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For a liquid film heated below and cooled above, the quasistatic temperature of a conductively cooled film makes a thicker region colder at the surface. If surface tension decreases with temperature, Marangoni stress draws fluid towards thicker regions, competing with smoothing by capillary pressure. Writing ${\gamma }_s={\gamma }_0+bh$ gives $a_c={\gamma }_0/(3\mu )$, $a_m=b/(2\mu )$. A balanced long-wave scaling gives $H_{\tau }+(H^2{H}_X{)}_X+(H^3{H}_{XXX}{)}_X=0$. The dispersion relation $s=k^2-k^4$ has unstable band $0<|k|<1$ and fastest normal mode at $|k|=1/\sqrt{2}$. This thermal mechanism does not require surfactant.

Thermochemical equilibrium requires every chemical reaction to have zero Gibbs free-energy change. For ideal reactants, the equilibrium constant is $K=\exp [-{\Delta }_r{G}^{\circ }/(RT)]$ after standard-state activity factors are included.

A water layer across which temperature changes rapidly with depth. In a reduced gravity description of the upper ocean, its displacement is represented by the interface between warm surface water and colder deep water. Deepening the eastern tropical Pacific thermocline makes upwelling less effective at cooling the surface during El Niño.

A thermocouple uses two different electrical conductors and the Seebeck effect to infer a measuring-junction temperature from a voltage and a known reference temperature. The reference material and the reference temperature are distinct choices: B specifies the comparison Seebeck coefficient, while $T_c$ specifies the lower limit of the integral. The two-material experiment in Seebeck effect gives the geometry, sign convention, and calibration limitations.

The useful measurement is a pair: the raw terminal electric potential difference and the thermocouple reference junction temperature. For the A-positive terminal order and homogeneous-wire assumptions of Seebeck effect, write the open-circuit electromotive force asHere $T_h$ is the thermocouple measuring junction temperature and $T_c$ is the common temperature of the reference connections. The subscript $h$ names the measuring end; it need not be hotter. A voltmeter senses the complete connected thermoelectric path through the wires and its leads, in the electrochemical potential convention already described under Seebeck effect. It does not measure an absolute temperature, the individual Seebeck coefficient of A, or an isolated contact electric potential.

The thermocouple measuring junction joins A and B at the location whose temperature is to be inferred. Its own temperature is what the electrical measurement estimates. To infer a surrounding object's temperature, the junction must approach thermal equilibrium with that object: insufficient immersion, heat conduction along the wires, or a changing environment can keep the junction at another temperature. A stable electrical display alone does not establish that this thermal condition holds. The two-heat bath procedure and terminal-reversal controls under Seebeck effect provide the starting experiment; the practical issue here is the reference and conversion chain.

A thermocouple reference function gives the nominal thermoelectric electromotive force versus temperature for a specified material pair and reference temperature, conventionally $0{}^{\circ }\mathrm{C}$. Real Seebeck coefficients vary with temperature, sois generally nonconstant. A lookup calibration can use interpolation between tabulated values; a fitted calibration can use polynomials over stated intervals. Use the correct material type, units, reference and validity interval in both the forward and inverse conversions. The FP-TC-120 manual specifically requires the channel's thermocouple type to be configured and describes its use of NIST-175 reference characteristics.^[ref] A nominal reference curve is not evidence that an aged or inhomogeneous wire still follows it.

A local error estimate explains why the reference thermometer matters. If $\delta V$ is an additive voltage error and $\delta T_c$ is the error in the supplied reference temperature, differentiation of the compensation equation gives, for $F^{\prime }(T_h)\ne 0$,A small slope $F^{\prime }(T_h)$ magnifies voltage errors. In the stipulated $40\mu V/K$ model, a $40\mu \mathrm{V}$ offset gives a $1\mathrm{K}$ inferred-temperature error, and a $1\mathrm{K}$ reference error gives a $1\mathrm{K}$ inferred-temperature error. These are sensitivities of the chosen model, not universal performance figures for real sensors.

The thermocouple reference junction is the connection region where the two thermoelements meet the measurement leads. In ordinary copper-lead wiring there are two material transitions, A–Cu and B–Cu, rather than a second welded A–B junction. They must have a common known temperature $T_c$ for the simple expression $E(T_h,T_c)$ to apply. An ice-water heat bath can establish a nominal $0{}^{\circ }\mathrm{C}$ reference; cold-junction compensation instead measures $T_c$ locally. The word “cold” describes the reference role and imposes no requirement that $T_c<T_h$.

Thermocouple voltage calibration relates an actual assembled sensor's electromotive force to independently established junction temperatures. It must be distinguished from selecting a nominal letter-type conversion curve. In Thermocouples Calibrations Services by National Institute of Standards and Technology (opening method description and bath-calibration table), NIST describes comparison with a maintained reference thermocouple, comparison with a standard platinum resistance thermometer in controlled heat baths, and fixed-point calibration at the freezing points of zinc, aluminum, silver and gold on the International Temperature Scale of 1990. These are independent thermal references, unlike a test that checks only a voltage-input circuit.^[ref]

For a concrete electrical check, the SCC-TC guide describes shorting the input after 30 minutes of powered operation and adjusting its offset to a zero-voltage readout. Its separate calibrator procedure requires equal connection temperatures at the calibrator and module, measures the supplied voltage, converts it to temperature, and compares with the calibrator setting. Such an injected-signal test checks the readout chain; it does not expose a real measuring junction to an independently known bath temperature.^[ref]

Useful additional checks follow from the model: log the raw voltage, the reference temperature, and the final inferred temperature separately; hold $T_h$ stable while changing $T_c$ and test whether cold-junction compensation preserves the inferred $T_h$; compare against an independent thermometer at several stable heat bath settings. Check immersion and repeatability instead of assuming a steady display establishes thermal equilibrium. These are proposed diagnostic checks, not new experimental results. The equal-temperature and polarity controls are already described under Seebeck effect.

