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Equation of State01:07

Equation of State

1.6K
The equation of state is an equation that relates physical quantities, such as pressure, volume, temperature, and the number of moles, of a thermodynamics system with each other. The equation relating physical quantities with each other can be a simple mathematical expression or too complicated to express in mathematical form. In either case, a relationship between physical quantities exists. If the equation of state cannot be expressed in a mathematical form, then experimental data and...
1.6K
State Space Representation01:27

State Space Representation

159
The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
159
Transfer Function to State Space01:23

Transfer Function to State Space

184
State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an...
184
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

59
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
59
State Space to Transfer Function01:21

State Space to Transfer Function

164
The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
The transformation process begins with the state-space representation, characterized by the state equation and the output equation. These equations are typically represented as:
164
Free Energy Changes for Nonstandard States03:25

Free Energy Changes for Nonstandard States

10.8K
The free energy change for a process taking place with reactants and products present under nonstandard conditions (pressures other than 1 bar; concentrations other than 1 M) is related to the standard free energy change according to this equation:
 
where R is the gas constant (8.314 J/K·mol), T is the absolute temperature in kelvin, and Q is the reaction quotient. This equation may be used to predict the spontaneity of a process under any given set of conditions.
Reaction Quotient...
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Updated: May 23, 2025

Setting Limits on Supersymmetry Using Simplified Models
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Published on: November 15, 2013

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A universal equation-of-state model based on single variable functions.

Ti-Wei Xue1, Zeng-Yuan Guo2

  • 1Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China.

Scientific Reports
|March 12, 2025
PubMed
Summary

This study introduces a simple, universal equation of state (EOS) model using a macroscopic thermodynamic approach. It offers a new way to develop EOS theories, applicable to high-density and supercritical conditions.

Keywords:
Equation of stateIdeal gasSingle variable functionThermodynamics

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Area of Science:

  • Thermodynamics
  • Physical Chemistry
  • Materials Science

Background:

  • Numerous existing equations of state (EOS) are complex or rely on empirical coefficients.
  • These limitations hinder the application of current EOS theories across diverse material structures and intermolecular interactions.

Purpose of the Study:

  • To develop a simple and universal equation of state (EOS) model.
  • To establish a fully macroscopic thermodynamic approach for EOS development.
  • To provide a new framework for understanding and applying thermodynamic principles.

Main Methods:

  • Construction of two single-variable thermodynamic functions based on pressure and temperature.
  • Thermodynamic derivation of two new EOS models in P-V-T and P-S-T forms.
  • Utilizing a macroscopic thermodynamic approach without assumptions on material structure or intermolecular interactions.

Main Results:

  • Developed two novel EOS models with forms comparable in simplicity to the ideal gas EOS.
  • Ensured coefficients possess clear thermodynamic significance, enabling direct calculation without empirical fitting.
  • Demonstrated the model's capability to accurately characterize substance thermodynamic properties.

Conclusions:

  • The proposed universal EOS model offers a significant advancement over existing theories.
  • The model is suitable for high-density and supercritical applications, expanding EOS utility.
  • This work enriches fundamental thermodynamics and provides a new avenue for EOS theory development.