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Related Concept Videos

General Properties of Solutions02:12

General Properties of Solutions

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Many common substances around us exist as a solution, such as ocean water, air, and gasoline. All solutions are mixtures of substances that are composed of varying amounts of two or more types of atoms or molecules. A mixture with a non-uniform composition is a heterogeneous mixture, whereas a mixture with a uniform composition is a homogeneous mixture. The components that make the homogeneous mixture are evenly spread out and thoroughly mixed. 
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Solution Formation02:16

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There is no one solvent that can dissolve every type of solute. Some substances that readily dissolve in a certain solvent might be insoluble in a different solvent. A simple way to predict which substances dissolve in which solvent is the phrase "like dissolves like". This means that polar substances, such as salt and sugar, dissolve in a polar substance like water. In contrast, non-polar substances are more soluble in non-polar solvents such as carbon tetrachloride.
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Imagine adding a small amount of sugar to a glass of water, stirring until all the sugar has dissolved, and then adding a bit more. You can repeat this process until the sugar concentration of the solution reaches its natural limit, a limit determined primarily by the relative strengths of the solute-solute, solute-solvent, and solvent-solvent attractive forces. You can be certain that you have reached this limit because, no matter how long you stir the solution, undissolved sugar remains. The...
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The formation of a solution is an example of a spontaneous process, a process that occurs under specified conditions without energy from some external source.
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According to Raoult’s law, the partial vapor pressure of a solvent in a solution is equal or identical to the vapor pressure of the pure solvent multiplied by its mole fraction in the solution. However, Raoult's Law is only valid for ideal solutions. For a solution to be ideal, the solvent-solute interaction must be just as strong as a solvent-solvent or solute-solute interaction. This suggests that both the solute and the solvent would use the same amount of energy to escape to the...
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A solute is a component of a solution that is typically present at a much lower concentration than the solvent. Solute concentrations are often described with qualitative terms such as dilute (of relatively low concentration) and concentrated (of relatively high concentration).
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Closed-form solutions for the Lévy-stable distribution.

Karina Arias-Calluari1, Fernando Alonso-Marroquin1,2, Michael S Harré3

  • 1School of Civil Engineering, The University of Sydney, Sydney NSW 2006, Australia.

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Summary
This summary is machine-generated.

Researchers developed a uniform analytical approximation for the Lévy-stable distribution, crucial for modeling power laws with infinite variance in fields like finance and physics.

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Area of Science:

  • Mathematics and Physics
  • Statistical Mechanics
  • Financial Mathematics

Background:

  • The Lévy-stable distribution models power-law phenomena with infinite variance, essential in economics and statistical mechanics.
  • Existing numerical methods for this distribution face challenges due to its complex, non-explicit form.
  • A uniform analytical solution has been lacking, hindering broader application.

Purpose of the Study:

  • To develop a novel, uniform analytical approximation for the Lévy-stable distribution.
  • To address the computational challenges associated with the non-explicit nature of this distribution.
  • To provide a more accessible and accurate method for researchers utilizing Lévy-stable distributions.

Main Methods:

  • The study introduces a 'trans-stable' auxiliary function to manage numerical calculation issues.
  • A uniform solution is derived by asymptotically matching 'inner' and 'outer' power series expansions.
  • The proposed analytical approximation is validated against numerical results.

Main Results:

  • A new uniform analytical approximation for the Lévy-stable distribution has been successfully developed.
  • The trans-stable function effectively resolves numerical instabilities in calculations.
  • The approximation demonstrates high accuracy when compared to established numerical methods.

Conclusions:

  • The presented uniform analytical approximation offers a robust and accurate method for the Lévy-stable distribution.
  • This approach simplifies complex calculations, making the distribution more accessible for various applications.
  • The findings are significant for fields relying on power-law modeling, including finance and anomalous transport.