties_of_Matter/Atomic_and_Molecular_Properties/Atomic_Radii.

3. Blackstock, L.; Brewer, S.; Cinel, B.; Prema, D. 2.5: The Shape of Atomic Orbitals. In CHEM1500: Chemical Bonding and Organic Chemistry. LibreTexts, 2022. https://chem.libretexts.org/Courses/Thompson_Rivers_University/CHEM1500%3A_Chemical_Bonding_and_Organic_Chemistry/02%3A_Quantum_Theory_and_Electronic_Structure_of_Atoms/2.05%3A_The_Shape_of_Atomic_Orbitals.

4. Blackstock, L.; Brewer, S.; Jensen, A. PASS Chemistry Book CHEM 1510/1520; LibreTexts, 2023. https://chem.libretexts.org/Courses/Thompson_Rivers_University/PASS_Chemistry_Book_CHEM_1510%2F%2F1520.

5. Blackstock, L.; Brewer, S.; Jensen, A. 3.2: Question 3.E.09 PASS – Ranking Atomic Radius. In PASS Chemistry Book CHEM 1500. LibreTexts, 2024. https://chem.libretexts.org/Courses/Thompson_Rivers_University/PASS_Chemistry_Book_CHEM_1500/03%3A_Periodic_Relationships_Among_the_Elements/3.02%3A_Question_3.E.09_PASS_-_ranking_atomic_radius.

6. OpenStax. 6.E: Electronic Structure and Periodic Properties (Exercises). In Chemistry 1e (OpenSTAX). LibreTexts, 2022. 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).

7. Flowers, P.; Theopold, K; Langley, R.; Robinson, W. R. Ch. 6 Exercises. In Chemistry 2e. OpenStax, 2019. https://openstax.org/books/chemistry-2e/pages/6-exercises.

8. Thompson Rivers University Open Learning. Periodic Table Of Elements Black And White [adapted from Dmarcus100].