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For an ideal dilute chemical solution, moving one mole of a species with signed ion charge number $z$ from outside to inside changes its electrochemical potential byHere $R$ is the molar gas constant, $T$ the uniform absolute temperature, $F$ the Faraday constant, and $c$ the concentration. This assumes the same reference chemical potential on both sides. At thermodynamic equilibrium, $\Delta \tilde{\mu }=0$, henceThe Nernst equilibrium potential uses thermodynamic activities in the general case; replacing them by concentrations is an ideal-solution approximation.

Choose an original passive model: only ${\mathrm{K}}^{+}$ permeates ($z=+1$), other ions cannot cross, there are no pumps or imposed electric currents, and there is no net water transfer. Stipulate reservoir concentrations $c_{\mathrm{in}}=140\mathrm{mM}$ and $c_{\mathrm{out}}=5\mathrm{mM}$, with $T=310\mathrm{K}$, $R=8.314\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ and $F=96485\mathrm{C}\mathrm{mo}{\mathrm{l}}^{-1}$. Treat reservoir concentrations as effectively unchanged by charging. ThenThe concentration ratio is dimensionless; $J/C=\mathrm{V}$ supplies the voltage unit. At $V_m=0$, $\Delta \tilde{\mu }>0$ for inward transport, so outward transport is favored. At $V_m=E_{\mathrm{K}}$ it vanishes. A more negative $V_m$ favors inward transport. These sign checks explain the result physically.

Keeping the other chosen inputs fixed, doubling $c_{\mathrm{out}}$ to $10\mathrm{mM}$ raises $E_{\mathrm{K}}$ by $(RT/F)\ln 2\approx 18.52\mathrm{mV}$. Equal inside and outside concentrations give $E_{\mathrm{K}}=0$; reversing their ratio reverses its sign. In finite reservoirs, transport changes both concentrations; the exact final thermodynamic equilibrium must also satisfy material conservation. The fixed-reservoir calculation is useful when only a small fraction of the available ions moves.

In thermal equilibrium, nondegenerate conduction electrons and valence electron holes share one chemical potential. Multiplying $n=N_c{e}^{-(E_c-\mu )/(k_{B}T)}$ by $p=N_v{e}^{-(\mu -E_v)/(k_{B}T)}$ cancels that chemical potential:This relation remains valid with semiconductor doping under the stated assumptions. Equality of the two individual densities, $n=p=n_i$, additionally requires the intrinsic neutral case. Separate electron and hole quasi-Fermi levels in nonequilibrium need not obey this equilibrium product.

With one parabolic valley and two spin states at each band edge, the prefactor multiplying $(k_{B}T{)}^3$ in the product is $4(m_c{m}_h{)}^{3/2}/(2\pi {\hslash }^2{)}^3$. Both populations must be nondegenerate; the Pauli reduction of the pinned-edge mass-action product quantifies a failure of the classical approximation.

Single-electron superconducting tunneling transfers individual electrons across a barrier into or out of quasiparticle states of a material with superconductivity. In a normal-metal–insulator–superconductor junction, abbreviated NIS, only one electrode has superconductivity. Such a junction can reveal a superconducting energy gap without supporting a DC Josephson effect between two superconducting reservoirs.

For an elementary interpretation, assume an opaque barrier, approximately constant normal-electrode density of states, and an energy-independent tunneling matrix element over the range probed. Write the superconducting density of states divided by its normal-state value as $n_s(E)$, with $E$ measured from its chemical potential. A signed voltage convention chosen for positive normal-electrode injection giveswhere $f$ is the Fermi-Dirac distribution at a common temperature and $R_N$ is the normal-state junction electrical resistance. This current convention is local to the NIS measurement. The expression counts occupied initial and empty final states; it is a quasiparticle transport model, not a current-phase relation of a superconducting weak link.

