Thermodynamic Potentials in Materials Science: Internal Energy, Enthalpy, Helmholtz and Gibbs Free Energy
Thermodynamic Potentials in Materials Science: Internal Energy, Enthalpy, Helmholtz and Gibbs Free Energy
Thermodynamic potentials provide a compact way to determine which processes are energetically favourable under specified experimental conditions. In materials science, they are central to understanding phase stability, chemical reactions, defect formation, oxygen nonstoichiometry, hydration of proton-conducting oxides, and the temperature dependence of functional properties.
The key is not simply to memorize several energy functions. Each potential is constructed for a particular set of controlled variables. Choosing the correct one turns an equilibrium problem into a clear minimization criterion.
1. The thermodynamic state and the first law
For a simple, closed, compressible system containing a fixed amount of material, the first law can be written as
[ dU = q + w ]
where (U) is internal energy, (q) is heat supplied to the system, and (w) is work supplied to the system. With only pressure–volume work,
[ dU = q - P,dV ]
For a reversible change, (q_{}=T,dS), giving
[ dU = T,dS-P,dV ]
For a multicomponent system in which the amounts of the chemical species can change, the differential becomes
[ dU=T,dS-P,dV+_i _i,dn_i ]
Here, (S) is entropy, (V) is volume, (n_i) is the amount of component (i), and (_i) is its chemical potential. This expression assumes equilibrium thermodynamics and pressure–volume work only; additional work terms are needed for magnetic, electrical, elastic, or interfacial contributions when they are relevant.
The equation identifies the natural independent variables of internal energy:
[ U=U(S,V,n_1,n_2,) ]
The corresponding derivatives are
[ T=(){V,n_i}, -P=(){S,n_i}, i=(){S,V,n_{ji}} ]
These derivatives connect measurable or physically interpretable quantities to the energy landscape of a material.
2. Why do we need different thermodynamic potentials?
In a laboratory experiment, entropy and volume are rarely the variables that are directly controlled. More commonly, a sample is held at approximately constant temperature and pressure, or at constant temperature and volume. A useful thermodynamic function should therefore use the variables that match the experimental boundary conditions.
Thermodynamic potentials are obtained from internal energy by Legendre transforms. A Legendre transform replaces one natural variable with its conjugate variable. For example, replacing entropy (S) with temperature (T) introduces the term (-TS).
The four principal potentials are:
| Potential | Definition | Natural variables |
|---|---|---|
| Internal energy, (U) | (U) | (S,V,n_i) |
| Enthalpy, (H) | (U+PV) | (S,P,n_i) |
| Helmholtz free energy, (F) or (A) | (U-TS) | (T,V,n_i) |
| Gibbs free energy, (G) | (H-TS=U+PV-TS) | (T,P,n_i) |
The symbols (F) and (A) are both used for Helmholtz free energy. The notation (F) is common in physics, while (A) is common in some chemistry and engineering texts.
3. Internal energy: (U)
Internal energy is the total microscopic energy stored in a system, including contributions from atomic and molecular motion, bonding, electronic states, and other microscopic degrees of freedom.
Its differential for the simple system introduced above is
[ dU=T,dS-P,dV+_i_i,dn_i ]
For an isolated system, total energy is conserved. The second law states that spontaneous change proceeds toward a state of maximum entropy, subject to the relevant conservation constraints. Thus, for an isolated system at fixed (U), (V), and composition, equilibrium corresponds to maximum entropy.
In materials research, internal energy is useful in atomistic calculations and energy comparisons. However, comparing internal energies alone does not generally establish phase stability at finite temperature because entropy and, where relevant, pressure–volume contributions must also be considered.
4. Enthalpy: (H)
Enthalpy is defined as
[ H=U+PV ]
Differentiating and substituting the first-law expression gives
[ dH=T,dS+V,dP+_i_i,dn_i ]
For fixed composition, the natural variables are entropy and pressure:
[ H=H(S,P) ]
Enthalpy is particularly useful for processes conducted at constant pressure. At constant pressure, with only pressure–volume work, the heat exchanged in a process equals the enthalpy change:
[ q_P=H ]
This is why reaction enthalpies, hydration enthalpies, and heat effects are commonly reported as (H).
Important distinction: an exothermic reaction has (H<0), but this does not by itself prove that the reaction is spontaneous. The entropy contribution must also be included.
