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ATP synthase couples downhill proton motion across a cell membrane to synthesis of ATP from ADP and inorganic phosphate. Its membrane-embedded $F_o$ sector contains an $a$ subunit with offset proton-access half-channels and a rotating $c$-subunit ring. Reversible protonation of sites on the ring permits rotation relative to the stationary $a$ subunit. The central rotor transmits this motion through the $\gamma$ shaft to the soluble $F_1$ head. A peripheral stator holds the head relative to the membrane sector. The common bacterial schematic has an ${\alpha }_3{\beta }_3$ head, catalytic sites mainly on the three $\beta$ subunits, and a central $\gamma /ϵ$ rotor; accessory stator components depend on the organism.

The binding-change mechanism cycles the catalytic sites between open, loose-binding and tight-binding conformations. Rotor orientation changes their affinities for ADP, phosphate and ATP; the sites operate out of phase. Tight binding favors formation of bound ATP, and mechanical rotation changes the conformation to permit product release. In the ideal tightly coupled cycle, one revolution gives three catalytic turnovers:The proton count is not universally fixed: an ideal ring with $n_c$ proton-carrying subunits gives $n_c$ protons per revolution and $n_c/3$ protons per synthesized ATP, before additional transport costs.

For a proton moving from side 1 to side 2, its molar electrochemical potential change isThe chemical gradient and membrane electric potential difference together supply the proton-motive force. Synthesis requires the proton free-energy release to meet the positive Gibbs free energy of ATP synthesis. The coupling is reversible: isolated $F_1$ supplied with ATP can hydrolyse it and rotate its shaft, while the intact enzyme can use hydrolysis-driven rotation to pump protons against their gradient. The sketch is structural and schematic; it does not reconstruct the absent stepping trace.

The protein primary structure is the ordered protein sequence along a polypeptide, from its N-terminus to its C-terminus. In cells, information in a DNA coding sequence is copied into messenger RNA by transcription. During translation, the ribosome reads successive three-base codons; transfer RNA molecules match these by their anticodons and supply the corresponding amino acids. The genetic code is degenerate: different codons can encode the same amino acid. A start establishes the reading frame and a stop terminates translation. Peptide bonds link the amino acids into the backbone, leaving chemically different side chains. Thus the sequence specifies the order of side-chain interactions, rather than a literal coordinate list for the folded molecule.

Protein secondary structure describes local backbone arrangements. An alpha helix is stabilized by a regular pattern of backbone hydrogen bonds, typically between the carbonyl of residue $i$ and the amide of residue $i+4$. Beta sheets assemble extended strands through inter-strand hydrogen bonding, with parallel or antiparallel strand orientations. Turns and loops connect these elements; not every region belongs to a helix or sheet. The peptide bond is approximately planar, while backbone torsion angles provide conformational freedom subject to steric restrictions.

Protein tertiary structure is the complete three-dimensional organization of one chain, often arranged into protein domains. Protein quaternary structure describes the organization of multiple chains in an assembly. Soluble globular proteins often bury nonpolar side chains and expose polar ones; membrane proteins can instead expose hydrophobic surfaces to the surrounding lipid. Other stabilizing interactions include hydrogen bonds, electrostatic interactions, Van der Waals forces and sometimes covalent disulfide bonds. These competing contributions define active sites, binding pockets and mechanically useful structures; folded proteins are dynamic ensembles, not perfectly rigid objects.

The connection from sequence to structure is thermodynamic and kinetic. Folding usually decreases the chain's conformational entropy, but favorable interactions and changes in solvent entropy, including the hydrophobic effect, can lower the total Gibbs free energy. Under suitable conditions the native ensemble is favored over unfolded alternatives. A change in sequence can therefore change the balance of accessible conformations and alter folding or function. Reversible refolding of a denatured protein in an appropriate environment demonstrates that the sequence can contain the information needed for its native structure.^[ref]

The environment is part of the specification: temperature, pH, ionic conditions, ligands and post-translational modifications matter. Molecular chaperones can prevent aggregation and help chains escape kinetic traps without providing a residue-by-residue structural template. Some sequences are intrinsically disordered proteins or disordered regions, and some require partners for a stable fold. Thus primary structure strongly constrains structure, but does not guarantee one unique rigid shape independent of conditions.