An original analytic illustration uses the ideal fully gapped BCS theory spectrum at zero temperature, with positive gap edge $\Delta$:For positive $V$, the occupation difference restricts the integral to $0<E<eV$. Since $d\sqrt{E^2-{\Delta }^2}/dE=E/\sqrt{E^2-{\Delta }^2}$, integration and odd extension giveThe threshold is $\Delta /e$ for this NIS model. In the corresponding zero-temperature single-electron process between two fully gapped superconductors, creating excitations on both sides instead requires $|eV|\ge {\Delta }_1+{\Delta }_2$. That quasiparticle threshold does not forbid coherent pair electric current below it. At finite temperature, thermally populated quasiparticles soften the simple threshold argument.

At zero temperature in the NIS approximation, $R_{N}d{I}_{\mathrm{qp}}/dV=n_s(eV)$ away from the singular gap edge. Thus a voltage sweep probes the excitation density of states. The ideal divergence is rounded in real measurements, and at nonzero temperature the derivative samples a thermally broadened range of energies. For $|eV|>\Delta$, the model has $V{I}_{\mathrm{qp}}>0$: electrical work creates excitations and is dissipated on relaxation. This distinguishes its steady transport from the conservative Josephson coupling energy.

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]

The chemical potential assigned to a specified standard state. With dimensionless thermodynamic activity $a$, the per-particle convention is $\mu ={\mu }^{\circ }+k_{B}T\ln a$; a molar convention uses $RT$. The standard state changes both ${\mu }^{\circ }$ and the activity definition without changing observable equilibrium.

A standard state is a specified reference state used to define a species' standard chemical potential ${\mu }_i^{\circ }$. The reference convention, including temperature, must be consistent when calculating thermodynamic activities, reaction quotients, and equilibrium constants.

Substitutional semiconductor doping replaces selected host atoms with impurity atoms on ordinary lattice sites. A shallow donor impurity can supply a mobile conduction electron and remain positively charged; a shallow acceptor impurity can capture an electron and leave a mobile valence hole, remaining negatively charged. In nondegenerate, substantially ionized material, donor-dominated doping raises the chemical potential toward the conduction edge and acceptor-dominated doping lowers it toward the valence edge. Dopants do not create net bulk charge in a neutral sample: fixed impurity charge compensates the mobile carrier charge. Freeze-out, compensation and degeneracy qualify the simplest relation between dopant and carrier densities.

A density of states $g(ϵ)=g_0[1+\alpha \cos (2\pi ϵ/\Delta )]$ has zero-temperature oscillatory grand potential $g_0\alpha {\Delta }^2\cos (2\pi \mu /\Delta )/(4{\pi }^2)$. Its Fermi-window convolution multiplies that oscillation by $R_T$ when the lower band edge is far below the chemical potential. The Fourier transform of the even window removes its sine component and gives the title factor. At high $X$, $R_T\sim 2X{e}^{-X}$; at small $X$, $R_T=1-X^2/6+O(X^4)$. The corresponding thermal change relative to zero temperature has amplitude proportional to $1-R_T$, rather than $R_T$.

Consider a planar surface with outward normal along $z$. For an auxiliary equilibrium electron gas just outside it, let $n_v$ be the total number density, counting both signs of $v_z$. Its Maxwell-Boltzmann velocity distribution has normal-component densityAn electron with $v_z>0$ crosses more area per unit time when it moves faster. The outgoing number flux therefore uses $v_z{f}_z$, rather than just the probability of $v_z>0$:This integral contributes only $T^{1/2}$. Holding a classical bulk number density fixed and imposing a normal kinetic energy threshold $W$ would instead give $\Gamma =n\sqrt{k_{B}T/(2\pi m_e)}{e}^{-W/(k_{B}T)}$; that argument alone does not produce $T^2$.

For the ideal surface model, measure external electron energies from the metal's chemical potential $\mu$. A freely propagating external state has $E-\mu =W+p^2/(2m_e)$. When $W\gg k_{B}T$, its Fermi-Dirac distribution reduces to the Maxwell–Boltzmann statistics tail. Counting two spin states per phase space cell of volume $h^3$ givesEach of the three Gaussian integrals supplies $T^{1/2}$. Combining this $T^{3/2}$ state count with the $T^{1/2}$ crossing factor givesThe auxiliary atmosphere supplies an equilibrium state-counting argument; an operating collector removes the outgoing electrons. It need not maintain a real equilibrium gas throughout the gap. Unit transmission and the ideal free electron state count are substantive assumptions, so this calculation does not determine an arbitrary real surface's measured $A$.