5. Helmholtz free energy: (F) or (A)
The Helmholtz free energy is defined as
[ F=U-TS ]
Its differential is
[ dF=-S,dT-P,dV+_i_i,dn_i ]
For fixed composition, its natural variables are temperature and volume:
[ F=F(T,V) ]
At constant temperature and volume, equilibrium minimizes the Helmholtz free energy. A spontaneous change under these constraints proceeds in the direction of decreasing (F), until equilibrium is reached.
Helmholtz free energy is useful for systems in which volume is constrained, such as certain computational simulations, confined systems, and calculations performed in a fixed simulation cell.
The entropy term matters because the most stable state is not necessarily the state with the lowest internal energy. A state with higher energy may be favoured at elevated temperature if it has sufficiently greater entropy.
6. Gibbs free energy: (G)
For many materials experiments, temperature and pressure are controlled. The appropriate potential is Gibbs free energy:
[ G=H-TS=U+PV-TS ]
Its differential is
[ dG=-S,dT+V,dP+_i_i,dn_i ]
At constant temperature and pressure, and for a system allowed to change internally while satisfying the relevant conservation constraints, equilibrium corresponds to a minimum in total Gibbs free energy.
For a process at constant (T) and (P):
- (G<0): the forward process is thermodynamically favourable.
- (G>0): the reverse process is thermodynamically favourable.
- (G=0): the system is at equilibrium with respect to that process.
These statements describe thermodynamic direction, not reaction rate. A process with negative (G) may still be extremely slow because of a large kinetic barrier.
The Gibbs free-energy equation
For a process at a specified temperature,
[ G=H-TS ]
where (H) is the enthalpy change and (S) is the entropy change. If these quantities vary substantially with temperature, their temperature dependence must be included rather than treating them as constants.
The equation illustrates four common cases:
| (H) | (S) | Temperature tendency |
|---|---|---|
| Negative | Positive | Favourable at all temperatures, under the stated approximation |
| Positive | Negative | Unfavourable at all temperatures, under the stated approximation |
| Negative | Negative | More favourable at lower temperature |
| Positive | Positive | More favourable at higher temperature |
A short numerical example
Consider a hypothetical process with
[ H= -40 , S= -80 ]
First, convert entropy to consistent energy units:
[ S=-0.080 ]
At (T=300 ),
[ G=-40-(300)(-0.080) =-16 ]
The process is thermodynamically favourable at 300 K. At (T=600 ),
[ G=-40-(600)(-0.080) =+8 ]
Under the constant-(H), constant-(S) approximation, the process becomes unfavourable at 600 K. The crossover temperature is
[ T_{}= =500 ]
This example shows why temperature can change phase stability or reaction direction even when the chemical composition is unchanged.
7. Chemical potential: the link to composition and defects
For a multicomponent system, the chemical potential of species (i) is defined by
[ i= (){T,P,n_{ji}} ]
It is the change in Gibbs free energy associated with adding an infinitesimal amount of that species while holding temperature, pressure, and the other component amounts fixed.
Chemical potential is central to diffusion, phase equilibrium, defect chemistry, and electrochemical reactions. A species tends to redistribute in a way that reduces the appropriate total free energy. At equilibrium between two phases () and (), the chemical potential of each species that can transfer between them must be equal:
[ _i=_i ]
For a gas, chemical potential depends on fugacity; for an ideal gas, fugacity can be approximated by partial pressure. For a component in a condensed phase, it is commonly expressed in terms of activity:
[ _i=_i^+RTa_i ]
where (_i^) is the chemical potential in the chosen standard state, (R) is the gas constant, and (a_i) is dimensionless activity.
For oxygen in an ideal-gas approximation,
[ {}(T,p{}) = _{}^(T) +RT() ]
The reference pressure (p^) makes the logarithm dimensionless. This relation is essential when interpreting how oxygen partial pressure changes the equilibrium concentrations of oxygen vacancies and electronic carriers in oxides.
8. Why thermodynamic potentials matter for proton-conducting oxides
In a proton-conducting oxide, hydration is commonly represented using Kröger–Vink notation as
[ ]
The equilibrium is governed by the Gibbs free-energy change:
[ G_{} = H_{} -TS_{} ]
The equilibrium position depends on temperature, water-vapour activity, and the chemical potentials of the participating species. The enthalpy and entropy contributions help explain why proton incorporation is often favoured at lower temperatures, while dehydration becomes increasingly important as temperature rises. The actual behaviour depends on composition, defect interactions, and gas conditions.
