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Diffusion01:21

Diffusion

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Diffusion is a type of passive transport. In passive transport, a substance tends to move from an area of high concentration to an area of low concentration until the concentration is equal across the space. For example, take the diffusion of substances through the air. When someone opens a perfume bottle in a room filled with people, the perfume is at its highest concentration in the bottle and is at its lowest at the edges of the room. The perfume vapor will diffuse, or spread away, from the...
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Diffusion01:12

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Diffusion is the passive movement of substances down their concentration gradients—requiring no expenditure of cellular energy. Substances, such as molecules or ions, diffuse from an area of high concentration to an area of low concentration in the cytosol or across membranes. Eventually, the concentration will even out, with the substance moving randomly but causing no net change in concentration. Such a state is called dynamic equilibrium, which is essential for maintaining overall...
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Diffusion on Chromatography Columns01:07

Diffusion on Chromatography Columns

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In column chromatography, when an analyte is introduced as a narrow band at the top of the column, the solutes begin to separate and broaden, developing a Gaussian profile. This broadening occurs due to various factors, such as longitudinal diffusion.
Longitudinal diffusion occurs when the solute molecules in the mobile phase diffuse from the more concentrated center of the chromatographic band to the more dilute regions on either side, both towards and against the flow direction. This...
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IR Frequency Region: X–H Stretching01:24

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In IR spectroscopy, signals produced by the X−H bonds (such as C−H, O−H, or N−H) can be observed in the frequency range of  2700–4000 cm–1. The C−H stretching vibration forms sharp bands in the region 2850–3000 cm–1. The presence of the O−H stretching vibration leads to the forming of an absorption band in the frequency range 3650–3200 cm−1. At the same time, N−H stretching can be confirmed by absorption bands in...
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Interference and Diffraction02:18

Interference and Diffraction

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Interference is a characteristic phenomenon exhibited by waves. When two electromagnetic waves interact with their peaks and troughs coinciding, a resulting wave with enhanced amplitude is produced. This is known as constructive interference. In this case, the two waves interacting are in phase with each other.
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The X̄ Chart00:58

The X̄ Chart

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The  x̄ chart is a statistical tool for monitoring the means in a process.
The x̄ chart, often known as the individual control chart, is a crucial tool in statistical process control. It is designed to monitor process behavior and performance over time and is widely used in various industries to ensure that processes are operating at their optimum capacity and within specified limits.
A x̄ chart is constructed by plotting individual measurements of a quality...
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Related Experiment Video

Updated: Nov 27, 2025

In Situ Monitoring of Diffusion of Guest Molecules in Porous Media Using Electron Paramagnetic Resonance Imaging
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In Situ Monitoring of Diffusion of Guest Molecules in Porous Media Using Electron Paramagnetic Resonance Imaging

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Diffusion on Middle-ξ Cantor Sets.

Alireza Khalili Golmankhaneh1, Arran Fernandez2, Ali Khalili Golmankhaneh3

  • 1Department of Physics, Urmia Branch, Islamic Azad University, Urmia, Iran.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces a new calculus, Cζ-calculus, for middle-ξ Cantor sets. This fractional calculus framework enables solving differential equations on fractal sets and analyzing diffusion behaviors.

Keywords:
Cζ-calculusHausdorff dimensiondiffusion on fractalmiddle-ξ Cantor setsrandom walkstaircase function

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Area of Science:

  • Fractal Geometry
  • Non-integer Calculus
  • Mathematical Physics

Background:

  • Standard calculus is insufficient for functions with fractal support due to their complex nature.
  • Generalized Cantor sets, specifically middle-ξ Cantor sets, exhibit self-similarity and dimensions exceeding their topological ones.
  • Local fractional derivatives are necessary to analyze functions on fractal supports.

Purpose of the Study:

  • To generalize Cζ-calculus to middle-ξ Cantor sets.
  • To develop a novel calculus applicable to fractal structures.
  • To solve differential equations and analyze diffusion phenomena on these fractal sets.

Main Methods:

  • Generalization of Cζ-calculus for middle-ξ Cantor sets with 0<ξ<1.
  • Application of local fractional derivatives.
  • Solving differential equations on fractal domains.
  • Analysis of super-, normal, and sub-diffusion conditions.

Main Results:

  • A new calculus framework, Cζ-calculus, has been successfully generalized for middle-ξ Cantor sets.
  • Differential equations defined on these fractal sets have been solved.
  • Illustrative examples demonstrate the application and effectiveness of the proposed calculus.
  • Conditions governing different types of diffusion (super-, normal, sub-) on fractal sets are established.

Conclusions:

  • The generalized Cζ-calculus provides a powerful tool for analyzing fractal phenomena.
  • This work extends the applicability of fractional calculus to complex fractal geometries.
  • The findings contribute to understanding diffusion processes on fractal sets.