Thermodynamic potential explained

A thermodynamic potential (or more accurately, a thermodynamic potential energy)^{[1] [2]} is a scalar quantity used to represent the thermodynamic state of a system. Just as in mechanics, where potential energy is defined as capacity to do work, similarly different potentials have different meanings. The concept of thermodynamic potentials was introduced by Pierre Duhem in 1886. Josiah Willard Gibbs in his papers used the term fundamental functions. While thermodynamic potentials cannot be measured directly, they can be predicted using computational chemistry.^[3]

One main thermodynamic potential that has a physical interpretation is the internal energy . It is the energy of configuration of a given system of conservative forces (that is why it is called 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 . In other words, each thermodynamic potential is equivalent to other thermodynamic potentials; each potential is a different expression of the others.

In thermodynamics, external forces, such as gravity, are counted as contributing to total energy rather than to thermodynamic potentials. For example, the working fluid in a steam engine sitting on top of Mount Everest has higher total energy due to gravity than it has at the bottom of the Mariana Trench, but the same thermodynamic potentials. This is because the gravitational potential energy belongs to the total energy rather than to thermodynamic potentials such as internal energy.

Description and interpretation

Five common thermodynamic potentials are:^[4]

where = temperature, = entropy, = pressure, = volume. is the number of particles of type in the system and is the chemical potential for an -type particle. The set of all are also included as natural variables but may be ignored when no chemical reactions are occurring which cause them to change. The Helmholtz free energy is in ISO/IEC standard called Helmholtz energy^[1] or Helmholtz function. It is often denoted by the symbol, but the use of is preferred by IUPAC,^[5] ISO and IEC.^[6]

These five common potentials are all potential energies, but there are also entropy potentials. The thermodynamic square can be used as a tool to recall and derive some of the potentials.

Just as in mechanics, where potential energy is defined as capacity to do work, similarly different potentials have different meanings like the below:

• Internal energy is the capacity to do work plus the capacity to release heat.
• Gibbs energy^[2] is the capacity to do non-mechanical work.
• Enthalpy is the capacity to do non-mechanical work plus the capacity to release heat.
• Helmholtz energy^[1] is the capacity to do mechanical work plus non-mechanical work.

From these meanings (which actually apply in specific conditions, e.g. constant pressure, temperature, etc.), for positive changes (e.g.,), we can say that is the energy added to the system, is the total work done on it, is the non-mechanical work done on it, and is the sum of non-mechanical work done on the system and the heat given to it.

Note that the sum of internal energy is conserved, but the sum of Gibbs energy, or Helmholtz energy, are not conserved, despite being named "energy". They can be better interpreted as the potential to perform "useful work", and the potential can be wasted.^[7]

Thermodynamic potentials are very useful when calculating the equilibrium results of a chemical reaction, or when measuring the properties of materials in a chemical reaction. The chemical reactions usually take place under some constraints such as constant pressure and temperature, or constant entropy and volume, and when this is true, there is a corresponding thermodynamic potential that comes into play. Just as in mechanics, the system will tend towards a lower value of a potential and at equilibrium, under these constraints, the potential will take the unchanging minimum value. The thermodynamic potentials can also be used to estimate the total amount of energy available from a thermodynamic system under the appropriate constraint.

In particular: (see principle of minimum energy for a derivation)^[8]

• When the entropy and "external parameters" (e.g. volume) of a closed system are held constant, the internal energy decreases and reaches a minimum value at equilibrium. This follows from the first and second laws of thermodynamics and is called the principle of minimum energy. The following three statements are directly derivable from this principle.
• When the temperature and external parameters of a closed system are held constant, the Helmholtz free energy decreases and reaches a minimum value at equilibrium.
• When the pressure and external parameters of a closed system are held constant, the enthalpy decreases and reaches a minimum value at equilibrium.
• When the temperature, pressure and external parameters of a closed system are held constant, the Gibbs free energy decreases and reaches a minimum value at equilibrium.

Natural variables

For each thermodynamic potential, there are thermodynamic variables that need to be held constant to specify the potential value at a thermodynamical equilibrium state, such as independent variables for a mathematical function. These variables are termed the natural variables of that potential.^[9] The natural variables are important not only to specify the potential value at the equilibrium, but also because if a thermodynamic potential can be determined as a function of its natural variables, all of the thermodynamic properties of the system can be found by taking partial derivatives of that potential with respect to its natural variables and this is true for no other combination of variables. If a thermodynamic potential is not given as a function of its natural variables, it will not, in general, yield all of the thermodynamic properties of the system.

