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Calculating pH for Titration Solutions: Strong Acid/Strong Base
A titration is carried out for 25.00 mL of 0.100 M HCl (strong acid) with 0.100 M of a strong base NaOH. The pH at different volumes of added base solution can be calculated as follows:
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Chemical equations represent the identities and relative quantities of substances involved in a chemical reaction. The substances undergoing reaction are called reactants, and their formulas are placed on the left side of the equation. The substances generated by the reaction are called products, and their formulas are placed on the right side of the equation. Plus signs (+) separate individual reactant and product formulas, and an arrow (→) separates the reactant and product (left and right)...
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Pulse dynamics in reaction-diffusion equations with strong spatially localized impurities.

Arjen Doelman1, Peter van Heijster2, Jianhe Shen3

  • 1Mathematisch Instituut, Leiden University, 2300 RA Leiden, The Netherlands.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|March 7, 2018
PubMed
Summary

This study introduces a geometric singular perturbation framework to analyze localized structures in reaction-diffusion systems with nonlinear impurities. It reveals how impurities affect stability and bifurcations, with different behaviors in scalar versus two-component systems.

Keywords:
Hopf bifurcationdefect systemsexistencelocalized patternsmultiple scalesstability

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Area of Science:

  • Mathematical physics
  • Nonlinear dynamics
  • Chemical kinetics

Background:

  • Reaction-diffusion equations model complex spatiotemporal phenomena.
  • Localized structures, like pulses, are crucial in various scientific fields.
  • Impurities can significantly alter the behavior of these systems.

Purpose of the Study:

  • To develop a general geometric singular perturbation framework.
  • To investigate the impact of localized nonlinear impurities on localized structures.
  • To analyze the existence, stability, and bifurcations of these structures.

Main Methods:

  • Utilizing a multiple-scale analysis.
  • Deriving algebraic conditions for existence and stability.
  • Employing a geometric singular perturbation approach.

Main Results:

  • Algebraic conditions for pinned single- and multi-pulse solutions derived.
  • Explicit control over the spectrum of (multi-)pulse solutions achieved.
  • Hopf bifurcations are impossible in the scalar case but possible in two-component systems.

Conclusions:

  • The developed framework effectively analyzes impurity effects on localized structures.
  • System dimensionality and impurity interaction critically influence stability and bifurcations.
  • The study provides new insights into nonlinear wave and pattern stability.