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Related Concept Videos

Atomic Nuclei: Types of Nuclear Relaxation01:28

Atomic Nuclei: Types of Nuclear Relaxation

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Nuclear relaxation restores the equilibrium population imbalance and can occur via spin–lattice or spin–spin mechanisms, which are first-order exponential decay processes.
In spin–lattice or longitudinal relaxation, the excited spins exchange energy with the surrounding lattice as they return to the lower energy level. Among several mechanisms that contribute to spin–lattice relaxation, magnetic dipolar interactions are significant. Here, the excited nucleus transfers...
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Concentration and Rate Law03:03

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The rate of a reaction is affected by the concentrations of reactants. Rate laws (differential rate laws) or rate equations are mathematical expressions describing the relationship between the rate of a chemical reaction and the concentration of its reactants.
For example, in a generic reaction aA + bB ⟶ products, where a and b are stoichiometric coefficients, the rate law can be written as:
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The free energy change for a process taking place with reactants and products present under nonstandard conditions (pressures other than 1 bar; concentrations other than 1 M) is related to the standard free energy change according to this equation:
 
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Reaction Quotient...
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Reaction Quotient02:35

Reaction Quotient

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The status of a reversible reaction is conveniently assessed by evaluating its reaction quotient (Q). For a reversible reaction described by m A + n B ⇌ x C + y D, the reaction quotient is derived directly from the stoichiometry of the balanced equation as
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Atomic Nuclei: Nuclear Relaxation Processes01:23

Atomic Nuclei: Nuclear Relaxation Processes

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In the absence of an external magnetic field, nuclear spin states are degenerate and randomly oriented. When a magnetic field is applied, the spins begin to precess and orient themselves along (lower energy) or against (higher energy) the direction of the field. At equilibrium, a slight excess population of spins exists in the lower energy state. Because the direction of the magnetic field is fixed as the z-axis,  the precessing magnetic moments are randomly oriented around the z-axis.
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Temperature Dependence on Reaction Rate02:55

Temperature Dependence on Reaction Rate

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The Collision Theory
Atoms, molecules, or ions must collide before they can react with each other. Atoms must be close together to form chemical bonds. This premise is the basis for a theory that explains many observations regarding chemical kinetics, including factors affecting reaction rates.
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Exponential Change of Relaxation Rate by Quenched Disorder.

Jan Meibohm1, Sabine H L Klapp1

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Disorder in complex systems can surprisingly speed up or slow down relaxation rates. This study analyzes Brownian particle relaxation in harmonic potentials with quenched Gaussian disorder, finding non-monotonic effects and tunable relaxation speeds.

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Area of Science:

  • Statistical Physics
  • Complex Systems Dynamics
  • Computational Physics

Background:

  • Brownian motion is fundamental to understanding particle dynamics in various systems.
  • Rugged energy landscapes, common in complex systems, pose challenges for predicting relaxation behavior.
  • Quenched Gaussian disorder provides a tractable model for studying these landscapes.

Purpose of the Study:

  • To determine the asymptotic relaxation rate of a Brownian particle in a harmonic potential with quenched Gaussian disorder.
  • To investigate how disorder properties influence the relaxation rate's mean and variance.
  • To analyze the probability distribution of the relaxation rate in the weak disorder limit.

Main Methods:

  • Analytical derivation of the asymptotic relaxation rate.
  • Mathematical modeling of a Brownian particle in a disordered harmonic potential.
  • Analysis of non-monotonic functions for mean and variance of relaxation rate.
  • Derivation of the probability distribution in the weak disorder limit.

Main Results:

  • The asymptotic relaxation rate exhibits non-monotonic dependence on disorder parameters for various disorder types.
  • Disorder can exponentially increase or decrease the relaxation rate compared to the unperturbed case.
  • In the weak disorder limit, the relaxation rate distribution is Gaussian with universal mean and variance limits.

Conclusions:

  • Disorder can significantly accelerate or decelerate relaxation in complex systems.
  • Controlled disorder offers a mechanism for tuning the relaxation speed.
  • The findings provide insights into the behavior of systems with rugged energy landscapes.