In particular, electrons inside a metal generally form a degenerate Fermi gas. A classical distribution for the escaping tail does not imply a classical distribution for the whole interior. Richardson himself emphasized this distinction in his lecture.^[ref]

The thermodynamic limit describes a sequence of larger systems, not a single enormous specimen: particle number $N$ and volume $V$ tend to infinity with number density $n=N/V$ fixed. Hold the chosen intensive and extensive thermodynamic quantities such as temperature $T$ and an applied magnetic field fixed along that sequence. In the canonical ensemble the controls are $T,N,V$; in the grand canonical ensemble one instead fixes $T$ and chemical potential, and the limiting number density is a result, possibly with a jump at phase coexistence. These are different specifications of a limiting experiment.

The useful limit is usually an intensive and extensive thermodynamic quantities normalization of the Helmholtz free energy, not a finite limit of the total extensive quantity itself:Here $Z_{N,V}$ is the canonical partition function, $k_B$ is the Boltzmann constant, $f$ is free-energy density, and $a$ is Helmholtz free energy per particle. Thus $F_{N,V}=Vf+o(V)$ when the limit exists; finite systems need not be exactly extensive quantities because of the surface correction to thermodynamic free energy.

Specify the interactions and the growing containers too. For a conventional fluid, use a fixed stable pair potential with a tempered pair potential tail, and containers whose boundary layer of any fixed thickness occupies a vanishing fraction of the volume; uniformly enlarged cubes work, a tube of fixed cross-section does not give the same bulk limit. For a finite-range Ising model, keep the lattice spacing and bond coupling fixed while enlarging all intended spatial directions. Ordinary bounded wall interactions or periodic boundary conditions can then give the same bulk free-energy density, although boundary conditions can select different spontaneous symmetry breaking states. This is a set of hypotheses, not a theorem that every interaction has a boundary-independent limit.^[ref]

The Curie–Weiss model below deliberately uses a different limiting prescription: its all-to-all coupling is divided by $N$. Without that factor its interaction energy would grow as $N^2$, rather than as an extensive quantity. The model has no spatial wall or correlation length to compare with a container; it is an exactly summable illustration of finite-size rounding, not a quantitative model of the fluid experiment below.

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.

Tonicity describes how a surrounding chemical solution changes cellular volume through osmosis. It depends on which solutes remain effectively excluded on the timescale of interest, not simply on the amount of dissolved material. The following comparisons take a functional cell membrane, an initially unpressurized biological cell, and effectively excluded solutes as the reference; rising turgor pressure can oppose entry in a walled biological cell.

Operationally, hypotonic solutions cause sustained swelling, hypertonic solutions sustained shrinking, and isotonic solutions no sustained volume change in the specified passive comparison. State the reference biological cell, membrane properties and observation interval. A permeating solute can transiently change water potential while crossing; after its chemical potential equilibrates, its initial unequal concentrations cannot be treated as a permanently retained osmotic load. Thus equal osmolarities need not imply isotonic solutions. Conversely, a stationary walled biological cell may have unequal osmolarities balanced by turgor pressure.^[ref]

A triple point is a temperature–pressure pair at which three distinct bulk thermodynamic phases of a one-component substance coexist in thermodynamic equilibrium. In the familiar solid–liquid–gas case, their three phase coexistence curves meet. Other combinations, such as two crystal polymorphs and a liquid, also qualify: “triple” counts thermodynamic phases, rather than states-of-matter categories.

