Rice-Ramsperger-Kassel (RRK) theory unimolecular reactions

RRK theory provides the physical mechanism for a time delay between activation and reaction, and in doing so, it successfully explains why unimolecular reactions appear first-order at high pressure but transition to second-order at low pressure

1. Rice-Ramsperger-Kassel (RRK) Theory NAME: AYAAN ARIF SHAIKH ROLL NO: 024 CLASS: MSC-1 SUBJECT: Physical 1

2. Table of Contents The Unimolecular Reaction Paradox The Lindemann-Hinshelwood Mechanism The Conceptual Core of RRK Theory An Assessment of the RRK Theory Critical Limitations and the Transition to RRKM Theory Conclusion References 2

3. Introduction : The Unimolecular Reaction Paradox Early 20th-century experiments showed first-order kinetics for unimolecular gas reactions (Rate = k[A]), contradicting the expectation of second-order kinetics if activation energy came from bimolecular collisions (which depend on [A]²). Perrin's Radiation Hypothesis (1919): Proposed that molecules gained activation energy by absorbing infrared radiation from vessel walls, explaining first-order kinetics by bypassing collision dependence. Flaws in the Radiation Hypothesis: This hypothesis was quickly disproven; Langmuir calculated insufficient radiation density, and Lindemann noted experimental inconsistencies with light intensity. Return to Collision-Based Activation: The failure of the radiation hypothesis forced a return to the idea that molecules energize each other through collisions. The Unanswered Question: The core challenge remained: how could a collision-based activation mechanism ultimately result in first-order reaction kinetics? This required a new model that accounted for a delay between energy acquisition and reaction. 3

4. Lindemann-Hinshelwood Mechanism 4 Lindemann's Multi-Step Mechanism (1921): Frederick Lindemann (and J.A. Christiansen) resolved the unimolecular paradox by proposing a multi-step activation process Key Insight : Energized Intermediate (A): * His breakthrough introduced a crucial "time-lag" between the initial energizing collision and the actual chemical reaction, involving a transient, energized intermediate species (A*). Activation (Bimolecular Collision) Deactivation (Bimolecular Collision) Reaction (Unimolecular Transformation) An "ordinary" reactant (A) becomes "activated" (A*) by gaining internal vibrational energy through a bimolecular collision with another molecule (M ). The activated molecule (A*) can be deactivated back to an ordinary molecule (A) by losing its excess energy via a bimolecular collision with another molecule (M), making the activation reversible. The activated molecule (A*) undergoes a unimolecular transformation to form the product (P).

5. Hinshelwood's Contribution and Critical Shortcomings 1)Hinshelwood's Refinement: *Proposed activation energy is distributed among 's' internal vibrational oscillators. *Improved quantitative fit for k1​ but discrepancies persisted. 2)Quantitative Failure in Fall-Off Region: *Failed to predict fall-off curves, especially for complex molecules, accurately. *Required "absurd numbers of degrees of freedom" to fit data (e.g., azomethane). 3)Incorrect Low-Pressure Behavior: Predicted a linear 1/ k uni ​ vs. 1/[M] plot. Experimental data consistently showed significant curvature. 4) Fundamental Conceptual Flaw The model assumed all energized A* molecules were identical, with a constant reaction probability (k2​) This necessitated replacing constant k2​ with an energy-dependent rate constant k(E), paving the way for RRK theory. Experimentation suggested reaction rate should depend on internal energy (k(E)). 5 .

6. 6 The Conceptual Core of RRK Theory In 1927-28, Rice, Ramsperger, and Kassel (RRK) resolved the Lindemann-Hinshelwood model's flaws by postulating that an energized molecule's reaction rate, K(E), is a function of its internal vibrational energy (E) , not a constant. This introduced the microcanonical rate constant .

7. Energized vs. Activated Molecules 7 Feature Energized Molecule (A*) Activated Molecule (A‡) Definition A molecule with total internal energy E ≥ E0 ​ (critical reaction energy). An energized molecule (A*) where ≥ E0 ​ is concentrated in the specific bond/mode relevant to the reaction coordinate at a given moment . Energy State Possesses sufficient total energy. Possesses sufficient localized energy in the reactive mode. Occurrence Achieved after an energizing collision (before IVR fully randomizes). Achieved stochastically due to IVR, when energy fluctuates into the reaction coordinate. Reactivity Does not immediately react. Reaction is a statistical outcome. Represents the specific configuration at the point of reaction . Relation to IVR Represents the specific configuration at the point of reaction . Represents the specific configuration at the point of reaction .

8. The Mathematical Formulation of the RRK Model 8

9. An Assessment of the RRK Theory: Major Successes and Significance A Landmark Improvement in Kinetic Theory. *The RRK theory significantly advanced the understanding of unimolecular reactions by successfully explaining the pressure dependence and fall-off curve, which previous models could not . *It resolved quantitative failures of earlier theories by introducing an energy-dependent reaction rate . Incorporating Molecular Structure 9 The RRK theory connected reaction rates to molecular structure and complexity via the empirical parameters 's', explaining differences in behavior between molecules of varying sizes and bridging the gap between macroscopic kinetics and microscopic molecular vibrations .

