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Related Concept Videos

Fast Reactions01:27

Fast Reactions

Fast reactions occurring in times shorter than the time needed to mix reactants pose a unique challenge for investigation. In a liquid-phase continuous-flow system, reactants A and B are swiftly pushed into the mixing chamber, where mixing occurs within 1 ms. The reaction mixture then flows through an observation tube, and one measures light absorption to determine species concentrations at various points of the tube. This method is most appropriate when relatively large volumes of reactants...
Reaction Mechanisms03:06

Reaction Mechanisms

Chemical reactions often occur in a stepwise fashion, involving two or more distinct reactions taking place in a sequence. A balanced equation indicates the reacting species and the product species, but it reveals no details about how the reaction occurs at the molecular level. The reaction mechanism (or reaction path) provides details regarding the precise, step-by-step process by which a reaction occurs.
For instance, the decomposition of ozone appears to follow a mechanism with two steps:
Behavior of Gas Molecules: Molecular Diffusion, Mean Free Path, and Effusion03:48

Behavior of Gas Molecules: Molecular Diffusion, Mean Free Path, and Effusion

Although gaseous molecules travel at tremendous speeds (hundreds of meters per second), they collide with other gaseous molecules and travel in many different directions before reaching the desired target. At room temperature, a gaseous molecule will experience billions of collisions per second. The mean free path is the average distance a molecule travels between collisions. The mean free path increases with decreasing pressure; in general, the mean free path for a gaseous molecule will be...
Reaction Mechanisms: The Steady-State Approximation01:26

Reaction Mechanisms: The Steady-State Approximation

The steady-state approximation, also referred to as the quasi-steady-state approximation to differentiate it from a true steady state, is a widely used method for simplifying calculations in complex reaction mechanisms. This approach is particularly useful when dealing with multi-step reactions that involve reverse reactions or several steps, which can significantly increase mathematical complexity and make the reactions nearly unsolvable analytically.The steady-state approximation operates on...
Rate-Determining Steps03:08

Rate-Determining Steps

Relating Reaction Mechanisms
In a multistep reaction mechanism, one of the elementary steps progresses significantly slower than the others. This slowest step is called the rate-limiting step (or rate-determining step). A reaction cannot proceed faster than its slowest step, and hence, the rate-determining step limits the overall reaction rate.
The concept of rate-determining step can be understood from the analogy of a 4-lane freeway with a short-stretch of traffic-bottleneck caused due to...
Diffusion01:12

Diffusion

Diffusion is the passive movement of substances down their concentration gradients—requiring no expenditure of cellular energy. Substances, such as molecules or ions, diffuse from an area of high concentration to an area of low concentration in the cytosol or across membranes. Eventually, the concentration will even out, with the substance moving randomly but causing no net change in concentration. Such a state is called dynamic equilibrium, which is essential for maintaining overall...

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Updated: Jun 3, 2026

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
06:55

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level

Published on: September 26, 2016

Multiparticle diffusion limited [Formula: see text] reaction in small volumes.

Zoran Konkoli1

  • 1Department of Microtechnology and Nanoscience-MC2, Bionano Systems Laboratory, Chalmers University of Technology, Sweden.

Journal of Physics. Condensed Matter : an Institute of Physics Journal
|March 17, 2011
PubMed
Summary

This study analyzes particle annihilation in clusters, finding that particle numbers stabilize over time. The decay rate depends on particle and system size, offering insights into reaction dynamics.

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Last Updated: Jun 3, 2026

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
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Area of Science:

  • Statistical Physics
  • Chemical Kinetics
  • Mathematical Modeling

Background:

  • Investigates multiparticle annihilation reactions where particles A form clusters of size k.
  • Considers arbitrary reaction order k > 2, spatial dimension d, and system size L.
  • Particles diffuse with constant D and annihilate with a position-dependent rate σ.

Purpose of the Study:

  • To analytically investigate a multiparticle reaction model with cluster annihilation.
  • To derive and solve equations of motion for many-point densities.
  • To determine the effective reaction rate and long-time behavior of particle density.

Main Methods:

  • Rephrased the master equation using field theory language.
  • Derived equations of motion for many-point densities.
  • Solved approximate equations analytically in the diffusion-controlled limit.

Main Results:

  • Derived an explicit expression for the effective reaction rate using Laplace transforms.
  • Demonstrated that particle numbers saturate to a constant value at large times.
  • Identified the exponential decay constant as a non-algebraic function of particle size a and system size L.

Conclusions:

  • The multiparticle annihilation model exhibits saturation in particle number.
  • The saturation dynamics are governed by an effective reaction rate dependent on particle and system geometry.
  • Analytical solutions provide a framework for understanding complex reaction-diffusion systems.