- Thermodynamic potential
A thermodynamic potential is a
scalar potentialfunction used to represent the thermodynamic stateof a system. One main thermodynamic potential which has a physical interpretation is the internal energy, U. It is the energy of configuration of a given system of conservative forces (that is why it is a potential) and only has meaning with respect to a defined set of references (or data). Expressions for all other thermodynamic energy potentials are derivable via Legendre transforms from an expression for U. In thermodynamics, certain forces, such as gravity, are typically disregarded when formulating expressions for potentials. For example, while all the working fluidin a steam enginemay have higher energy due to gravity while sitting on top of Mt. Everestthan it would at the bottom of the Mariana trench, the gravitational potential energyterm in the formula for the internal energy would usually be ignored because "changes" in gravitational potential within the engine during operation would be negligible. Five common thermodynamic energy potentials are [Alberty (2001) p1353] :
Note that the infinitesimals on the right hand side of each of the above equations are of the natural variables of the potential on the left hand side.Similar equations can be developed for all of the other thermodynamic potentials of the system. There will be one fundamental equation for each thermodynamic potential, resulting in a total of fundamental equations.
The equations of state
We can use the above equations to derive some differential definitions of some thermodynamic parameters. If we define Φ to stand for any of the thermodynamic potentials, then the above equations are of the form:
where and are conjugate pairs, and the are the natural variables of the potential . From the
chain ruleit follows that:
Where is the set of all natural variables of except . This yields expressions for various thermodynamic parameters in terms of the derivatives of the potentials with respect to their natural variables. These equations are known as equations of state since they specify parameters of the thermodynamic state [Callen (1985) p37] . If we restrict ourselves to the potentials U,F,H and G, then we have:
where, in the last equation, is any of the thermodynamic potentials U, F, H, G and are the set of natural variables for that potential, excluding . If we use all potentials, then we will have more equations of state such as
and so on. In all, there will be "D" equations for each potential resulting in a total of "D" 2"D" equations of state. If the "D" equations of state for a particular potential are known, then the fundamental equation for that potential can be determined. This means that all thermodynamic information about the system will be known, and that the fundamental equations for any other potential can be found, along with the corresponding equations of state.
The Maxwell relations
Again, define and to be conjugate pairs, and the to be the natural variables of some potential . We may take the "cross differentials" of the state equations, which obey the following relationship:
From these we get the
Maxwell relations[Alberty (2001) p1353] [Callen (1985) p181] . There will be "(D-1)/2" of them for each potential giving a total of "D(D-1)/2" equations in all. If we restrict ourselves the U, F, H, G
Using the equations of state involving the chemical potential we get equations such as:
and using the other potentials we can get equations such as:
Again, define and to be conjugate pairs, and the to be the natural variables of the internal energy.Since all of the natural variables of the internal energy "U" are extensive quantities
it follows from Euler's homogeneous function theorem that the internal energy can be written as:
From the equations of state, we then have:
Substituting into the expressions for the other main potentials we have:
As in the above sections, this process can be carried out on all of the other thermodynamic potentials. Note that the Euler integrals are sometimes also referred to as fundamental equations.
The Gibbs-Duhem relation
Deriving the Gibbs-Duhem equation from basic thermodynamic state equations is straightforward [Moran & Shapiro, p538] [Alberty (2001) p1354] [Callen (1985) p60] . The
Gibbs free energycan be expanded locally at equilibrium in terms of the thermodynamic state as:
With the substitution of two of the
Maxwell relationsand the definition of chemical potential, this is transformed into:
The chemical potential is just another name for the partial molar Gibbs free energy, and as such:::
Subtracting yields the Gibbs-Duhem relation:
The Gibbs-Duhem is a relationship among the intensive parameters of the system. It follows that for a simple system with components, there will be independent parameters, or degrees of freedom. For example, a simple system with a single component will have two degrees of freedom, and may be specified by only two parameters, such as pressure and volume for example. The law is named after
Josiah Willard Gibbsand Pierre Duhem.
Changes in these quantities are useful for assessing the degree to which a chemical reaction will proceed. The relevant quantity depends on the reaction conditions, as shown in the following table. "Δ" denotes the change in the potential and at equilibrium the change will be zero.
Most commonly one considers reactions at constant "p" and "T", so the Gibbs free energy is the most useful potential in studies of chemical reactions.
* cite journal
author=Alberty, R. A.
url = http://www.iupac.org/publications/pac/2001/pdf/7308x1349.pdf
title = Use of Legendre transforms in chemical thermodynamics
journal=Pure Appl. Chem.
year=2001 | volume=Vol. 73 | issue=8 | pages=1349–1380
first = Herbert B. | last = Callen | authorlink = Herbert Callen | year = 1985
title = Thermodynamics and an Introduction to Themostatistics | edition = 2nd Ed.
publisher = John Wiley & Sons | location = New York | id = ISBN 0-471-86256-8
first = Herbert B. | last =Moran | first=Michael J. |coauthors=Shapiro, Howard N.
title = Fundamentals of Engineering Thermodynamics | edition = 3rd Ed.| id = ISBN 0-471-07681-3
* [http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thepot.html Thermodynamic Potentials] - Georgia State University
* [http://arxiv.org/pdf/physics/0004055.pdf Chemical Potential Energy: The 'Characteristic' vs the Concentration-Dependent Kind]
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