For the same substance, the chemical potential in each thermodynamic phase must agree:There are two independent equality conditions for two controls, temperature and pressure. Generically they select an isolated pair $(T_t,p_t)$; the third pairwise equality follows from the first two. Equivalently, the phase rule gives $F=1-3+2=0$. This counts independent intensive controls, not phase fractions: changing the amounts of the three thermodynamic phases need not change the ideal bulk temperature and pressure, provided all three remain present. Different sets of crystal polymorphs can give different triple points for the same substance.^[ref]

At a liquid-gas critical point, the two fluid thermodynamic phases instead merge, and their mass density difference and latent heat vanish. It is the endpoint of the liquid–gas phase coexistence curve, rather than a meeting of three distinct thermodynamic phases. The regular independent-equality count used for a triple point cannot explain this merger. A triple point can retain finite mass density and entropy differences between all three thermodynamic phases.^[ref]

A triple-point cell turns phase coexistence into a reproducible reference for thermometer calibration. The NIST water-cell demonstration uses a sealed glass vessel containing water of specified isotopic composition and an open central thermometer well. Cooling this well with solid carbon dioxide and alcohol freezes an ice mantle while leaving liquid water and its vapor. The coolant is outside the sealed sample; the sample's vapor space does not communicate with room air.^[ref]

In the actual NIST cell-comparison experiment, newly frozen mantles were compared with an aged reference mantle in a controlled maintenance bath. A warm aluminium rod melted a thin inner layer around the well, allowing the mantle to rotate freely and establishing an ice–water interface close to the sensor. A standard platinum resistance thermometer and a resistance bridge measured the electrical resistance ratio to a calibrated, temperature-controlled reference resistor. Readings at two currents permitted extrapolation to zero electrical power, correcting self-heating; measured immersion depths supplied hydrostatic pressure corrections. Comparing preparations over time tested stabilization rather than assuming freshly frozen ice was ready.^[ref]

The inference is a reproducible fixed-point response for the thermometer, not a direct reading of three chemical potentials. The reference cell anchors the comparison; a water point alone does not determine a thermometer's response at every other temperature. The ideal $F=0$ argument excludes impurity and isotope variation, finite interface effects and unintended heat transfer; real cells also require adequate immersion and stable electrical references. Repeatability is evidence that these effects are controlled, not a reason to omit their uncertainty.

Since 20 May 2019 the kelvin has been defined through a fixed numerical value of the Boltzmann constant. The thermodynamic temperature of the water triple point is consequently a measured quantity with uncertainty, rather than an exact value imposed by the kelvin definition. Its continued use as an assigned reference on the International Temperature Scale of 1990 is a separate convention.^[ref][ref]

Map states with unequal escape rates ${\Gamma }_f\gg {\Gamma }_s$ onto distinct charge outcomes after an interval between their characteristic escape times. Both removal energies can lie above the reservoir chemical potential, so their splitting need not be thermally resolved. A useful rate difference must have a physical origin: a demonstrated single-dot example uses a two-electron singlet state and triplet state with different occupied orbitals. A bare single-electron Zeeman pair in the same orbital does not automatically provide the required rate contrast.^[ref]

Two separately conserved bosonic species have two condensate fields and independent chemical potentials. A contact interaction energy density $U_{11}{n}_1^2/2+U_{22}{n}_2^2/2+U_{12}{n}_1{n}_2$ gives ${\mu }_1=U_{11}{n}_1+U_{12}{n}_2$ and ${\mu }_2=U_{12}{n}_1+U_{22}{n}_2$ in a uniform mixture. Coherent interconversion would change the conservation constraints and cannot be included while retaining independent particle-number constraints unchanged.

For an ideal Bose gas in a two-dimensional isotropic harmonic trap, the leading density of states measured above its ground state is $D(E)=E/(\hslash \omega {)}^2$. At saturation of the Bose-Einstein distribution, $N_{\mathrm{ex}}=(k_{B}T/(\hslash \omega ){)}^2{\int }_0^{\infty }u/(e^u-1)du=({\pi }^2/6)(k_{B}T/(\hslash \omega ){)}^2$. The finite excited-state capacity gives the displayed critical temperature and condensate fraction $N_0/N=1-(T/T_c{)}^2$ below it. The appropriate trapped-gas thermodynamic limit fixes $N{\omega }^2$ as $N\to \infty$ and $\omega \to 0$; a finite trap exhibits finite-size rounding. The number density is inhomogeneous, and the physical chemical potential approaches the oscillator ground state energy, not an unshifted zero by convention.