This framework connects thermodynamics to measurable quantities such as water uptake, proton concentration, and conductivity. It also clarifies an important distinction: a thermodynamically favourable hydration reaction does not guarantee rapid hydration. Surface exchange, bulk diffusion, microstructure, and reaction kinetics can limit how quickly equilibrium is approached.
9. Why thermodynamic potentials matter for thermoelectric materials
Thermoelectric behaviour involves coupled transport of heat and charge. The Seebeck coefficient can be interpreted in terms of the response of the carrier electrochemical potential to a temperature gradient, but it should not be identified universally with a single entropy-per-carrier value without specifying the transport regime and model.
Thermodynamic derivatives also underpin relations between thermoelectric coefficients. For example, the Kelvin relation connects the Peltier coefficient () and Seebeck coefficient (S):
[ =ST ]
with consistent sign conventions. The temperature dependence of thermodynamic and transport properties is therefore important when analysing thermoelectric measurements.
The Gibbs free energy is also relevant to the chemical and phase stability of thermoelectric compounds at operating temperature. Oxidation, decomposition, volatilization, and secondary-phase formation can change composition and degrade performance even if the initial transport properties are promising.
For a broader treatment of the transport properties themselves, see Thermoelectric Performance: Why the Three Transport Properties Must Be Read Together.
10. Choosing the correct potential in practice
The most reliable approach is to identify the variables controlled by the experiment or calculation before selecting a potential.
| Experimental or theoretical condition | Useful potential | Equilibrium criterion |
|---|---|---|
| Isolated system, fixed (U,V,n_i) | Entropy (S) | Maximum entropy |
| Fixed (S,V,n_i) | Internal energy (U) | Minimum (U) |
| Fixed (S,P,n_i) | Enthalpy (H) | Minimum (H) |
| Fixed (T,V,n_i) | Helmholtz (F) | Minimum (F) |
| Fixed (T,P,n_i) | Gibbs (G) | Minimum (G) |
For functional oxides, constant-temperature and constant-pressure reasoning is often a useful starting point, but the gas atmosphere introduces chemical reservoirs. When oxygen or water exchanges with the sample, the appropriate thermodynamic description must account for the chemical potentials of those reservoirs. In open systems, a semi-grand or grand-potential formulation may be more convenient than minimizing the Gibbs energy at fixed composition.
11. Common mistakes to avoid
- Using (H) as a synonym for spontaneity. Spontaneity at constant temperature and pressure is determined by (G), not by enthalpy alone.
- Ignoring units. Entropy is often reported in J mol(^{-1}) K(^{-1}), while enthalpy is reported in kJ mol(^{-1}). Convert them before evaluating (TS).
- Confusing equilibrium with kinetics. A negative Gibbs free-energy change does not specify how quickly a reaction proceeds.
- Using the wrong controlled variables. Gibbs free energy is not the universal potential for every boundary condition; Helmholtz free energy is appropriate at fixed (T,V).
- Ignoring composition and chemical reservoirs. Defect concentrations in oxides depend on oxygen and water chemical potentials, not only on temperature.
- Treating standard-state quantities as actual-condition quantities. Standard Gibbs energies must be corrected for activities, pressures, and composition when evaluating a real system.
12. Summary
Thermodynamic potentials are not competing definitions of the same quantity. They are related functions designed for different constraints:
- (U) describes energy in terms of entropy and volume.
- (H=U+PV) is useful for constant-pressure heat processes.
- (F=U-TS) is the natural potential at constant temperature and volume.
- (G=H-TS) is the natural potential at constant temperature and pressure.
For materials science, their practical value lies in connecting energy, entropy, temperature, pressure, and chemical composition. This makes them indispensable for understanding phase stability, defect equilibria, oxygen nonstoichiometry, proton hydration, and the chemical stability of thermoelectric materials.
Further reading
- H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley.
- D. R. Gaskell and D. E. Laughlin, Introduction to the Thermodynamics of Materials, 6th ed., CRC Press.
- P. Atkins, J. de Paula, and J. Keeler, Atkins’ Physical Chemistry, Oxford University Press.
- A. R. West, Solid State Chemistry and its Applications, 2nd ed., Wiley.