Thermodynamic potential

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.

One main thermodynamic potential that 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 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 $U$ . 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.

( $U$ ) is the capacity to do work plus the capacity to release heat.

Internal energy

^[2] ( $G$ ) is the capacity to do non-mechanical work.

Gibbs energy

( $H$ ) is the capacity to do non-mechanical work plus the capacity to release heat.

Enthalpy

^[1] ( $F$ ) is the capacity to do mechanical work plus non-mechanical work.

Helmholtz energy

Five common thermodynamic potentials are:^[3]

where $T$ = temperature, $S$ = entropy, $p$ = pressure, $V$ = volume. $N i$ is the number of particles of type $i$ in the system and $μ i$ is the chemical potential for an $i$ -type particle. The set of all $N i$ 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 $F$ , but the use of $A$ is preferred by IUPAC,^[4] ISO and IEC.^[5]

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:

From these meanings (which actually apply in specific conditions, e.g. constant pressure, temperature, etc.), for positive changes (e.g., $Δ U > 0$ ), we can say that $Δ U$ is the energy added to the system, $Δ F$ is the total work done on it, $Δ G$ is the non-mechanical work done on it, and $Δ H$ 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.^[6]

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)^[7]

Stability Conditions[edit]

As the internal energy is a convex function of entropy and volume, the stability condition requires that the second derivative of internal energy with entropy or volume to be positive. It is commonly expressed as $d^{2}U>0$ . Since the maximum principle of entropy is equivalent to minimum principle of internal energy, the combined criteria for stability or thermodynamic equilibrium is expressed as $d^{2}U>0$ and $dU=0$ for parameters, entropy and volume. This is analogous to $d^{2}S<0$ and $dS=0$ condition for entropy at equilibrium.^[19] The same concept can be applied to the various thermodynamic potentials by identifying if they are convex or concave of respective their variables.

${\biggl (}{\partial ^{2}F \over \partial T^{2}}{\biggr )}_{V,N}\leq 0$ and ${\biggl (}{\partial ^{2}F \over \partial V^{2}}{\biggr )}_{T,N}\geq 0$

Where Helmholtz energy is a concave function of temperature and convex function of volume.

${\biggl (}{\partial ^{2}H \over \partial P^{2}}{\biggr )}_{S,N}\leq 0$ and ${\biggl (}{\partial ^{2}H \over \partial S^{2}}{\biggr )}_{P,N}\geq 0$

Where enthalpy is a concave function of pressure and convex function of entropy.

${\biggl (}{\partial ^{2}G \over \partial T^{2}}{\biggr )}_{P,N}\leq 0$ and ${\biggl (}{\partial ^{2}G \over \partial P^{2}}{\biggr )}_{T,N}\leq 0$

Where Gibbs potential is a concave function of both pressure and temperature.

In general the thermodynamic potentials (the internal energy and its Legendre transforms), are convex functions of their extrinsic variables and concave functions of intrinsic variables. The stability conditions impose that isothermal compressibility is positive and that for non-negative temperature, $C_{P}>C_{V}$ .^[20]

Coomber's relationship

Alberty, R. A. (2001). (PDF). Pure Appl. Chem. 73 (8): 1349–1380. doi:10.1351/pac200173081349.

"Use of Legendre transforms in chemical thermodynamics"

(1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 978-0-471-86256-7.

Callen, Herbert B.

Moran, Michael J.; Shapiro, Howard N. (1996). (3rd ed.). New York; Toronto: J. Wiley & Sons. ISBN 978-0-471-07681-0.

Fundamentals of Engineering Thermodynamics

McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, 0-07-051400-3

ISBN

Thermodynamics, From Concepts to Applications (2nd Edition), A. Shavit, C. Gutfinger, CRC Press (Taylor and Francis Group, USA), 2009, 9781420073683

ISBN

Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, 0-356-03736-3

ISBN

Elements of Statistical Thermodynamics (2nd Edition), L.K. Nash, Principles of Chemistry, Addison-Wesley, 1974, 0-201-05229-6

ISBN

Statistical Physics (2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008, 9780471566588

ISBN

– Georgia State University

Thermodynamic potential

Internal energy

Gibbs energy

Enthalpy

Helmholtz energy

Stability Conditions[edit]

Coomber's relationship

"Use of Legendre transforms in chemical thermodynamics"

Callen, Herbert B.

Fundamentals of Engineering Thermodynamics

ISBN

ISBN

ISBN

ISBN

ISBN

Thermodynamic Potentials

Chemical Potential Energy: The 'Characteristic' vs the Concentration-Dependent Kind