Gibbs Duhem Equation

Dive into the intriguing realm of engineering thermodynamics with a comprehensive exploration of the Gibbs Duhem Equation. This fundamental principle offers key insights into the behaviour of substances in mixtures, proving itself essential in the toolbox of every engineering student and professional. To truly harness its power, you must thoroughly understand its origins, practical uses, calculation methods, and associated concepts. So, join us on this journey through the meaning, applications, and intricacies of the Gibbs Duhem Equation.

Explore our app and discover over 50 million learning materials for free.

- Design Engineering
- Engineering Fluid Mechanics
- Engineering Mathematics
- Engineering Thermodynamics
- Absolute Temperature
- Adiabatic Expansion
- Adiabatic Expansion of an Ideal Gas
- Adiabatic Lapse Rate
- Adiabatic Process
- Application of First Law of Thermodynamics
- Availability
- Binary Cycle
- Binary Mixture
- Bomb Calorimeter
- Carnot Cycle
- Carnot Theorem
- Carnot Vapor Cycle
- Chemical Energy
- Chemical Potential
- Chemical Potential Ideal Gas
- Clausius Clapeyron Equation
- Clausius Inequality
- Clausius Theorem
- Closed System Thermodynamics
- Coefficient of Thermal Expansion
- Cogeneration
- Combined Convection and Radiation
- Combined Cycle Power Plant
- Combustion Engine
- Compressor
- Conduction
- Conjugate Variables
- Continuous Combustion Engine
- Continuous Phase Transition
- Convection
- Dead State
- Degrees of Freedom Physics
- Differential Convection Equations
- Diffuser
- Diffusion Equation
- Double Tube Heat Exchanger
- Economizer
- Electrical Work
- Endothermic Reactions
- Energy Degradation
- Energy Equation
- Energy Function
- Enthalpy
- Enthalpy of Fusion
- Enthalpy of Vaporization
- Entropy Change for Ideal Gas
- Entropy Function
- Entropy Generation
- Entropy Gradient
- Entropy and Heat Capacity
- Entropy and Irreversibility
- Entropy of Mixing
- Equation of State of a Gas
- Equation of State of an Ideal Gas
- Equations of State
- Exergy
- Exergy Analysis
- Exergy Efficiency
- Exothermic Reactions
- Expansion
- Extensive Property
- External Combustion Engine
- Feedwater Heater
- Fins
- First Law of Thermodynamics Differential Form
- First Law of Thermodynamics For Open System
- Flow Process
- Fluctuations
- Forced Convection
- Four Stroke Engine
- Free Expansion
- Free Expansion of an Ideal Gas
- Fundamental Equation
- Fundamentals of Engineering Thermodynamics
- Gases
- Gibbs Duhem Equation
- Gibbs Free Energy
- Gibbs Paradox
- Greenhouse Effect
- Heat
- Heat Capacity
- Heat Equation
- Heat Exchanger
- Heat Generation
- Heat Pump
- Heat and Work
- Helmholtz Free Energy
- Hydrostatic Transmission
- Initial Conditions
- Intensive Property
- Intensive and Extensive Variables
- Internal Energy of a Real Gas
- Irreversibility
- Isentropic Efficiency
- Isentropic Efficiency of Compressor
- Isentropic Process
- Isobaric Process
- Isochoric Process
- Isolated System
- Isothermal Process
- Johnson Noise
- Joule Kelvin Expansion
- Joule-Thompson Effect
- Kinetic Theory of Ideal Gases
- Landau Theory of Phase Transition
- Linear Heat Conduction
- Liquefaction of Gases
- Macroscopic Thermodynamics
- Maximum Entropy
- Maxwell Relations
- Mechanism of Heat Transfer
- Metastable Phase
- Moles
- Natural Convection
- Nature of Heat
- Negative Heat Capacity
- Negative Temperature
- Non Equilibrium State
- Nuclear Energy
- Nucleation
- Nusselt Number
- Open System Thermodynamic
- Osmotic Pressure
- Otto Cycle
- Partition Function
- Peng Robinson Equation of State
- Polytropic Process
- Potential Energy in Thermodynamics
- Power Cycle
- Power Plants
- Pressure Volume Work
- Principle of Minimum Energy
- Principles of Heat Transfer
- Quasi Static Process
- Ramjet
- Real Gas Internal Energy
- Reciprocating Engine
- Refrigeration Cycle
- Refrigerator
- Regenerative Rankine Cycle
- Reheat Rankine Cycle
- Relaxation Time
- Reversibility
- Reversible Process
- Rotary Engine
- Sackur Tetrode Equation
- Specific Volume
- Steady State Heat Transfer
- Stirling Engines
- Stretched Wire
- Surface Thermodynamics
- System Surroundings and Boundary
- TdS Equation
- Temperature Scales
- Thermal Boundary Layer
- Thermal Diffusivity
- Thermodynamic Equilibrium
- Thermodynamic Limit
- Thermodynamic Potentials
- Thermodynamic Relations
- Thermodynamic Stability
- Thermodynamic State
- Thermodynamic System
- Thermodynamic Variables
- Thermodynamics of Gases
- Thermoelectric
- Thermoelectric Effect
- Thermometry
- Third Law of Thermodynamics
- Throttling Device
- Transient Heat Transfer
- Triple Point and Critical Point
- Two Stroke Diesel Engine
- Two Stroke Engine
- Unattainability
- Van der Waals Equation
- Vapor Power System
- Variable Thermal Conductivity
- Wien's Law
- Zeroth Law of Thermodynamics
- Materials Engineering
- Professional Engineering
- Solid Mechanics
- What is Engineering