Cells store usable free energy in chemical disequilibria: reduced carbon compounds, redox carriers such as NADH, ATP relative to ADP and inorganic phosphate, ion concentration gradients and membrane voltage. These reservoirs are maintained away from equilibrium; the useful quantity is the chemical or electrochemical potential difference, not simply a bond's isolated energy. Some ATP-consuming reactions can be driven because a coupled overall reaction decreases Gibbs free energy, even when the uncoupled synthesis would not.

Oxidation of nutrients transfers reducing equivalents to electron carriers and then to an electron-transport chain. Coupled proton pumping converts part of the redox free-energy drop into a protonmotive force. ATP synthase can convert that force into ATP chemical potential, and ATP hydrolysis drives biosynthesis, transport and mechanical motion. Substrate-level phosphorylation is another ATP source; photosynthetic cells can initially supply the disequilibrium with absorbed light. These are flows with unavoidable dissipation and heat release, not independent energy-creation processes. A reservoir is depleted unless metabolism replenishes it.

Two examples are sodium dodecyl sulfate, with a negatively charged sulfate head and a hydrocarbon tail, and cetrimonium bromide, with a positively charged ammonium head and a hydrocarbon tail. Both are amphiphiles: one region is hydrophilic and another is hydrophobic.

In water, the hydrophobic effect favors association of the tails into a core, reducing their exposed area while the hydrated heads remain outside. This can lower the total Gibbs free energy despite loss of translational entropy. Head-group repulsion, interfacial cost and chain-packing constraints oppose unlimited growth, so finite micelles are favored under suitable conditions above the critical micelle concentration.

The appropriate geometric measure is the amphiphile packing parameter:For a spherical micelle with $n$ tails and core radius $r_c$, the volume and area relations are $nv=4\pi r_c^3/3$ and $n{a}_0=4\pi r_c^2$. Dividing gives $r_c=3v/a_0$, so a sphere fits without overextending a tail if $P\le 1/3$. Here $P<1/3$ andThus the packing model predicts spherical micelles. This is a preferred aggregate geometry, not a claim that concentration, temperature or ionic conditions can never alter the morphology.

For either phase, the entropy differential at fixed composition isThe pressure derivative uses the Maxwell relation from $dG=-SdT+VdP$. Differentiating the equality of entropies along coexistence gives the general relationFor a differentiable coexistence curve with pressure varying, equality of Gibbs free energies also implies $-\Delta SdT+\Delta VdP=0$. Since $\Delta S=0$, the two phases then share $V$. Thus, with $\Delta C_P\ne 0$,This is the Ehrenfest entropy-continuity relation. Entropy equality alone would give $T\Delta (V{\alpha }_T)/\Delta C_P$ without the shared-volume condition; a constant-pressure boundary or vanishing discontinuities needs separate limiting analysis.

Gibbs free energy and its minimum. For fixed atom number $N$, there are $N+n$ sites andconfigurations with $n$ indistinguishable vacancies. Their configurational entropy is $k_B\ln W$. Relative to the vacancy-free Gibbs free energy $G_0$, creating $n$ vacancies costs energy $nϵ$ and pressure-volume work $npv$. Put $A=ϵ+pv>0$. ThenAt leading thermodynamic order, Stirling's approximation givesThe minimum obtained from the stationary point below isIf the perfect crystal's internal energy and other entropy are chosen as zero, its pressure term is $G_0=pNv$. More generally the information given fixes the vacancy contribution rather than the absolute bulk free energy. The pressure-volume term must be retained because the number of sites, and hence the volume, changes.

Equilibrium vacancy population. Differentiating the extensive free energy at fixed $N,T,p$ givesThe stationary point is consequently the unique interior minimum. Write $q=e^{-A/(k_{B}T)}$; its condition is $n/(N+n)=q$, giving the requested thermodynamic result for equilibrium vacancies with variable lattice size,Substitution back into $G$ cancels the terms containing $n_{\ast }$ and yields the boxed minimum. In the dilute-defect limit $A\gg k_{B}T$, $n_{\ast }\simeq N{e}^{-A/(k_{B}T)}$. The Bose-like denominator is a consequence of the changing number of lattice sites and combinatorial entropy; it does not assert that vacancies here are an ideal gas of bosonic particles.