Limits arise separately from the thermocouple reference function, sensor-wire changes, thermal contact, terminal temperature differences, and the voltage-input circuit. In particular, the SCC-TC guide warns of connector-to-module temperature gradients and voltage drops caused by the fault-detection electric current flowing through lead electrical resistance.^[ref] A fitted curve or compensation algorithm cannot make all these errors vanish. No single accuracy is assigned to every thermocouple.

For a matched receiver accepting one spatial mode and one polarization of waves, the thermal contribution to the available spectral power of a blackbody isThis expression excludes a zero-point term; it is the thermally emitted contribution. Defining the Rayleigh-Jeans law equivalent antenna temperature by $dP/d\nu =k_B{T}_A$ gives the displayed relation. With $x=h\nu /(k_{B}T)$,Thus $T_A\simeq T$ requires $x\ll 1$. It is an approximation with a frequency-dependent error, rather than a redefinition of thermodynamic temperature.

For the stipulated $T=3\mathrm{K}$ and $\nu =4.080\mathrm{GHz}$, $h\nu /k_B\simeq 0.19581\mathrm{K}$ and $T_A\simeq 2.90316\mathrm{K}$. Replacing $T_A$ by $T$ here would differ by about $3.23\%$. At much higher frequencies, the approximation fails more strongly. A real instrument averages over its measured frequency response and antenna beam pattern, so using a single central frequency adds a narrow-band approximation.

A thermodynamic cycle returns a working substance to its initial state. Its net internal-energy change is zero, so the net work output equals net heat input.

Local equilibrium associates the outgoing heat flux $q$ with outgoing entropy flux $q/\theta$ at positive temperature $\theta$. Its divergence appears in the Lagrangian entropy inequality, with the appropriate nominal flux in reference coordinates.

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.

For a simple compressible system at fixed composition and particle number, the internal pressure is the partial derivative of internal energy with respect to volume at fixed temperature. From $dU=TdS-pdV$ and the Maxwell relation $(\partial S/\partial V{)}_T=(\partial p/\partial T{)}_V$, it has the measurable expression in the title. It is distinct from the mechanical pressure $p$.

For an ideal gas, $p=nRT/V$ gives ${\pi }_T=0$, so its internal energy depends only on temperature at fixed amount of gas. For the Dieterici equation $p=nRT\exp [-an/(VRT)]/(V-nb)$, direct differentiation givesThus an equation of state determines this volume sensitivity even without determining the full caloric function $U(T,V)$.

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.

A thermodynamic process changes a thermodynamic system between states. A quasistatic process supplies equilibrium intermediate states and therefore a path on equilibrium coordinates. An irreversible thermodynamic process can have equilibrium endpoints while lacking a uniform intermediate temperature or pressure. Endpoint values still determine changes in a state function; computing heat and work requires the actual interaction history.

Thermodynamics studies energy transfers and the constraints on macroscopic changes. Its vocabulary connects internal energy, entropy, temperature, pressure and volume to heat and work. The first law of thermodynamics balances energy; the Second law of thermodynamics decides which changes can occur spontaneously and limits how much work a heat engine can deliver. These principles let us bound performance before choosing a material or a detailed machine design. Applications include power generation, propulsion and refrigerators. ^[ref]

Start by choosing a system boundary. A closed system can exchange energy but no matter; an isolated system exchanges neither. A state function, such as internal energy or entropy, depends on the equilibrium state. A process specifies how that state changes. Thus a body's temperature is a property it has, whereas heat records energy crossing its boundary during a process. A body contains internal energy, rather than a stored quantity of heat.

Two insulated bodies initially at different positive temperatures approach thermal equilibrium when brought into thermal contact. Their combined energy is conserved, but their combined entropy increases. They do not spontaneously separate again into a hotter body and a colder one. This familiar asymmetry makes thermodynamics interesting: conserving energy leaves many imaginable changes available, and entropy supplies a further constraint. The same distinction underlies both entropy maximization under thermal contact and the thermal efficiency limit in Carnot's theorem.

For a film described by $dU=TdS+\gamma dA$, with material content fixed, its Helmholtz free energy obeys $dF=-SdT+\gamma dA$. Equality of mixed partial derivatives gives the Maxwell relation $(\partial S/\partial A{)}_T=-(\partial \gamma /\partial T{)}_A$. If $\gamma$ depends only on temperature, write ${\gamma }^{\prime }=d\gamma /dT$; thenHere $C_A$ is the heat capacity at constant area. For $C_A>0$ and ${\gamma }^{\prime }<0$, reversible adiabatic stretching cools the film, since $dT/dA=T{\gamma }^{\prime }/C_A<0$, although positive mechanical input can increase its total internal energy. If $\gamma ={\gamma }_0-\beta T$ with constant positive $\beta$ and constant $C_A$, the isentropic process satisfies $C_A\ln (T/T_{\ast })+\beta A=\mathrm{constant}$, where $T_{\ast }$ is an arbitrary fixed reference temperature. The work convention uses the total area conjugate to $\gamma$; a two-surface film described by projected area requires the corresponding factor of two consistently throughout.

An equilibrium thermodynamic state specifies sufficient independent macroscopic properties, together with material identity and constraints, to determine the other equilibrium properties. For a uniform fixed-composition ideal gas, its amount of substance, temperature and volume determine its pressure through the ideal gas law. These properties specify a state, without specifying a thermodynamic process that prepared it.^[ref]