Free energy
Physical size  

Surname 
Free energy, Helmholtz energy 

Size type  energy  
Formula symbol  

The free energy (also Helmholtz potential , Helmholtz free energy or Helmholtz energy according to Hermann von Helmholtz ) is a thermodynamic potential . It has the dimension of an energy , is denoted by the symbol or and measured in the unit Joule . It is the Legendre transform of the internal energy with respect to entropy and is calculated as the internal energy of the system minus the product of the absolute temperature and the entropy of the system:
 .
The free energy is an extensive quantity .
The molar free energy (unit: J / mol ) is the free energy related to the amount of substance :
 .
The specific free energy (unit: J / kg) is the free energy related to the mass :
 .
The molar and the specific free energies are intense quantities .
meaning
A system, the temperature and volume of which are kept constant, assumes, of all the achievable states with this temperature and this volume, the state of equilibrium in which the free energy has the smallest possible value.
If a system changes to a lowerenergy state with the same temperature via a reversible process , the difference in the free energies of the two states indicates what proportion of the energy to be dissipated can be used for work .
In theoretical thermodynamics, free energy is a fundamental function , from which the entire thermodynamic information about the system can be derived. The prerequisite, however, is that it is given as a function of the variables temperature , volume and amount of substance of the chemical components contained in the system. These are the "natural variables" of free energy. It can also be applied as a function of other variables, but then no longer contains the complete thermodynamic information.
The free energy is a Legendre transform of the internal energy. The internal energy is also a fundamental feature when used as a function of their natural variables , , is given. The transition to other sets of variables requires the application of a Legendre transformation if it is to occur without loss of information. The transformation of the internal energy is a fundamental function of natural variables , , generated returns the expression , so the free energy. The term following from the Legendre transformation compensates for the loss of information that would otherwise be associated with the variable change.
The free energy is not to be confused with the free enthalpy or Gibbs energy .
Minimum principle of free energy
According to the second law of thermodynamics , a closed system among the attainable states takes as a state of equilibrium that which has the highest entropy for the given internal energy . From this maximum principle of entropy, a minimum principle of internal energy can be derived: If the entropy is kept constant, a system assumes the state of equilibrium that has the lowest internal energy.
A similar minimum principle exists for the free energy: A system, the temperature and volume of which are kept constant, assumes the state of equilibrium in which the free energy has the smallest possible value of all achievable states with this temperature and this volume.
To prove this, consider a system whose temperature is kept at a constant value. This can be done, for example, in that the system under consideration is in contact via a heatpermeable wall with a second system that invariably has the desired temperature (in thermodynamic expression: a heat reservoir ). If necessary, the system under consideration can exchange heat with the heat reservoir via a heat flow through the contact wall until it has adjusted its temperature to that of the reservoir again.
In the course of any process, the entropies of the system and the heat reservoir usually change. According to the Second Law of Thermodynamics , the entropy of the closed overall system formed from the system and the heat reservoir increases or at best remains the same:
 ,
or
 .
The “larger” symbol applies to processes that increase the entropy of the overall system and therefore run on their own initiative. The equals sign applies when the overall system has assumed the greatest possible entropy under the given conditions and is in a state of thermal equilibrium (it also applies to reversible processes that do not generate entropy).
The change in entropy of the reservoir is by definition related to the heat flowing into the reservoir and the temperature of the reservoir
 .
Because the reservoir and the system under consideration exchange heat exclusively with one another , and since the system and the reservoir have the same temperature according to the assumption, is . Hence it follows from the above inequality
 .
It was thus possible to formulate the entropy criterion, which actually considers the entropies of the system and the reservoir, exclusively using quantities of the system under consideration, which greatly simplifies the application. Since a distinction is no longer necessary, the indices on the quantities of the system are now omitted and the inequality reads
 (Clausius' inequality).
Furthermore, it is now assumed that the system does not exchange work with its environment. Volumechanging work is suppressed for this purpose by the volume of the system is kept constant ( isochoric process , ). The system is also designed in such a way that it cannot do any other kind of work. Then the internal energy of the system can only change by exchanging heat with the reservoir ( ), and this follows from Clausius' inequality
or moved
 .
On the other hand, the change in the free energy of the system is according to its definition
 ,
which in the present case is due to the assumed constancy of the temperature ( )
simplified. A comparison of the marked equations finally yields the following statement:
 .
The "smaller" symbol applies to processes that run voluntarily. The equals sign applies as soon as the system has reached equilibrium (or in the case of a reversible process).