10. 10 The Conceptual Legacy: Energy-Dependent Reactivity *Impact on Collision Models * Core Contribution: Energy-Dependent Reactivity RRK theory established that microscopic reactivity is fundamentally dependent on internal energy. Introduced the microcanonical rate constant, k(E) , a cornerstone of modern chemical dynamics. *Broad Applicability Indispensable for understanding phenomena in high-temperature combustion, atmospheric chemistry, photochemistry, and mass spectrometry . Provided indirect justification for the "strong collision assumption" (single collision deactivates A*) in the Lindemann mechanism, implying efficient energy transfer. However, RRK's energy dependence also highlighted limitations of "strong collision" models. Paved the way for "weak collision" models, where energy is transferred in smaller, incremental steps, a key area in modern kinetics.

11. Critical Limitations and the Transition to RRKM Theory 11 Despite its successes, RRK theory is fundamentally a classical model and suffers from several significant limitations that ultimately necessitated its refinement. The Flaws of a Classical Model Continuous Energy: It treats vibrational energy as a continuous variable, completely ignoring the quantum mechanical nature of molecules, which possess discrete, quantized vibrational energy levels. Neglect of Zero-Point Energy: The classical framework fails to account for the zero-point energy of the vibrational modes, which is the minimum possible energy a quantum mechanical oscillator can have. This is a significant omission, as it affects the true amount of energy available to overcome the reaction barrier. Oversimplification The RRK theory inaccurately predicts reaction rates by oversimplifying vibrational modes as identical harmonic oscillators, neglecting the diverse and often anharmonic nature of molecular vibrations . Ad-hoc Parameters The RRK theory is limited by its reliance on empirically adjusted parameters, 's' and 'A', which are fitted to experimental data rather than derived from fundamental principles, thus restricting its predictive power and general applicability .

12. Comparative Analysis of Unimolecular Reaction Theories 12 Feature Lindemann-Hinshelwood Theory Rice-Ramsperger-Kassel (RRK) Theory Rice-Ramsperger-Kassel-Marcus (RRKM) Theory Molecular Model Structureless point mass (Lindemann); s abstract internal modes (Hinshelwood). A collection of s identical, coupled, classical harmonic oscillators. A realistic model including all vibrational frequencies and rotational modes. Nature of Reaction Step Unimolecular decay of A* with a constant, energy-independent rate k₂. Statistical localization of energy in a critical oscillator. Rate k(E) is a continuous function of total energy E. Passage over a potential energy barrier (Transition State). Rate k(E) is calculated from the ratio of the sum of states at the transition state to the density of states of the reactant. Key Parameters k₁, k₋₁, k₂. Critical energy E₀, pre-exponential factor A, number of effective oscillators s. Molecular vibrational frequencies, moments of inertia, barrier height E₀, properties of the transition state Primary Strengths Introduced the correct three-step mechanism and explained the pressure dependence qualitatively. First theory to incorporate molecular complexity and energy-dependent reactivity, providing a much better fit for fall-off curves. A predictive, first-principles theory with no adjustable parameters. Quantitatively accurate and widely used. Primary Weaknesses Quantitatively incorrect fall-off curves. No link to molecular structure. Classical model, oversimplified molecular picture, reliance on adjustable parameters (s, A). Not predictive. Computationally intensive. Relies on the accuracy of potential energy surface calculations and the assumption of rapid IVR. Treatment of Energy A simple threshold is required. Classical, continuous distribution of energy among oscillators. Quantum mechanical, discrete energy levels (state counting).

13. Conclusion: The Enduring Legacy of RRK Theory The Rice-Ramsperger-Kassel (RRK) theory, although mostly replaced by the more advanced and precise RRKM theory, still stands as a significant accomplishment in the evolution of chemical kinetics. It offered the initial effective theoretical framework that could semi-quantitatively explain the intricate pressure dependence of unimolecular reactions, a challenge that previous models had failed to address. Its genuine and enduring legacy, however, resides in the significant conceptual changes it brought forth. Through the representation of the molecule as a collection of coupled oscillators and the assumption that energy circulates freely throughout, RRK theory formulated the foundations of intramolecular vibrational energy redistribution (IVR) and energy-dependent reactivity as key concepts in reaction dynamics. The idea of a microscopic rate constant, k(E), has fundamentally transformed chemists' perspectives on the elementary stages of a chemical reaction. RRK theory serves as a crucial link between the classical, phenomenological models of the early 20th century and the contemporary quantum-statistical theories that prevail in the field today. It serves as a demonstration of the strength of statistical reasoning in chemistry and continues to be a crucial educational resource for presenting the fundamental concepts of unimolecular reaction rate theory 13

