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Related Concept Videos

Factors Affecting Solubility04:01

Factors Affecting Solubility

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Compared with pure water, the solubility of an ionic compound is less in aqueous solutions containing a common ion (one also produced by dissolution of the ionic compound). This is an example of a phenomenon known as the common ion effect, which is a consequence of the law of mass action that may be explained using Le Chȃtelier’s principle. Consider the dissolution of silver iodide:
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Solubility Equilibria03:07

Solubility Equilibria

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Solubility equilibria are established when the dissolution and precipitation of a solute species occur at equal rates. These equilibria underlie many natural and technological processes, ranging from tooth decay to water purification. An understanding of the factors affecting compound solubility is, therefore, essential to the effective management of these processes. This section applies previously introduced equilibrium concepts and tools to systems involving dissolution and precipitation.
The...
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Physical Properties Affecting Solubility02:19

Physical Properties Affecting Solubility

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Solutions of Gases in Liquids
As for any solution, the solubility of a gas in a liquid is affected by the attractive intermolecular forces between solute and solvent species. Unlike solid and liquid solutes, however, there is no solute-solute intermolecular attraction to overcome when a gaseous solute dissolves in a liquid solvent since the atoms or molecules comprising a gas are far separated and experience negligible interactions. Consequently, solute-solvent interactions are the sole...
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Solubility of Ionic Compounds02:55

Solubility of Ionic Compounds

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Solubility is the measure of the maximum amount of solute that can be dissolved in a given quantity of solvent at a given temperature and pressure. Solubility is usually measured in molarity (M) or moles per liter (mol/L). A compound is termed soluble if it dissolves in water.
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Solubility03:00

Solubility

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Solution, Solubility, and Solubility Equilibrium
A solution is a homogeneous mixture composed of a solvent, the major component, and a solute, the minor component. The physical state of a solution—solid, liquid, or gas—is typically the same as that of the solvent. Solute concentrations are often described with qualitative terms such as dilute (of relatively low concentration) and concentrated (of relatively high concentration).
In a solution, the solute particles (molecules,...
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Rigid Body Equilibrium Problems - I00:49

Rigid Body Equilibrium Problems - I

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A rigid body is said to be in static equilibrium when the net force and the net torque acting on the system is equal to zero. To solve for rigid body equilibrium problems, do the following steps.
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The Rigid Tube as an Alternative in Controlling the Problematic Airway
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Two exactly soluble models of rigidity percolation.

M F Thorpe1, R B Stinchcombe

  • 1Rudolf Peierls Centre for Theoretical Physics, University of Oxford, , 1 Keble Road, Oxford OX1 3NP, UK.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|January 1, 2014
PubMed
Summary
This summary is machine-generated.

We present exactly soluble models for rigidity percolation, offering benchmarks for other methods. Our findings reveal distinct transition behaviors in hierarchical and Bethe lattices, impacting the understanding of mechanical stability in disordered systems.

Keywords:
hierarchicallatticepercolationrenormalizationrigiditytree

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Materials Science

Background:

  • Rigidity percolation models are crucial for understanding mechanical stability in disordered materials.
  • Floppy modes (F) are analogous to free energy in bond dilution problems involving rigidity.
  • Two-dimensional vector displacements are considered in these pathological lattice models.

Purpose of the Study:

  • To provide exactly soluble models for rigidity percolation as benchmarks for numerical and approximate methods.
  • To analyze the critical behavior and transition orders in different lattice structures.
  • To compare theoretical predictions with simulation results.

Main Methods:

  • Renormalization group calculations for hierarchical lattices.
  • Algebraic scaling transformations.
  • Maxwell equal area construction for Bethe lattices.
  • Pebble game algorithm for random-bond lattices.

Main Results:

  • Hierarchical lattices exhibit a second-order phase transition with an unstable critical point at =4.41.
  • Order parameter exponents (β=0.0775) and scaling laws (dν=3.533) were determined for hierarchical lattices.
  • Bethe lattices show a first-order rigidity transition at =3.94, below the Maxwell constraint prediction.
  • Maxwell equal area construction accurately predicts the first-order transition, aligning with simulation data.

Conclusions:

  • Exactly soluble models provide valuable insights into rigidity percolation phenomena.
  • The study highlights the differences in transition behavior between hierarchical and Bethe lattices.
  • Theoretical predictions are validated by simulation results, reinforcing the models' utility.