The maximum principle for the entropy of the overall system thus leads to the fact that the free energy of the system under consideration assumes a minimum on the subset of the states with constant temperature and constant volume. If the system is not yet in equilibrium, it will move (if isothermal and isochoric conditions exist and the system is not doing any mechanical or other work) into states of lower free energy. The equilibrium is reached when the free energy has the smallest possible value under the given conditions.
If one wanted to determine the state of equilibrium directly with the help of the (general and always valid) entropy criterion, the maximum of the total entropy would have to be determined, i.e. the sum of the entropies of the system under investigation and its surroundings. Therefore, not only the change in the system entropy in the event of a change of state would have to be considered, but also the entropy change that the system generates by reacting to the environment there. The free energy criterion is a reformulation of the entropy criterion, which only includes properties of the system under consideration and which automatically takes into account the effect on the environment (under isothermal and isochoric conditions) by the term because is under the given conditions . When using the free energy criterion, the determination of the (isothermal and isochoric) equilibrium state can be limited to the consideration of the system, which makes the investigations noticeably easier.
The atmosphere can often serve as a heat reservoir for a real physical or chemical process. Because of their large volume, their temperature does not change significantly when a system transfers heat to them. The prerequisites for the applicability of the minimum principle of free energy are fulfilled when a system is located in a rigid vessel (so that the volume is kept constant) and is in thermal contact with the atmosphere (so that the temperature is kept constant).
In laboratory practice, however, such systems in rigid containers are less common than systems that are exposed to atmospheric pressure. The atmosphere then serves not only as a heat, but also as a “pressure reservoir”: It keeps temperature and pressure constant. The thermodynamic potential that takes on a minimum under these conditions is the Gibbs energy .
Free energy and maximum work
If a system changes from a state to a state with lower internal energy, the energy difference must be dissipated. This can be done by dissipating heat or performing mechanical (or chemical, electrical, magnetic, ...) work . The total amount of energy to be dissipated is clearly defined by the initial and final value of the internal energy state variable, but it can be divided into heat and work differently depending on the process control (heat and work are not state variables, but process variables ).
As a rule, however, not every arbitrary division is possible. A change in the entropy of the system can also be associated with the change in state. If, for example, the entropy of the final state is smaller than the entropy of the initial state, then not only the energy but also entropy must be removed. Since heat transports entropy, but not work, the dissipated energy must consist of at least as much heat as is required to dissipate the entropy difference. Only the remaining energy is available to be dissipated in the form of usable work. If the temperature of the system remains unchanged during the change of state ( isothermal change of state ), the maximum possible work that can be done by the system during the transition from to is identical to the negative difference in the free energies of the states and .
To prove this, consider a (positive or negative) change in the internal energy of a system, which, according to the conservation of energy required by the First Law of Thermodynamics , is identical to the sum of the (positive or negative) added heat and the (positive or negative) applied to the system Work :
 .
According to the Second Law of Thermodynamics , the Clausius inequality derived in the previous section applies to the relationship between the (positive or negative) change in entropy of the system and the heat supplied or removed
 .
The equals sign applies to a reversible process in which no entropy is generated. If you insert the inequality in the form into the energy conservation equation, you get
and (changed) the work done on the system
 .
Looking at the work done from the point of view of the system, reverse its sign and the work done by the system is
 .
So the work done by the system is less than or at most equal to the expression on the righthand side of the equation. The greatest possible work that the system can do results when the equal sign is valid, i.e. with reversible process management:
 .
The connection between this maximum work and the free energy remains to be shown. The change in free energy
is simplified with isothermal process management ( )
 ,
so that in the isothermal case
 ,
which was to be proved.
The last equation also shows:
 If the process is not associated with a change in entropy ( ), all of the energy to be dissipated can be obtained in the form of work:
 If the process is associated with a decrease in entropy in the system ( ), the term becomes negative and the maximum work that can be gained is less than . Part of the energy to be dissipated must transport the entropy to be dissipated as a heat flow.
 If the process is connected with an increase in entropy in the system ( ) it becomes positive and the maximum work that can be gained becomes greater than the decrease in internal energy . This case is possible if heat is supplied to the system from outside during the process: The supplied entropy or part of it remains in the system and additional energy is available to be dissipated as (entropyfree) work. (The entropy of the system could also increase due to irreversible processes, but the above considerations are limited to the reversible case, which allows the maximum gain in work.)
The type of work done is not specified. For example, it can be lifting work or volume change work . If volume change work is to be excluded because it is at the expense of other forms of work that are to be used, the temperature of the system as well as its volume must be kept constant.
