derivation: chemistry: Half-Life Formula for Zero-Order Reactions

[bhpayne]

• https://byjus.com/chemistry/first-order-reactions/
• https://byjus.com/chemistry/half-life/
• https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/02%3A_Reaction_Rates/2.10%3A_Zero-Order_Reactions

From https://www.google.com/search?q=Derivation+of+Half-Life+Formula+for+Zero-Order+Reactions

To derive the half-life formula for a zero-order reaction, we start with the integrated rate law for a zero-order reaction, which is: [A] = [A]₀ - kt; then, substitute [A] = [A]₀/2 (since at half-life, the concentration is halved) and solve for t (which becomes t₁/₂): [1, 2, 3]

• Start with the zero-order rate law: [1, 3, 4]
• [A] = [A]₀ - kt [1, 3, 4]

• Substitute [A] with [A]₀/2 for half-life: [1, 3, 5]
• [A]₀/2 = [A]₀ - kt₁/₂ [1, 3, 5]

• Rearrange to solve for t₁/₂: [1, 3, 4]
• kt₁/₂ = [A]₀ - [A]₀/2 [1, 3, 4]
• kt₁/₂ = [A]₀/2 [1, 3, 6]

• Isolate t₁/₂: [1, 7, 8]
• t₁/₂ = [A]₀ / (2k) [1, 7, 8]

Therefore, the half-life formula for a zero-order reaction is: t₁/₂ = [A]₀ / (2k), where [A]₀ is the initial concentration and k is the rate constant. [1, 7, 8]
Key points to remember: [2, 3, 9]

• In a zero-order reaction, the rate is independent of the reactant concentration, meaning the rate remains constant throughout the reaction. [2, 3, 9]
• The half-life of a zero-order reaction is directly proportional to the initial concentration. [5, 7, 8, 10, 11]
• This means that as the initial concentration increases, the half-life also increases. [5, 7, 8]

• [1] https://www.toppr.com/guides/chemistry/zero-order-reaction/
• [2] https://www.chem.purdue.edu/gchelp/howtosolveit/Kinetics/Halflife.html
• [3] https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/02%3A_Reaction_Rates/2.04%3A_Half-lives
• [4] https://byjus.com/chemistry/zero-order-reaction/
• [5] https://www.turito.com/blog/chemistry/half-life
• [6] https://www.khanacademy.org/science/ap-chemistry-beta/x2eef969c74e0d802:kinetics/x2eef969c74e0d802:concentration-changes-over-time/v/zero-order-reactions
• [7] https://byjus.com/chemistry/half-life/
• [8] https://unacademy.com/content/neet-ug/study-material/chemistry/derivation-of-half-life-only-for-zero-and-first-order-reactions/
• [9] https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/02%3A_Reaction_Rates/2.10%3A_Zero-Order_Reactions
• [10] https://www.doubtnut.com/qna/647235318
• [11] https://www.vedantu.com/jee-main/chemistry-half-life-of-zero-order-reaction