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Related Concept Videos

Phase Diagrams02:39

Phase Diagrams

51.0K
A phase diagram combines plots of pressure versus temperature for the liquid-gas, solid-liquid, and solid-gas phase-transition equilibria of a substance. These diagrams indicate the physical states that exist under specific conditions of pressure and temperature and also provide the pressure dependence of the phase-transition temperatures (melting points, sublimation points, boiling points). Regions or areas labeled solid, liquid, and gas represent single phases, while lines or curves represent...
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Isochoric and Isobaric Processes01:21

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A thermodynamic process that occurs at constant volume is called an isochoric process. According to the first law of thermodynamics, heat supplied or removed from the system is partially utilized to perform work and change the internal energy of the system. However, in an isochoric process, the volume remains constant. Hence, the work done by the system is zero. Therefore, the exchange of heat changes the internal energy of the system only. 
Suppose 1000 g of water is heated from 40...
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Phase Diagram01:24

Phase Diagram

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A phase diagram is a graphical representation of the physical states of a substance under different conditions of temperature and pressure. It shows the boundaries between solid, liquid, and gas phases and the conditions at which these phases coexist in equilibrium. An area in a phase diagram represents a single phase, whereas lines or phase boundaries represent the equilibrium between two phases.In the phase diagram of water, the boundary line between the solid and liquid states illustrates...
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Phase Diagram01:19

Phase Diagram

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The phase of a given substance depends on the pressure and temperature. Thus, plots of pressure versus temperature showing the phase in each region provide considerable insights into the thermal properties of substances. Such plots are known as phase diagrams. For instance, in the phase diagram for water (Figure 1), the solid curve boundaries between the phases indicate phase transitions (i.e., temperatures and pressures at which the phases coexist).
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Phase Changes01:19

Phase Changes

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Phase transitions play an important theoretical and practical role in the study of heat flow. In melting or fusion, a solid turns into a liquid; the opposite process is freezing. In evaporation, a liquid turns into a gas; the opposite process is condensation.
A substance melts or freezes at a temperature called its melting point and boils or condenses at its boiling point. These temperatures depend on pressure. High pressure favors the denser form of the substance, so typically, high pressure...
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Phase Transitions01:21

Phase Transitions

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A phase transition is the process in which a substance changes from one state of matter to another, like from a solid to a liquid, liquid to gas, or vice versa, at a specific temperature and under given pressure conditions. This change is spontaneous and is affected by alterations in temperature and pressure. These parameters impact the strength of the forces between molecules (intermolecular forces) in the substance.During a phase transition, both the initial and final phases of the substance...
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Isochrons and phaseless sets.

J Guckenheimer1

  • 1Division of Natural Sciences, University of California, 95064, Santa Cruz, CA, USA.

Journal of Mathematical Biology
|March 18, 2017
PubMed
Summary
This summary is machine-generated.

This study explores mathematical models of biological clocks, proving the existence of isochrons and detailing their properties. These findings advance our understanding of dynamical systems and circadian rhythms.

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Area of Science:

  • Mathematics
  • Dynamical Systems Theory
  • Chronobiology

Background:

  • Arthur Winfree's phase resetting experiments provide a foundation for mathematical modeling of biological clocks.
  • These models raise complex questions within the field of dynamical systems.

Purpose of the Study:

  • To address mathematical questions arising from Winfree's biological clock models.
  • To demonstrate the existence of isochrons in these systems.
  • To establish key properties of isochrons.

Main Methods:

  • Analysis of mathematical models derived from biological clock experiments.
  • Theoretical investigation of dynamical systems properties.
  • Proof of existence for isochrons.

Main Results:

  • The existence of isochrons is mathematically shown.
  • Fundamental properties of these isochrons are established.
  • Theoretical framework for understanding biological clock dynamics is advanced.

Conclusions:

  • The mathematical framework supports the understanding of biological clock behavior.
  • Isochrons are confirmed as a significant feature in these dynamical systems.
  • This work bridges mathematical theory with biological phenomena.