It is not necessary for the entire process to be isothermal. It is sufficient if the start and end temperatures of the process are the same.
The connection with the maximum available work explains the name of this thermodynamic potential: The difference for two states of the same temperature is that part of the energy change that is available for the external work performance in a reversible from following process, i.e. can be used "freely" (see also section → History ).
Free energy as a fundamental function
If one considers a system whose properties are given by the state variables entropy , volume and substance quantities of the chemical components, then the internal energy of the system, expressed as a function of the stated state variables (namely all extensive variables of the system), is
a fundamental function of the system. It describes the system completely; all thermodynamic properties of the system can be derived from it.
However, these variables are often unfavorable for practical work and it would be preferable to have the temperature or the pressure in the list of variables. In contrast to the usual procedure, however, a variable change in the present case must not be made by a simple substitution, otherwise information will be lost. If, for example, the entropy is to be replaced by the temperature , the functions and could be eliminated in order to obtain a function of the form . However, since the temperature is defined thermodynamically as the partial derivative of the internal energy according to the entropy
this formulation would be synonymous with a partial differential equation for , which would only define functions up to indefinite. This would still be a description of the system under consideration, but it would no longer be a complete description and thus no longer a fundamental function.
A Legendre transformation must be carried out to change variables while maintaining the complete information . For example, if you want to go to the list of variables , the transformation is:
 .
The Legendre transform is called free energy. It is in turn a fundamental function if it is given as a function of the variables  these are the natural variables of free energy. It can also be expressed as a function of other variables, but is then no longer a fundamental function.
The origin of the free energy from a Legendre transformation explains the additive term : It compensates for the loss of information that would otherwise be associated with the change in variables.
Fundamental functions, which have the dimension energy, are also called thermodynamic potentials . The free energy is therefore a thermodynamic potential.
Derivatives of free energy
If you start from the internal energy as a function of its natural variables and form its total differential, you get:
 .
The partial derivations occurring here are interpreted in thermodynamics as the definitions of temperature , pressure and chemical potential of the th substance :
so that the differential can also be written as
 .
On the one hand, the total differential of free energy as a function of its natural variables is formal
 .
and on the other hand, using their definition
so that it follows from the comparison of the coefficients in the marked equations
 ,
such as
and
 .
At the same time, the derivation shows how the subtraction of the term changes the list of independent variables from in by removing the term dependent from and adding a dependent term from the total differential .
The second of the marked equations is a "differential fundamental function", namely the differential free energy as a function of its natural variables:
 .
In addition, it may be necessary to introduce further products of work coefficients and work coordinates for the description of the system, e.g. B. for electromagnetic fields see the next section.
Thermodynamics with electromagnetic fields
Taking into account electrical and magnetic fields , the internal energy is given by:
With
  electric field strength
  electrical flux density times volume
  magnetic field strength
  magnetic flux density times volume
The free energy is now defined by:
whereby the electromagnetic fields in the volume under consideration are assumed to be homogeneous. The total differential is:
For constant volume, particle number and electric field this becomes:
Depending on the requirements, the electromagnetic quantities can also be subjected to a further Legendre transformation, i.e.
with the differential
history
The property of this thermodynamic potential to describe the maximum work that a system can perform has already been explained by James Clerk Maxwell in his work Theory of Heat (1871). Josiah Willard Gibbs coined the English term "available energy" in 1873 for the potential.
The German term "free energy" was introduced by Hermann von Helmholtz in the first part of his work on "Thermodynamics of chemical processes" presented to the Royal Prussian Academy of Sciences in 1882 :
“If we consider that chemical forces can produce not only heat but also other forms of energy, the latter even without the need for any change in temperature in the interacting bodies corresponding to the magnitude of the output, such as For example, in the work of galvanic batteries: it does not seem to me to be questionable that in chemical processes, too, a distinction must be made between the parts of their kinship forces capable of free transformation into other forms of work and the part which can only be produced as heat. I shall allow myself to call these two parts of energy briefly the free and the bound energy . We shall see later that the processes that occur automatically from the state of rest and when the uniform temperature of the system is kept constant and proceed without the help of an external worker can only proceed in such a direction that the free energy decreases. "
See also
Web links
 The 2nd law of thermodynamics . Clear explanation of the thermodynamic relationships at Kiel University.
literature
 Callen: Thermodynamics and an Introduction to Thermostatistics . Wiley & Sons. ISBN 9780471862567
 Ulrich Nickel: Textbook of Thermodynamics. A clear introduction. 3rd, revised edition. PhysChem, Erlangen 2019, ISBN 9783937744070 .