There is a finite-size correction to a variable-site vacancy model if “average” is interpreted exactly. The microscopic count written above gives the relative isothermal-isobaric sumsoIts relative equilibrium free energy is $-k_{B}T\ln \mathcal{Y}=(N+1)k_{B}T\ln (1-q)$. Thus the source's $N$ numerator is correct to leading extensive order, but is not the exact finite-$N$ mean for that stated configuration count. A free-energy minimum is likewise not generally identical to the mean of a finite distribution. These distinctions vanish relatively as $N\to \infty$.

Thermal volume. Each atom or vacancy occupies volume $v$, so at the same thermodynamic accuracyFor the exact counting model, replace $n_{\ast }$ by $(N+1)q/(1-q)$ in $v(N+⟨n⟩)$. The vacancy-free limit is $Nv$ as $T\to 0$. This model isolates expansion from vacancies, taking $v$ and $ϵ$ constant; it omits ordinary phonon thermal expansion. Its growth at high temperature should not be extrapolated to arbitrarily large vacancy fractions in a stable crystal, and $A>0$ is essential for a normalizable variable-site ensemble.

First solve the equation of state for the volume:The Gibbs free energy has differential $dG=-SdT+Vdp+\mu dN$. At fixed temperature and particle number, integrate $dG=Vdp$ between the two pressures:ConsequentlyUse $S=-(\partial G/\partial T{)}_{p,N}$, keeping both endpoint pressures fixed while differentiating the difference:Equivalently, the Maxwell relation $(\partial S/\partial p{)}_{T,N}=-(\partial V/\partial T{)}_{p,N}$ gives $-N{k}_B/p+N{A}^{\prime }(T)$, whose pressure integral is the same result. These are the Gibbs free-energy change for a pressure-linear gas correction identities. The physical path requires $p_0,p_1>0$ and positive volume throughout. The correction $A(T)$ has units of volume per particle, so each term in $\Delta G$ is an energy and each term in $\Delta S$ has units of energy per kelvin.

A phase diagram partitions a space of thermodynamic variables into regions occupied by different equilibrium thermodynamic phases and records the phase coexistence curves between them. For a pure, one-component substance, a pressure–temperature phase diagram gives one point for each pair $(T,p)$. In a single-thermodynamic phase region both controls can vary independently. On a two-thermodynamic phase phase coexistence curve, they obey one relation; at a triple point, both are fixed. These statements assume bulk thermodynamic equilibrium, fixed composition and no additional applied fields.^[ref]

At fixed temperature and pressure, the stable bulk thermodynamic phase minimizes Gibbs free energy for a fixed amount of substance. For a pure substance this is a comparison of chemical potentials on a common normalization. Two candidate thermodynamic phases coexist where ${\mu }_{\alpha }(T,p)={\mu }_{\beta }(T,p)$ and neither is exceeded in stability by a third thermodynamic phase. Differentiating their equality gives the Clapeyron equation. Thus a phase coexistence curve is a locus of equal stability, rather than a trajectory that every heated sample must follow.^[ref]

The following original cartoon uses dimensionless positive coordinates $\theta =T/T_{\ast }$ and $\pi =p/p_{\ast }$, with arbitrary reference scales $T_{\ast },p_{\ast }$. It represents no particular substance and contains no measured data. Stipulate a triple point at $(1,1)$, a liquid-gas critical point at $(3,5)$, and these drawing rules:They specify sublimation, liquid–gas phase coexistence and melting boundaries for the picture. They are not an equation of state, do not predict latent heat or mass density, and do not supply a quantitative critical theory. In particular, the melting slope is merely an illustrative choice.

For example, at fixed $\pi =3$ the melting boundary is at $\theta =1.2$ and the liquid–gas boundary is at $\theta =2$. Heating from $(\theta ,\pi )=(1.6,3)$ to $(3.6,3)$ therefore starts in liquid, crosses phase coexistence at $(2,3)$, and ends in gas. In an equilibrium conversion at that crossing, added heat can change the phase fraction at fixed temperature through latent heat. The pressure–temperature point alone does not specify how much of each thermodynamic phase is present: for that one needs an additional amount or volume constraint.

