Thermodynamic potential
A thermodynamic potential is a scalar quantity used to represent the thermodynamic state of a system. Similarly to the potential energy of the conservative gravitational field, defined as capacity to do work, various thermodynamic potentials have similar meanings. The author of the term of thermodynamic potentials is Pierre Duhem in an 1886 work. Josiah Willard Gibbs in his papers used the term fundamental functions. Effects of changes in thermodynamic potentials can sometimes be measured directly, while their absolute magnitudes can only be assessed using computational chemistry or similar methods.
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 and only has meaning with respect to a defined set of references. 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: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 or Helmholtz function. It is often denoted by the symbol, but the use of is preferred by IUPAC, ISO and IEC.
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 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 is the capacity to do mechanical work plus non-mechanical work.
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.
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:
- When the entropy and "external parameters" 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
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. Using IUPAC notation in which the brackets contain the natural variables, we have:
| Thermodynamic potential name | Formula | Natural variables |
| Internal energy | ||
| Helmholtz free energy | ||
| Enthalpy | ||
| Gibbs energy |
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 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.
Fundamental equations
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. 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: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.
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:
where
- is temperature,
- is entropy,
- is pressure, and
- is volume,
This leads to the standard differential form of the internal energy in case of a quasistatic reversible change:
Since, and are thermodynamic functions of state, 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:
Here the are the generalized forces corresponding to the external variables.
Applying Legendre transforms repeatedly, the following differential relations hold for the four potentials :
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:
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:where and are conjugate pairs, and the are the natural variables of the potential. From the chain rule it follows that:
where is the set of all natural variables of except that are held as constants. 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. If we restrict ourselves to the potentials , , and , then we have the following equations of state :
where, in the last equation, is any of the thermodynamic potentials, and are the set of natural variables for that potential, excluding. If we use all thermodynamic potentials, then we will have more equations of state such as
and so on. In all, if the thermodynamic space is dimensions, then there will be equations for each potential, resulting in a total of equations of state because thermodynamic potentials exist. If the 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 because the fundamental equations for any other potential can be found via the Legendre transforms and the corresponding equations of state for each potential as partial derivatives of the potential can also be found.