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Jetzt kostenlos anmeldenNie wieder prokastinieren mit unseren Lernerinnerungen.

Jetzt kostenlos anmeldenDive into the intriguing realm of engineering thermodynamics with a comprehensive exploration of the Gibbs Duhem Equation. This fundamental principle offers key insights into the behaviour of substances in mixtures, proving itself essential in the toolbox of every engineering student and professional. To truly harness its power, you must thoroughly understand its origins, practical uses, calculation methods, and associated concepts. So, join us on this journey through the meaning, applications, and intricacies of the Gibbs Duhem Equation.

- \( \mu_{i} \) is the chemical potential.
- \( n_{i} \) is the amount (in moles) of the ith component.

The chemical potential \( \mu_{i} \) represents the change in a system’s potential energy with the addition of a particle of species i while keeping the particles of other species constant.

Gibbs is also famous for the Gibbs free energy concept, a thermodynamic potential that reveals the maximum reversible work a system can perform at constant temperature and pressure.

- Thermodynamic Equilibrium: A state of balance where there are no observable changes as time goes by.
- Chemical Potential: It represents the change in potential energy of a system with the addition of a particle of species i, while keeping the particles of other species constant.
- Internal Energy: The total energy that's associated with the kinetic energy due to the motion of molecules and the potential energy associated with the vibrational and electric energy of atoms within molecules.

Term |
Definition |

Thermodynamic Equilibrium | A state of balance where there are no observable changes as time goes by |

Chemical Potential | Change in potential energy of a system with the addition of a particle of species i, while keeping the particles of other species constant |

Internal Energy | Total energy that's associated with the kinetic energy due to the motion of molecules and the potential energy associated with the vibrational and electric energy of atoms within molecules |

For instance, let's consider a mixture of two substances A and B, which are in vapour-liquid equilibrium. Suppose that the substances do not generate azeotropes. In an isothermal process, the Gibbs Duhem equation can be written as:

\[dn_{A}d\mu_{A} + dn_{B}d\mu_{B} = 0\]

In essence, this shows that when a small amount of substance A is added to the system at constant temperature and pressure, the chemical potentials of A and B both change to maintain the Gibbs Duhem equation.

**Step 1**: Define the total composition of the system, represented by \(n_i\) where \(i\) is the component.**Step 2**: Write down the Gibbs Duhem equation for the system at a fixed temperature and pressure, \(\sum_{i} n_i d\mu_i = 0\).**Step 3**: Introduce the appropriate values for chemical potential for each component in the mixture.**Step 4**: Perform the required algebraic manipulations to solve the equations.**Step 5**: Interpret the solutions - changes in the chemical potential of individual components can indicate how the system's equilibrium will be affected by changes in the amounts of these components.

For instance, one study used the Gibbs Duhem equation to better understand the properties of seawater. It modelled the seawater as a mixture of water and salt, and then used the Gibbs Duhem equation to understand how changes in their relative quantities would shift the equilibrium of the system.

For a simple system: Gibbs Duhem Equation →\[\sum_{i}n_{i}d \mu_{i} = 0\] Here, \(n_{i}\) is the amount of the ith component. \(\mu_{i}\) is the chemical potential of the ith component.Contributing to the development of processes in diverse industries, the Gibbs Duhem equation aids in controlling vital parameters and refining processes for maximum efficiency and safety.

For an oil reservoir at a known temperature and pressure, using the Gibbs Duhem Equation, one can calculate the mole fraction of gases such as methane, ethane, and others in the petroleum reserves. This detailed knowledge helps to plan the extraction process more efficiently and ensure safety regulations are adhered to.

- \(n_{i}\) is the number of moles of the \(i^{th}\) component
- \(d\mu_{i}\) denotes the change in chemical potential for the \(i^{th}\) component

- Ignoring the principle of 'independent variables': In the Gibbs Duhem equation, the variables are not fully independent. The equation practically bounds them due to which you cannot alter one variable without affecting at least one other.
- Overlooking the condition of constant temperature and pressure: The Gibbs Duhem equation is valid only under isothermal and isobaric conditions. Attempting to apply it to a system wherein temperature or pressure is changing can lead to incorrect results.
- Not considering each component in a mixture: Neglecting any substance in a system can drastically affect the results, given that the Gibbs Duhem equation works for the complex equilibria of mixtures.