Now connect exactly the same endpoints by raising $\pi$ from $3$ to $6$ at $\theta =1.6$, moving to $\theta =3.6$ at $\pi =6$, then lowering $\pi$ to $3$. The first leg stays on the liquid side because ${\theta }_{\mathrm{melt}}$ ranges only from $1.2$ to $1.5$ and ${\pi }_{\mathrm{vap}}(1.6)=2.2<3$. The second leg lies above the entire liquid–gas boundary, whose largest drawn pressure coordinate is $5$; the third lies beyond its largest temperature coordinate, $3$. No leg crosses a first-order phase transition boundary. The middle leg passes through a supercritical fluid region, with $\theta >3$ and $\pi >5$, and the final leg returns to a gaslike state continuously.

Avoiding a first-order phase transition does not mean that no heat or work is exchanged. Along the detour, mass density, enthalpy and heat capacity may change substantially but continuously. The liquid-gas critical point itself has singular behavior, so the detour goes around it rather than through it. This connected fluid region is the striking lesson of a phase diagram: a familiar change from liquid to gas need not involve boiling across a sharp boundary.

An equilibrium phase diagram also does not predict how rapidly a conversion occurs. Metastability and a barrier described by classical nucleation theory can delay the observed phase transition. Nor should a binary-composition phase diagram be read as the same picture: its coexistence regions and lever rule encode composition and phase fractions, and can occupy an area even though pure-substance two-thermodynamic phase coexistence occupies a curve in the pressure–temperature plane.

Many ionic crystals have high melting temperatures compared with small molecular solids, because reorganizing their strongly interacting ions costs energy. But melting leaves a dense liquid with interactions; it does not perform the complete separation defining a lattice energy. At fixed pressure, the transition occurs when the Gibbs free energies of the solid and liquid match. The relevant enthalpy and entropy differences both matter, so lattice energy alone is not a universal ordering of melting temperatures. Strong covalent bonds in extended networks can also give high melting temperatures.^[ref]

Protein folding is the process by which a polypeptide chain acquires an organized spatial arrangement. The interactions of its amino acid residues with one another and the surroundings favor some arrangements over others. Protein folding can bring chemically useful groups together at an active site without changing their order along the protein backbone. Its importance is practical: knowing a sequence alone does not tell us whether a sample has the structure needed for function. See Cooper: acquiring functional protein structure.

The ribonuclease A refolding experiment showed that an isolated protein could recover its native properties without a pre-existing folded template under suitable laboratory conditions. Anfinsen interpreted this through a thermodynamic hypothesis: the native arrangement minimizes the whole system's Gibbs free energy in the appropriate environment. His Nobel lecture was delivered on 11 December 1972 and published in Science in 1973. The specified environment is part of the claim, not a disposable qualification. See Anfinsen, Principles that Govern the Folding of Protein Chains, Science 181, 223–230 (1973): Nobel lecture, especially p. 223.

This does not imply that every protein rapidly folds unaided. Molecular chaperones can assist protein folding and reduce protein aggregation; required partners, surroundings and time still matter. Goloubinoff and colleagues' 1989 reconstitution experiments demonstrated assistance from two molecular chaperones and ATP in recovering an active enzyme from unfolded chains. This is experimental evidence that assistance can matter even when the chains already have their specified sequences. See Goloubinoff, Christeller, Gatenby and Lorimer, 1989: chaperone-assisted reconstitution of an active enzyme.

Here is an explicitly stipulated elementary kinetic model, not experimental data. A well-mixed ideal closed system has constant temperature and volume. Alongside an uncatalysed elementary reaction, allow an illustrative catalytic encounter that returns C immediately:Both channels obey the law of mass action. In the second, the forward reaction rate is ${\alpha }_f[\mathrm{A}][\mathrm{C}]$ and the reverse reaction rate is ${\alpha }_r[\mathrm{B}][\mathrm{C}]$. C cancels in each encounter, so its free concentration $c_C$ stays constant without an external supply. This toy elementary reaction encodes regeneration without binding; it is not a claim that real catalytic cycles universally consist of one encounter. A binding cycle can accumulate a reaction intermediate, in which case constant total C does not imply constant free $c_C$.