The set of natural variables for each of the above four thermodynamic potentials is formed from a combination of the,,, variables, excluding any pairs of conjugate variables; there is no natural variable set for a potential including the - or - variables together as conjugate variables for energy. An exception for this rule is the − conjugate pairs as there is no reason to ignore these in the thermodynamic potentials, and in fact we may additionally define the four potentials for each species.^[10] Using IUPAC notation in which the brackets contain the natural variables (other than the main four), we have:

Thermodynamic potential name	Formula	Natural variables
Internal energy	U[\mu _{j]} = U - \mu _{jN} _{j}	S,V,\{N _{i\ne} \},\mu _{j}
	F[\mu _{j]} = U - TS - \mu _{jN} _{j}	T,V,\{N _{i\ne} \},\mu _{j}
Enthalpy	H[\mu _{j]} = U + pV - \mu _{jN} _{j}	S,p,\{N _{i\ne} \},\mu _{j}
Gibbs energy	G[\mu _{j]} = U + pV - TS - \mu _{jN} _{j}	T,p,\{N _{i\ne} \},\mu _{j}

If there is only one species, then we are done. But, if there are, say, two species, then there will be additional potentials such as

U[\mu _{1,\mu} _{2]} = U - \mu _{1} N _{1-\mu} _{2} N _{2} and so on. If there are dimensions to the thermodynamic space, then there are unique thermodynamic potentials. For the most simple case, a single phase ideal gas, there will be three dimensions, yielding eight thermodynamic potentials.

The fundamental equations

See main article: Fundamental thermodynamic relation. The definitions of the thermodynamic potentials may be differentiated and, along with the first and second laws of thermodynamics, a set of differential equations known as the fundamental equations follow.^[11] (Actually they are all expressions of the same fundamental thermodynamic relation, but are expressed in different variables.) By the first law of thermodynamics, any differential change in the internal energy of a system can be written as the sum of heat flowing into the system subtracted by the work done by the system on the environment, along with any change due to the addition of new particles to the system:

d U = \delta Q - \delta W + \sum _{i} \mu _{idN} _{i}

where is the infinitesimal heat flow into the system, and is the infinitesimal work done by the system, is the chemical potential of particle type and is the number of the type particles. (Neither nor are exact differentials, i.e., they are thermodynamic process path-dependent. Small changes in these variables are, therefore, represented with rather than .)

By the second law of thermodynamics, we can express the internal energy change in terms of state functions and their differentials. In case of reversible changes we have:

\delta Q = T d S \delta W = p d V

where

is temperature,

is entropy,

is pressure,and is volume, and the equality holds for reversible processes.

This leads to the standard differential form of the internal energy in case of a quasistatic reversible change:

d U = T d S - p d V + \sum _{i} \mu _{idN} _{i}

Since, and are thermodynamic functions of state (also called state functions), the above relation also holds for arbitrary non-reversible changes. If the system has more external variables than just the volume that can change, the fundamental thermodynamic relation generalizes to:

dU = T d S - p d V + \sum _{j} \mu _{jdN} _{j} + \sum _{i} X _{i} d x _{i}

Here the are the generalized forces corresponding to the external variables .^[12]

Applying Legendre transforms repeatedly, the following differential relations hold for the four potentials (fundamental thermodynamic equations or fundamental thermodynamic relation):

d U	\	=		T d S	-	p d V	+ \sum _{i} \mu _{i} d N _{i}
d F	\	=	-	S d T	-	p d V	+ \sum _{i} \mu _{i} d N _{i}
d H	\	=		T d S	+	V d p	+ \sum _{i} \mu _{i} d N _{i}
d G	\	=	-	S d T	+	V d p	+ \sum _{i} \mu _{i} d N _{i}

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 differences between the four thermodynamic potentials can be summarized as follows:

d (pV) = d H - d U = d G - d F d (TS) = d U - d F = d H - d G

The equations of state

We can use the above equations to derive some differential definitions of some thermodynamic parameters. If we define to stand for any of the thermodynamic potentials, then the above equations are of the form:

d \Phi = \sum _{i} x _{idy} _{i}

where and are conjugate pairs, and the are the natural variables of the potential . From the chain rule it follows that:

x _{j = \left( (\partial \Phi)/(\partial y _{j})} \right) _{\{y _{i\ne} \} }

where

Notes and References

1. ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.4 Helmholtz energy, Helmholtz function
2. ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.5, Gibbs energy, Gibbs function
3. Nitzke . Isabel . Stephan . Simon . Vrabec . Jadran . 2024-06-03 . Topology of thermodynamic potentials using physical models: Helmholtz, Gibbs, Grand, and Null . The Journal of Chemical Physics . 160 . 21 . 10.1063/5.0207592 . 0021-9606.
4. Alberty (2001) p. 1353
5. Alberty (2001) p. 1376
6. ISO/IEC 80000-5:2007, item 5-20.4
7. Tykodi . R. J. . 1995-02-01 . Spontaneity, Accessibility, Irreversibility, "Useful Work": The Availability Function, the Helmholtz Function, and the Gibbs Function . Journal of Chemical Education . 72 . 2 . 103 . 10.1021/ed072p103 . 1995JChEd..72..103T . 0021-9584.
8. Callen (1985) p. 153
9. Alberty (2001) p. 1352
10. Alberty (2001) p. 1355
11. Alberty (2001) p. 1354
12. For example, ionic species N_j (measured in moles) held at a certain potential V_j will include the term \sum _{j} V _{j} d q _{j} = F\sum _{j} V _{j} z _{j} d N _{j} where F is the Faraday constant and z_j is the multiple of the elementary charge of the ion.