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Related Concept Videos

Titration Calculations: Strong Acid - Strong Base02:28

Titration Calculations: Strong Acid - Strong Base

33.8K
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:
(a) Titrant volume = 0 mL. The solution pH is due to the acid ionization of HCl. Because this is a strong acid, the ionization is complete and the hydronium ion molarity is 0.100 M. The pH of the solution is then:
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Strong Acid and Base Solutions03:22

Strong Acid and Base Solutions

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A strong acid is a compound that dissociates completely in an aqueous solution and produces a concentration of hydronium ions equal to the initial concentration of acid. For example, 0.20 M hydrobromic acid will dissociate completely in water and produces 0.20 M of hydronium ions and 0.20 M of bromide ions.
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Titration of a Strong Acid with a Strong Base01:23

Titration of a Strong Acid with a Strong Base

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During the titration of a strong acid with a strong base, pH calculations are primarily based on the concentration of residual hydronium or hydroxide ions. Initially, a strong acid like hydrochloric acid fully dissociates, creating hydronium and chloride ions, resulting in a low pH. The addition of a strong base like sodium hydroxide alters the concentration of hydronium ions by neutralizing them. As more base is added, the pH gradually increases. At the equivalence point, all hydronium ions...
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Uniform Distribution01:19

Uniform Distribution

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The uniform distribution is a continuous probability distribution of events with an equal probability of occurrence. This distribution is rectangular.
Two essential properties of this distribution are
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Uniform Circular Motion01:14

Uniform Circular Motion

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Uniform circular motion is a specific type of motion in which an object travels in a circle with a constant speed. For example, any point on a propeller spinning at a constant rate is undergoing uniform circular motion. The second, minute, and hour hands of a watch also undergo uniform circular motion. It is hard to believe that points on these rotating objects are actually accelerating, even though the rotation rate is constant. To understand this, we must analyze the motion in terms of...
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Non-uniform Circular Motion01:22

Non-uniform Circular Motion

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In uniform circular motion, the particle executing circular motion has a constant speed, and the circle is at a fixed radius. However, not all circular motion occurs at a constant speed. A particle can travel in a circle and speed up or slow down, showing an acceleration in the direction of motion. In that case, the motion is called non-uniform circular motion, and an additional acceleration is introduced, which is in the direction tangential to the circle. 
For example, such...
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Regular Care and Maintenance of a Zebrafish Danio rerio Laboratory: An Introduction
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On uniform regularity and strong regularity.

R Cibulka1, J Preininger2, T Roubal1

  • 1Faculty of Applied Sciences, Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic.

Optimization
|May 7, 2019
PubMed
Summary
This summary is machine-generated.

This study analyzes metric regularity properties for tracking solutions of differential generalized equations (DGEs). It introduces two inexact path-following methods with improved accuracy for control and optimization problems.

Keywords:
49J4049J5349k4090c31Control systemdiscrete approximationpath-followinguniform metric regularityuniform strong metric regularity

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Area of Science:

  • Optimization Theory
  • Numerical Analysis
  • Control Theory

Background:

  • Metric regularity and strong metric regularity are crucial for analyzing solution trajectories of generalized equations.
  • Differential generalized equations (DGEs) offer a unified framework for diverse problems in control and optimization, including differential variational inequalities and state-constrained systems.

Purpose of the Study:

  • To investigate uniform versions of metric regularity and strong metric regularity on compact subsets of Banach spaces, particularly along continuous paths.
  • To analyze the role of these regularity properties in path-following schemes for DGEs.
  • To develop and compare two inexact path-following methods for DGEs with different orders of grid error.

Main Methods:

  • Analysis of uniform metric regularity and strong metric regularity on compact subsets of Banach spaces.
  • Development of two inexact path-following algorithms for DGEs with grid error orders of O(h) and O(h^2).
  • Numerical experiments to compare the performance of the developed schemes on physics-related problems.

Main Results:

  • Established the importance of uniform regularity properties for path-following in DGEs.
  • Introduced and analyzed two inexact path-following methods with proven convergence properties.
  • Demonstrated the practical performance of the methods through numerical experiments.

Conclusions:

  • The study provides theoretical insights and practical algorithms for solving DGEs, relevant to control and optimization.
  • The developed path-following methods offer efficient ways to track solution trajectories of DGEs.
  • Further research can explore the application of these methods to more complex problems in applied mathematics and physics.