Adding the parallel reaction rates gives effective first-order reaction rate constantsBecause $k_f^u/k_r^u={\alpha }_f/{\alpha }_r=3$, each channel balances at the same $[\mathrm{B}]/[\mathrm{A}]=3$, and so does their sum. At $c_C=0$ the pair is $(0.030,0.010){\mathrm{s}}^{-1}$; at $c_C=0.001\mathrm{mol}{\mathrm{L}}^{-1}$ it is $(0.300,0.100){\mathrm{s}}^{-1}$. Both give $K=3$ with ideal thermodynamic activities and a common concentration standard state. At $T=300\mathrm{K}$ their compatible standard reaction change in Gibbs free energy is ${\Delta }_r{G}^{\circ }=-RT\ln 3\approx -2.7403\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Choosing an unrelated reverse rate constant would break this consistency.

Let $S=[\mathrm{A}]+[\mathrm{B}]$ be the conserved concentration, and define the dimensionless quantity $x=[\mathrm{B}]/S$. ThenSolving this first-order linear differential equation givesThe chemical relaxation time is $25\mathrm{s}$ without C and $2.5\mathrm{s}$ with the chosen C. From either $x_0=0$ or $x_0=1$, the exponential decay of the deviation leads to the same chemical equilibrium $x=3/4$. Reducing that deviation to one percent takes $\ln (100)\tau$: about $115.13\mathrm{s}$ versus $11.51\mathrm{s}$.

At this chemical equilibrium, conversion continues in both directions. For $S=1\mathrm{mol}{\mathrm{L}}^{-1}$, the uncatalysed channel has equal forward and reverse reaction rates $0.0075\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$. Adding C gives total directional reaction rates $0.075\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$ each, yet zero net reaction rate. Faster molecular exchange and unchanged composition coexist. This exact calculation explains how catalysis can matter enormously within an observation window while leaving the final equilibrium constant intact.

Relative to the specified solid, $Q_{\mathrm{sp}}<K_{\mathrm{sp}}$ favors dissolution if that solid is available; $Q_{\mathrm{sp}}>K_{\mathrm{sp}}$ favors precipitation; equality marks saturation. These comparisons identify a driving direction favored by thermodynamics, not a reaction rate.^[ref]

A proof outline comes from the reaction Gibbs free energy for dissolution:Here $R$ is the molar gas constant and $T>0$ the absolute temperature. If $Q_{\mathrm{sp}}>K_{\mathrm{sp}}$, the logarithm is positive, making the reverse transfer into solid favorable. At equality the bulk driving difference vanishes. Equality can also occur at the boundary where the last solid disappears, but it is not a universal constraint on every solid-free chemical solution.

Solubility specifies the amount that can remain dissolved at chemical equilibrium under stated conditions; it is different from how quickly a sample dissolves. Separating ions from an ionic crystal costs energy, but solvation establishes new interactions with the solvent. In water, hydration of ions couples charged ions to polar water molecules. The net enthalpy of dissolution can have either sign; entropy and concentration also enter the Gibbs free energy balance. Consequently, being a salt does not guarantee high solubility in water or any other solvent.^[ref]

Low solubility must also be distinguished from weak electrolyte behavior. A small amount of an ionic compound can dissolve essentially completely as ions, giving few charge carriers because little material dissolved. Calling that a weak electrolyte solely because its chemical solution conducts poorly would conflate two different questions.^[ref]

For transfer of the same species $X$ from a gaseous reference state into a solvent $S$, define the molar solvation free energy byHere ${\mu }^{\circ }$ is a standard chemical potential. Negative solvation free energy means that transfer into the stated solvent lowers the Gibbs free energy relative to the stated gaseous reference. Include solute-solvent interactions, making room in the solvent, and its reorganization; counting only a favorable pair interaction omits part of the transfer.^[ref]

State the temperature, gaseous reference pressure, dissolved reference concentration or other standard state convention, and species identity. Reference-state changes alter the quoted solvation free energy. For example, with an ideal dilute thermodynamic activity $a=c/c^{\circ }$, replacing $c^{\circ }$ by $\alpha c^{\circ }$ while preserving the physical chemical potential requires ${\mu }^{\circ }$ to increase by $RT\ln \alpha$. Thus the corresponding solvation free energy increases by $RT\ln \alpha$ if the gaseous reference stays fixed. This follows directly from $\mu ={\mu }^{\circ }+RT\ln a$, with $R$ the molar gas constant.

