Chemistry · Lesson
Thermodynamic potentials and their natural variables
Choose the thermodynamic potential that matches the imposed constraints. Read its differential as a relation between equilibrium states, with an explicit total, molar or particle basis.
Back to the Chemistry mapBy the end, you can
- Connect U, H, A and G through their definitions and natural variables.
- Choose an equilibrium criterion under stated reservoir and work conditions.
- Interpret a Maxwell identity with its fixed variables and units.
Start with a bulk equilibrium description
For a simple compressible bulk system, internal energy U is a function of entropy S, volume V and the amounts nᵢ of its species. Here nᵢ is measured in moles. On a smooth equilibrium-state branch, dU = T dS − p dV + Σᵢ μᵢ dnᵢ, where μᵢ is a molar chemical potential in J mol⁻¹. Additional surface, electrical or other work coordinates require additional terms.
The symbol d describes a change in a state function. This relation does not assert that heat equals T dS along every irreversible process. The familiar equality δq_rev = T dS refers to reversible heat transfer. A rapidly nonuniform body may also require spatial fields rather than one bulk temperature.
Replace a variable with the quantity controlled by a reservoir
Define enthalpy H = U + pV, Helmholtz energy A = U − TS and Gibbs energy G = U + pV − TS. The products pV and TS have energy units. Some texts call Helmholtz energy F; this chapter uses A to avoid confusing it with the Faraday constant.
These Legendre transforms exchange natural variables. H uses S,p,{nᵢ}; A uses T,V,{nᵢ}; G uses T,p,{nᵢ}. For example, dG = −S dT + V dp + Σᵢ μᵢ dnᵢ. At fixed composition, a pressure derivative of G gives V and a temperature derivative gives −S. Molar potentials divide the total by an explicitly chosen amount; they are not interchangeable with totals.
| Potential | Definition | Natural variables | Equilibrium criterion |
|---|---|---|---|
| U | Internal energy | S, V, {nᵢ} | At fixed S,V and conserved amounts: appropriate allowed energy minimum |
| H | U + pV | S, p, {nᵢ} | At fixed S,p and conserved amounts: appropriate allowed enthalpy minimum |
| A | U − TS | T, V, {nᵢ} | Fixed T,V: Helmholtz minimum |
| G | U + pV − TS | T, p, {nᵢ} | Fixed T,p: Gibbs minimum |
Bulk simple-compressible equilibrium relations for total U, H, A and G. Species amounts nᵢ are in moles, and μᵢ is a molar chemical potential. Species amounts are included in each natural-variable list. The minimum criteria require the stated constraints and work conditions; an isolated system instead uses the entropy maximum.
An equilibrium criterion belongs to its constraints
For an isolated system with fixed U,V and conserved constituent amounts, equilibrium maximizes entropy over the allowed internal variations. For a system coupled to a temperature reservoir at fixed T,V, the appropriate equilibrium minimum is A. At fixed T,p, it is G. These energy minimum criteria assume the corresponding constraints and permitted work, with no separately extracted non-volume work omitted from the balance.
Stable equilibrium requires comparison with allowed competing states; merely finding a stationary point is insufficient. During a reaction, individual species amounts can change while conserved elemental totals remain fixed. A decrease in the appropriate potential gives a thermodynamic direction, not a reaction speed or a guarantee that a nucleation barrier is crossed.
A Maxwell relation connects two responses
For smooth A(T,V,{nᵢ}), the fixed-composition differential is dA = −S dT − p dV. Equality of mixed second derivatives gives (∂S/∂V)T,{nᵢ} = (∂p/∂T)V,{nᵢ}. The notation after the derivative tells us what is held fixed.
For an ideal gas with fixed amount n, p = nRT/V, where R is the molar gas constant. The right side is therefore nR/V. The entropy derivative has the same units: J K⁻¹ m⁻³ equals Pa K⁻¹. Do not apply a smooth-branch derivative across a discontinuous phase change as if the branch remained differentiable.
Worked examples
Convert a consistent set of total quantities
For an invented equilibrium state, take U = 10.0 kJ, p = 1.00 × 10⁵ Pa, V = 2.00 × 10⁻³ m³, T = 3.00 × 10² K and S = 20.0 J K⁻¹. Calculate H, A and G.
- Convert U to 10000 J. The product pV is 200 J.
- The product TS is 6000 J.
- H = 10000 + 200 = 10200 J; A = 10000 − 6000 = 4000 J.
- G = H − TS = 4200 J. All terms here refer to the same total system.
H = 10.2 kJ, A = 4.0 kJ and G = 4.2 kJ for these supplied rounded inputs. These definitions alone do not determine a reaction or its rate.
Check the units on both sides of a Maxwell identity
For the ideal-gas model, use n = 1.00 mol and V = 0.0250 m³. What is (∂p/∂T)V,n and the matching (∂S/∂V)T,n?
- At fixed V,n, differentiate p = nRT/V with respect to T.
- Use R = 8.314462618 J mol⁻¹ K⁻¹.
- nR/V = 332.5785 J m⁻³ K⁻¹.
- One joule per cubic metre is one pascal.
Both derivatives are approximately 333 Pa K⁻¹, equivalently 333 J m⁻³ K⁻¹, within the stated ideal-gas model.
Try it yourself
A temperature reservoir holds T fixed and a rigid wall holds V fixed. Which energy potential supplies the equilibrium minimum under the stated ordinary work restrictions?
Show the answer and reasoning
Helmholtz energy A.
Its natural variables include T and V.
At fixed composition, which derivative of G gives volume?
Show the answer and reasoning
(∂G/∂p)T,{nᵢ} = V.
Read the coefficient of dp in dG.
What is the equilibrium criterion for an isolated system at fixed U,V and conserved amounts?
Show the answer and reasoning
Maximum entropy over the permitted internal variations.
An arbitrary fixed-T,p Gibbs criterion does not describe those constraints.
Does dU = T dS − p dV by itself imply δq = T dS for every irreversible path?
Show the answer and reasoning
No.
The state relation and reversible heat-transfer equality are different statements.
What to carry forward
- Name the constraints and the total or molar basis.
- Natural variables follow from the potential’s differential.
- Use smooth branches when taking mixed derivatives.
- Thermodynamic stability does not specify a relaxation rate.
References
Teaching resources and reference material supporting this lesson’s explanations and examples.