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Free Energy Changes for Nonstandard States

Free Energy Changes for Nonstandard States

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:

Calculation of First-Law Quantities II

Calculation of First-Law Quantities II

The first law of thermodynamics establishes that the change in internal energy of a system is given by ΔU = q + w, where q is the heat exchanged, and w is the work performed. For a perfect gas, both internal energy (U) and enthalpy (H) depend solely on temperature. Consequently, for any change of state, whether reversible or irreversible, the internal energy change is determined by integrating the heat capacity at constant volume, and the enthalpy change by integrating the heat capacity at...

Gibbs Free Energy

Gibbs Free Energy

One of the challenges of using the second law of thermodynamics to determine if a process is spontaneous is that it requires measurements of the entropy change for the system and the entropy change for the surroundings. An alternative approach involving a new thermodynamic property defined in terms of system properties only was introduced in the late nineteenth century by American mathematician Josiah Willard Gibbs. This new property is called the Gibbs free energy (G) (or simply the free...

Potential-Energy Criterion for Equilibrium

Potential-Energy Criterion for Equilibrium

Potential energy or potential function plays an essential role in determining the stability of a mechanical system. If a system is subjected to both gravitational and elastic forces, the potential function of the system can be expressed as the algebraic sum of gravitational and elastic potential energy. If the system is in equilibrium and is displaced by a small amount, then the work done on the system equals the negative of the change in the system's potential energy from the initial to the...

Free Energy and Equilibrium

Free Energy and Equilibrium

The free energy change for a process may be viewed as a measure of its driving force. A negative value for ΔG represents a driving force for the process in the forward direction, while a positive value represents a driving force for the process in the reverse direction. When ΔGrxn is zero, the forward and reverse driving forces are equal, and the process occurs in both directions at the same rate (the system is at equilibrium).
Recall that Q is the numerical value of the mass action expression...

Free Energy and Equilibrium

Free Energy and Equilibrium

The free energy change for a process may be viewed as a measure of its driving force. A negative value for ΔG represents a driving force for the process in the forward direction, while a positive value represents a driving force for the process in the reverse direction. When ΔG is zero, the forward and reverse driving forces are equal, and the process occurs in both directions at the same rate (the system is at equilibrium).
The reaction quotient, Q, is a convenient measure of the status of an...

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Related Experiment Video

Updated: Jun 28, 2026

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method

Published on: July 19, 2019

Quantum free-energy differences from nonequilibrium path integrals. I. Methods and numerical application.

Ramses van Zon¹, Lisandro Hernández de la Peña, Gilles H Peslherbe

¹Chemical Physics Theory Group, Department of Chemistry, University of Toronto, 80 Saint George Street, Toronto, Ontario, Canada.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|November 13, 2008

Summary

This summary is machine-generated.

This study introduces a novel method for calculating quantum system free-energy differences using nonequilibrium processes. The approach combines imaginary-time path integrals with work fluctuation relations, offering a new computational tool.

Related Experiment Videos

Last Updated: Jun 28, 2026

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method

Published on: July 19, 2019

Area of Science:

Quantum mechanics
Statistical mechanics
Computational physics

Background:

Calculating free-energy differences in quantum systems is computationally challenging.
Nonequilibrium work fluctuation relations offer alternative pathways for thermodynamic calculations.
Path-integral methods provide a powerful framework for quantum statistical mechanics.

Purpose of the Study:

To develop a method for computing free-energy differences in quantum systems.
To combine imaginary-time path-integral representations with nonequilibrium work fluctuation relations.
To establish a computational framework for quantum systems using fictitious Hamiltonian dynamics.

Main Methods:

The study combines the imaginary-time path-integral representation of the canonical partition function with nonequilibrium work fluctuation relations.
A fictitious Hamiltonian dynamics is associated with the path-integral representation, isomorphic to a classical field theory.
Two regularization methods are introduced to handle diverging energy in the classical field theory, limiting degrees of freedom.
A parameter-free smoothing procedure for work distribution functions is employed.

Main Results:

The Jarzynski nonequilibrium work relation and Crooks fluctuation relation are shown to formally hold when a control parameter in the fictitious Hamiltonian is changed.
The developed methods are numerically demonstrated for a quartic double-well potential with varying asymmetry.
The regularization techniques effectively manage the divergence of energy in the classical field theory.

Conclusions:

The proposed methods provide a viable approach for computing free-energy differences in quantum systems via nonequilibrium processes.
The combination of path integrals and fluctuation relations offers a powerful tool for quantum thermodynamics.
The numerical results validate the applicability and effectiveness of the developed computational techniques.