**Gibbs Duhem Equation Meaning:**The Gibbs Duhem equation is a law in thermodynamics that links the chemical potential of compounds in a mixture and is only applicable under constant temperature and pressure conditions. The equation provides an understanding of the behaviour of mixtures at equilibrium states and the variables influencing the system's performance.**Chemical Potential:**Known as change in potential energy of a system with the addition of a particle of species i, while keeping the particles of other species constant.**Thermodynamic Equilibrium:**A state of balance where there are no observable changes as time goes by.**Gibbs Duhem Equation Examples and Applications:**The equation has wide applicability, particularly in engineering thermodynamics and physical chemistry. It informs the understanding of mixtures, phase behaviour, and assists in controlling vital process parameters. For example, in petroleum engineering, the equation helps in understanding multi-component mixtures and in planning efficient extraction processes.**Gibbs Duhem Equation Formula:**For a simple system, the Gibbs Duhem equation is represented as \(\sum_{i}n_{i}d \mu_{i} = 0\). Here, \(n_{i}\) is the number of moles of the ith component and \(\mu_{i}\) is the chemical potential of the ith component. This equation stems from the derivation of Gibbs energy and effectively links fundamental thermodynamic quantities like pressure, temperature, and chemical potential within a system in equilibrium.

The Gibbs-Duhem equation is a thermodynamic equation that describes the relationship between changes in chemical potential for components in a mixture. It asserts that the sum of the chemical potential per component times the change in its molar fraction is equal to zero.

The Gibbs-Duhem equation provides a relationship between changes in the chemical potential of a mixture's components, given by the temperature and pressure. It essentially tells us that you can't change the composition of a phase without affecting the chemical potential of all components.

An example of the Gibbs Duhem equation is its use in the phase rule for multicomponent systems. It allows one to determine the number of independent intensive properties in a system at equilibrium, aiding in studies of phase behaviour in mixtures.

The Gibbs Duhem Equation is applicable for deriving the phase behaviour of mixtures in thermodynamics. It specifically assists in predicting the behaviour of multi-component, non-ideal systems by relating changes in temperature, pressure, and composition.

The Gibbs-Duhem relationship for a particle refers to the equation that demonstrates the interdependence of chemical potentials for components in a closed system, at equilibrium, while maintaining constant temperature and pressure. It is a derivation of the Gibbs-Duhem equation applied at a particle level.

What does the Gibbs Duhem equation fundamentally relate in the field of thermodynamics?

The Gibbs Duhem equation relates changes in the chemical potential for components in a system where temperature and pressure are held constant.

Who first introduced the Gibbs Duhem equation and what critical principle does it lay the foundation for?

Josiah Willard Gibbs and Pierre Duhem first introduced the Gibbs Duhem equation. It forms the basis for the Phase Rule, which provides the number of degrees of freedom in a system at equilibrium.

What is the meaning of chemical potential in terms of the Gibbs Duhem equation?

The chemical potential represents the change in a system’s potential energy with the addition of a particle of species i, while keeping the particles of other species constant.

What is the Gibbs Duhem equation used for in scientific and engineering realms?

The Gibbs Duhem equation is used to understand the behavior of mixtures at equilibrium states. It's instrumental in phase diagram analysis, predicting system behaviours at equilibrium, and gaining insights into mixtures undergoing phase transitions.

How does the Gibbs Duhem equation work in a system of two substances A and B in vapour-liquid equilibrium, if they do not generate azeotropes?

If a small amount of substance A is added to the system at constant temperature and pressure, the chemical potentials of A and B both change to maintain the Gibbs Duhem equation.

What is the step-by-step guide to solving a Gibbs Duhem equation?

First, define the total composition of the system. Second, write down the Gibbs Duhem equation for the system at a fixed temperature and pressure. Third, introduce the appropriate values for chemical potential for each component in the mixture. Four, perform the required algebraic manipulations to solve the equations. Lastly, interpret the solutions and their impact on the system's equilibrium.

Already have an account? Log in

Open in App
More about Gibbs Duhem Equation

The first learning app that truly has everything you need to ace your exams in one place

- Flashcards & Quizzes
- AI Study Assistant
- Study Planner
- Mock-Exams
- Smart Note-Taking

Sign up to highlight and take notes. It’s 100% free.

Save explanations to your personalised space and access them anytime, anywhere!

Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.

Already have an account? Log in

Already have an account? Log in

The first learning app that truly has everything you need to ace your exams in one place

- Flashcards & Quizzes
- AI Study Assistant
- Study Planner
- Mock-Exams
- Smart Note-Taking

Sign up with Email

Already have an account? Log in