 Schwabl: Statistical Mechanics . Jumper. ISBN 9783540310952
 C. Chipot, A. Pohorille: Free energy calculations  theory and applications in chemistry and biology. Springer, Berlin 2007, ISBN 9783540736172
 Huang, Kerson: Statistical Mechanics . Wiley. ISBN 9788126518494
Individual evidence
 ↑ ^{a } ^{b} E.R. Cohen, P. Giacomo: Symbols, Units, Nomenclature and Fundamental Constants in Physics. IUPAP , Document IUPAP 25 (SUNAMCO 871), 1987 Revision, 2010 Reprint, p. 31 ( PDF 438 kB).
 ↑ ^{a } ^{b} Entry on Helmholtz energy . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.H02772 Version: 2.3.3 ..
 ↑ HB Callen: Thermodynamics and an Introduction to Thermostatistics. 2nd ed., John Wiley & Sons, New York 1985, ISBN 0471862568 , pp. 132ff
 ^ PW Atkins: Physical Chemistry. VCH, Weinheim 1990, 2nd repr. 1st edition, ISBN 3527259139 , p. 107
 ^ PW Atkins: Physical Chemistry. VCH, Weinheim 1990, 2nd repr. 1st edition, ISBN 3527259139 , p. 115
 ↑ HB Callen: Thermodynamics and an Introduction to Thermostatistics. 2nd ed., John Wiley & Sons, New York 1985, ISBN 0471862568 , p. 156
 ^ PW Atkins: Physical Chemistry. VCH, Weinheim 1990, 2nd repr. 1st edition, ISBN 3527259139 , p. 114
 ^ PW Atkins: Physical Chemistry. VCH, Weinheim 1990, 2nd repr. 1st edition, ISBN 3527259139 , p. 116
 ^ PW Atkins: Physical Chemistry. VCH, Weinheim 1990, 2nd repr. 1st edition, ISBN 3527259139 , p. 117
 ↑ A. Sommerfeld: Lectures on Theoretical Physics , Volume V: Thermodynamics and Statistics. Harri Deutsch, Thun, Frankfurt / M. 1988, ISBN 3 87144 378 6 , pp. 43f.
 ↑ A. Sommerfeld: Lectures on Theoretical Physics , Volume V: Thermodynamics and Statistics. Harri Deutsch, Thun, Frankfurt / M. 1988, ISBN 3 87144 378 6 , p. 44
 ↑ HB Callen: Thermodynamics and an Introduction to Thermostatistics. 2nd ed., John Wiley & Sons, New York 1985, ISBN 0471862568 , p. 13
 ↑ HB Callen: Thermodynamics and an Introduction to Thermostatistics. 2nd ed., John Wiley & Sons, New York 1985, ISBN 0471862568 , pp. 63f
 ↑ HB Callen: Thermodynamics and an Introduction to Thermostatistics. 2nd ed., John Wiley & Sons, New York 1985, ISBN 0471862568 , pp. 137ff
 ↑ ^{a } ^{b} H.B. Callen: Thermodynamics and an Introduction to Thermostatistics. 2nd ed., John Wiley & Sons, New York 1985, ISBN 0471862568 , p. 146
 ↑ ^{a } ^{b} J.K. Fink: Physical Chemistry in Depth. Springer, Berlin Heidelberg 2009, ISBN 9783642010132 , p. 84
 ↑ HB Callen: Thermodynamics and an Introduction to Thermostatistics. 2nd ed., John Wiley & Sons, New York 1985, ISBN 0471862568 , p. 35
 ↑ ^{a } ^{b} H.J. Lange: The physics of weather and climate. Dietrich Reimer Verlag, Berlin 2002, ISBN 3496027479 , p. 28f
 ↑ E. Keszei: Chemical Thermodynamics. Springer, Berlin / Heidelberg 2012, ISBN 9783642198632 , pp. 13, 48
 ^ Thermodynamics of the mixed phases: With an introduction to the basics of thermodynamics, Rudolf Haase, ISBN 3662225468 , p. 55, Google Books
 ^ P. Duhem: Thermodynamics and Chemistry. John Wiley & Sons, New York 1903, p. 93f ( archive.org )
 ↑ H. Helmholtz: The thermodynamics of chemical processes. Meeting reports of the Royal Prussian Academy of Sciences in Berlin, 1882  first half volume (Berlin: Verl. D. Kgl. Akad. D. Wiss., 1882), p. 22 ( digitized version ), quotation: p. 23 ( digitized version )