At fixed temperature and pressure, compare fully specified initial and final states usingThe entropy change is for the whole selected system, including both solute and solvent. Dispersing a solute can increase accessible arrangements while orienting nearby solvent molecules restricts others. One picture of an ordered hydration shell cannot determine the sign of the total entropy change. A negative Gibbs free energy change favors the specified process; it does not determine its reaction rate.^[ref]

An illustrative enthalpy accounting path separates solute constituents, reorganizes the solvent, and establishes their interactions:This is a chosen decomposition between compatible states, not a measured sequence. If a gaseous-ion-to-chemical solution solvation term already includes solvent reorganization, do not add that reorganization a second time. Lattice enthalpy from solution calorimetry gives the alternative Hess's law route that combines lattice separation with the complete hydration of ions term.^[ref]

The invented inputs are $+100$ for separation and $+20$ for solvent reorganization, in $\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Interaction terms of $-150$ in solvent A and $-90$ in solvent B give $\Delta H_A=-30$ and $\Delta H_B=+30$. Stipulated total entropy changes of $-50$ and $+150\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ give $-T\Delta S_A=+15$ and $-T\Delta S_B=-45\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ at $300\mathrm{K}$. Both selected processes therefore have $\Delta G=-15\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. This deliberately constructed comparison shows why the signs of the enthalpy and Gibbs free energy changes need not agree; it does not predict either solvent's solubility.

For incremental dissolution at a particular composition, the relevant chemical potential difference also depends on thermodynamic activities. At chemical equilibrium that driving difference vanishes. A quoted standard Gibbs free energy change and the finite change between arbitrary mixtures answer different questions; use compatible standard states and the reaction quotient when discussing solubility.

The motivating calorimetry experiment is especially instructive. A weighed potassium chloride sample dissolves in a known amount of water in a platinum-lined, shielded calorimeter; a quartz-oscillator thermometer and measured electrical heating provide the readouts. Electrical calibration before and after the reaction, comparison with a resistance thermometer, and checks on residual water and particle size constrain errors. The inspected authored article reports positive enthalpy of solution at $298.15\mathrm{K}$ for the specified dilution, despite complete dissolution. This is a combined enthalpy balance, not an isolated solvation free energy or direct single-ion hydration of ions measurement. Shield tracking and final concentration matter; the existing lattice enthalpy from solution calorimetry discussion gives the numerical result and the conversion limits.^[ref]

Substrate-level phosphorylation forms ATP by enzyme-catalyzed transfer of a phosphate group from a phosphorylated substrate to ADP. In canonical glycolysis, two such transfers occur for each three-carbon route, giving four gross ATP per glucose. Unlike oxidative phosphorylation, these transfers do not require an electron transport chain.^[ref]

The pyruvate kinase example under Coupled-enzyme readout of glycolytic ATP formation separates the material transfer from its detection. A favourable coupled chemical reaction supplies the driving change in Gibbs free energy; the enzyme supplies a catalytic route. Counting ATP-forming transfers gives a gross yield, while subtracting upstream ATP use gives the net yield.

A thermodynamic system is in thermodynamic equilibrium when its macroscopic state has no spontaneous tendency to change through any exchange allowed by its constraints. For an isolated system, stable thermodynamic equilibrium maximizes entropy over the accessible states at fixed total energy and conserved material inventories. A wall can forbid an exchange, so always specify what can cross or move. Thermal, mechanical and chemical conditions answer different questions about that same constrained state. ^[ref]

For example, two compartments separated by a rigid, impermeable wall that transmits heat can reach thermal equilibrium while retaining different pressures and compositions. Allowing an unloaded piston to move adds a mechanical equilibrium condition. Allowing one chemical species to cross adds chemical potential equality for particle exchange; allowing a chemical reaction adds its chemical equilibrium condition. A fixed barrier can therefore support a constrained thermodynamic equilibrium that ceases to be an equilibrium when the barrier is removed.