14. 14 References Lindemann, F. A.; Arrhenius, S.; Langmuir, I.; Dhar, N. R.; Perrin, J.; Mcc . Lewis, W. C. Discussion on "the radiation theory of chemical action". Transactions of the Faraday Society Rice, O. K.; Ramsperger, H. C. Theories of unimolecular gas reactions at low pressures. Journal of the American Chemical Society Kassel, L. S. Studies in Homogeneous Gas Reactions I. The Journal of Physical Chemistry Marcus, R. A. Unimolecular Dissociations and Free Radical Recombination Reactions. The Journal of Chemical Physics Textbooks and Reviews Billing, G. D.; Mikkelsen, K. V. Introduction to Molecular Dynamics and Chemical Kinetics. Ilbert, R. G. Theory of Unimolecular and Recombination Reactions. Holbrook, K. A.; Pilling, M. J.; Robertson, S. H. Unimolecular Reactions. Houston, P. L. Chemical Kinetics and Reaction Dynamics. Laidler, K. J. Chemical Kinetics . Works cited Predicting pressure-dependent unimolecular rate constants using variational transition state theory with multidimensional tunneling combined with system-specific quantum RRK theory: a definitive test for fluoroform dissociation - RSC Publishing, https://pubs.rsc.org/en/content/articlehtml/2016/cp/c6cp02765b 10. (PDF) The Kinetics of Pressure-Dependent Reactions - ResearchGate, https://www.researchgate.net/publication/225089112_The_Kinetics_of_Pressure-Dependent_Reactions 11. Hinshel wood theory | PPTX - SlideShare, https://www.slideshare.net/slideshow/hinshel-wood-theory/234140978 12. RRK theory - University of Lethbridge, https://people.uleth.ca/~roussel/C4000statmech/RRK.pdf 13. RRKM theory - Wikipedia, https://en.wikipedia.org/wiki/RRKM_theory 14. Rice–Ramsperger–Kassel (RRK) theory - IUPAC Gold Book, https://old.goldbook.iupac.org/html/R/R05390.html 15. RRKM Theory | PDF | Chemical Reactions | Normal Mode - Scribd, https://www.scribd.com/document/183818741/RRKM-Theory 16. PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules - CL Jain, https://cljaincollege.org.in/wp-content/uploads/2023/03/Kinetics-Hinshelwood-and-RRKM-Models.pdf 17. Rice-Ramsperger-Kassel theory - Oxford Reference, https://www.oxfordreference.com/display/10.1093/oi/authority.20110803100419720 18. en.wikipedia.org, https://en.wikipedia.org/wiki/Intramolecular_vibrational_energy_redistribution#:~:text=Intramolecular%20vibrational%20energy%20redistribution%20(IVR,rates%20such%20as%20RRKM%20theory. 19. Intramolecular vibrational energy redistribution - Wikipedia, https://en.wikipedia.org/wiki/Intramolecular_vibrational_energy_redistribution 20. 8.1: Theoretical Tools for Studying Chemical Change and Dynamics - Chemistry LibreTexts , https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Theoretical_Chemistry_(Simons)/08%3A_Chemical_Dynamics/8.01%3A_Theoretical_Tools_for_Studying_Chemical_Change_and_Dynamics 21. Inefficient intramolecular vibrational energy redistribution for the H + HO2 reaction and negative internal energy, https://www.slideshare.net/slideshow/chemical-dynamics-introrrk-rrkm-theory/242760774 33. Calculating RRKM Rate Constants from Vibrational Frequencies and Their Dynamic Interpretation - The Mathematica Journal, https://www.mathematica-journal.com/2017/09/26/calculating-rrkm-rate-constants-from-vibrational-frequencies-and-their-dynamic-interpretation/ 34. Lindemann Hinshelwood RRK RRKM - YouTube, https://www.youtube.com/watch?v=nonVun44JwA 35. Advanced Molecular Dynamics and Chemical Kinetics - Wiley, https://www.wiley.com/en-us/Advanced+Molecular+Dynamics+and+Chemical+Kinetics-p-x000031982 36. Chemical kinetics Books - Alibris, https://www.alibris.com/search/books/subject/Chemical-kinetics 37. Comprehensive Chemical Kinetics: Unimolecular Kinetics, Part 1. The Reaction Step - Google Books, https://books.google.com/books/about/Comprehensive_Chemical_Kinetics.html?id=yjRGzwEACAAJ 38. Keith J Laidler - Chemical Kinetics-HarperCollins (1987) - College of Engineering and Applied Science, https://www.eng.uc.edu/~beaucag/Classes/AdvancedMaterialsThermodynamics/Books/Keith%20J%20Laidler%20-%20Chemical%20Kinetics-HarperCollins%20(1987).pdf

15. 15 I extend my sincere gratitude to Wilson College sincere thanks to Principal Dr. Jamson Masih & to Prof. (Dr.) H. Parbat, Head of Department Thank you for your time