Uniformity is not the definition of thermodynamic equilibrium. An isothermal atmosphere in hydrostatic equilibrium can have a common temperature while its pressure and mass density vary with height: the pressure gradient balances gravity. In an external field, the exchange condition must include the relevant potential energy; for charged matter this leads to the electrochemical potential. Relativistic gravitational corrections to thermal equilibrium are described separately by the Tolman temperature law.

A steady state only says that the macroscopic description is constant in time. A bar maintained between hot and cold thermal reservoirs has steady heat flux and produces entropy, so it is not in thermodynamic equilibrium. Even matching temperatures at one perfect-contact interface can coexist with heat transfer through the surrounding material; matching those two local values does not establish equilibrium of the whole apparatus. See thermal conduction for that spatial transport problem.

The useful simplification is that equilibrium conditions predict an endpoint without computing every collision or flow during relaxation. The first law of thermodynamics supplies the conserved quantities, and the Second law of thermodynamics supplies the direction of change. For a system held at fixed temperature and volume, the corresponding criterion is minimization of Helmholtz free energy; at fixed temperature and pressure, it is minimization of Gibbs free energy, with the material and exchange constraints still imposed.

A free energy is a thermodynamic potential that compares possible states under specified environmental constraints. For a closed system of fixed composition, treated as a simple compressible system with only pressure-volume work, stable thermodynamic equilibrium minimizes Helmholtz free energy at fixed temperature and volume, or Gibbs free energy at fixed temperature and pressure. The choice follows the experiment: a rigid sealed vessel and a sealed cylinder with a freely moving loaded piston impose different constraints. Both may exchange heat with a thermal reservoir. See W. C. Carter, MIT thermodynamics notes, sections 2-1.14 and 2-1.19.

The useful insight is that internal energy alone cannot rank these alternatives: exchanging heat also changes the surroundings' entropy. Subtracting $TS$ packages that accounting into a property of the chosen system. Adding $pV$ additionally accounts for displacing the surroundings at fixed pressure. A free energy is a state function; supplied heat describes a transfer during a process. Under the stated constraints, supplied heat is $Q=\Delta U$ at fixed volume or $Q=\Delta H$ at fixed pressure, whereas neither $\Delta F$ nor $\Delta G$ generally equals $Q$.

A lower free energy identifies a thermodynamic preference, without specifying a timescale. A barrier can trap a system in a local minimum, producing metastability although a lower global minimum exists. The enzyme example illustrates the corresponding distinction for chemical reactions: changing the route can change the reaction rate while preserving the endpoint Gibbs free energy difference and chemical equilibrium.

A state function used to describe a thermodynamic system with a chosen set of natural variables. Internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy use different variables and serve different environmental constraints.

The thermomechanical effect couples a temperature difference to a pressure difference across a superleak. For the closed-chamber model in the apparatus figure, consider neighboring equilibrium states with small velocities, negligible height difference and pure helium-4. Let ${\mu }_m$ be chemical potential per unit mass, in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$. The Gibbs free energy differential per unit mass givesThe superleak allows the superfluid to redistribute until the two sides have equal ${\mu }_m$. Setting $d{\mu }_m=0$ therefore givesThe sign matters: heating one side at initially equal pressure lowers its ${\mu }_m$; inflow raises its pressure until chemical balance is restored. The hotter side has the higher pressure. The integral follows the constant-${\mu }_m$ path; it need not be evaluated at fixed pressure when material properties vary appreciably.

For an original local example, stipulate constant $\rho =150\mathrm{kg}{\mathrm{m}}^{-3}$, $s=100\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$ and $T_h-T_c=0.020\mathrm{K}$. The predicted pressure difference is $300\mathrm{Pa}$. These are teaching assumptions, not helium measurements. This static thermodynamics relation predicts neither the required heater power nor the filling time: those require heat leakage, channel conductance and chamber heat capacity. An outlet would introduce kinetic energy and a flowing balance, so the static $300\mathrm{Pa}$ is not a predicted fountain height.