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A gas thermometer infers temperature from the equation of state of a confined gas. The constant-volume version measures pressure; the constant-pressure version measures volume. For an ideal gas,Here $N$ is particle number, $n$ is amount of substance, $k_B$ is the Boltzmann constant, and $R$ is the molar gas constant. The reference and test states must be internally equilibrated, and $T_0$ must be known on the absolute thermodynamic temperature scale. A pressure ratio cancels the unknown gas inventory and bulb size in the ideal constant-volume limit. This cancellation is the method's appeal: a mechanical measurement transfers an absolute temperature scale between different materials. The Maxwell speed distribution and ideal gas pressure from wall collisions explain the microscopic origin of the response. ^[ref]

For an original ideal-model calculation, choose $T_0=300\mathrm{K}$, $P_0=120\mathrm{kPa}$ and $P=160\mathrm{kPa}$. Then $T=400\mathrm{K}$. These are stipulated values, not observations. Both pressures must be absolute, referred to vacuum. If the surrounding atmospheric pressure were $100\mathrm{kPa}$, the corresponding gauge pressures would be $20$ and $60\mathrm{kPa}$; their ratio is three, which would incorrectly predict $900\mathrm{K}$. A pressure difference from the atmosphere cannot be substituted into the ideal gas law.

A motivating calibration experiment. Schooley's account of the NBS/NIST work describes a helium-filled metal bulb, a capillary, a capacitance diaphragm gauge, and a mercury pressure standard. Circulating bath liquid passed the bulb assembly and platinum resistance thermometers. The resistance probes checked the surrounding temperature; gas pressure supplied the independent thermodynamic comparison. In an early series, workers filled the cleaned bulb with purified helium at the highest test temperature, checked pressure over three hours, then measured at lower temperatures and near a reference at $0{}^{\circ }\mathrm{C}$. Six filling pressures allowed extrapolation toward zero pressure. ^[ref]

To turn this principle into a readable measurement protocol, use the following controls. These steps are a proposed teaching interpretation of the cited apparatus, rather than additional reported NBS observations:

The corresponding NBS apparatus used a protective case with an outer helium layer held close to the bulb pressure, reducing pressure deformation while aiding thermal contact. Thermal expansion was measured on bulb material. Gas in the capillary remained partly outside the hot region, requiring a measured tube temperature profile and geometry. Pure working gas also mattered: contaminants that sorb on surfaces can change the freely circulating inventory. These are physical effects to measure or bound, rather than scatter that disappears by averaging. ^[ref]

For illustration, neglect connecting-tube gas and describe a uniform bulb with a dilute virial expansion in particle number density ${\rho }_N=N/V$:Here $Z=PV/(N{k}_{B}T)$ and $B_2$ is the second virial coefficient in the particle-density convention; a molar-density convention rescales the coefficient. The second relation follows exactly from the definition of $Z$ for the same $N$, and shows the signs of the corrections. If the bulb expands by $0.1\%$ and $Z=1$ at both states, the toy pressure ratio above gives $400.4\mathrm{K}$, rather than $400\mathrm{K}$. Improved expansion and helium nonideality models really changed the temperatures inferred from the 1989 NIST data: Harvey's 2023 reanalysis found expansion updates of roughly $1$–$3\mathrm{mK}$ and nonideality updates of roughly $-0.1$ to $-0.5\mathrm{mK}$. Those are reported corrections to historical results, not the much larger illustrative correction just calculated. ^[ref]

With equal corrected bulb volumes, negligible or corrected connecting-tube gas, fixed reference and test temperatures, and a dilute classical single gas phase,Repeat the reference/test comparison for several sealed fills, reducing $N$ between comparisons but keeping it fixed within each pair. The extrapolation varies density at fixed endpoint temperatures; it does not cool one fill toward zero temperature. Real gases have different finite-density corrections but share the classical ideal gas intercept under these assumptions. The limit does not remove leaks, bulb expansion, gauge offsets, or the gas thermometer dead-space correction. At very low pressures, thermal transpiration can make remote sensor and bulb pressures differ through a nonisothermal capillary; this must be corrected too. The NBS work measured this pressure correction using a separate comparison of narrow and wider tubes. ^[ref]

At $x=1$, the toy models give $T_{\mathrm{app}}=407.92\mathrm{K}$ and $388.12\mathrm{K}$, respectively, although their imposed true test temperature is the same. Their zero-density intercept is the reusable quantity. Measuring several low-density points permits a fit and a check for curvature; asserting a straight line from one finite-density point supplies neither. A practical fit must balance nonideality against increasing relative pressure error and the capillary corrections, and keep its extrapolation uncertainty.

The Gross–Pitaevskii equation describes a dilute interacting condensate through a complex order parameter $\psi$, normalized so that $|\psi {|}^2$ is number density. Repulsive contact interactions have $g>0$. In a frame rotating at the condensate frequency, the operator includes $-\mu$, with chemical potential $\mu$. It is generated by a Hamiltonian functional; its homogeneous positive-density state has $n_0=\mu /g$ when $g,\mu >0$.

The Hall effect is a transverse electric potential difference produced when an electric current flows through a material in a transverse magnetic field. The contact orientation fixes its measured sign. The Drude conductivity and Hall experiment gives an explicit convention and the limitations of inferring a number density from it.

A hard-sphere gas is a classical model of identical impenetrable spheres, each of diameter $\sigma$ and radius $\sigma /2$. Its pair potential is $u(r)=+\infty$ for centre separation $r<\sigma$ and $u(r)=0$ for $r\ge \sigma$; the convention at exact contact has no effect on the integral. It isolates the effect of particle size without an attractive potential energy.

The upper-left panel is a cross-section of a three-dimensional forbidden region: the second centre cannot enter a sphere of radius $\sigma$ around the first centre. The particles themselves have radius $\sigma /2$. Consequently, with particle volume $v_{\mathrm{s}}$ and pair excluded volume $v_{\mathrm{ex}}$,The last equality follows directly from the Mayer f-function: $f=-1$ throughout the forbidden volume and zero outside it. Confusing particle radius with contact separation changes the answer by a factor of eight; omitting the unordered-pair factor changes it by a further factor of two.

Define the packing fraction $\eta =n{v}_{\mathrm{s}}$, where $n$ is number density. Then the virial expansion beginsFor example, at the toy value $\eta =0.01$, the leading correction predicts $Z\approx 1.04$; the omitted terms still matter if accuracy beyond this approximation is needed. The classical second virial coefficient is independent of temperature because the geometry of forbidden configurations does not change when the gas is warmed. The pressure itself still scales with temperature at fixed number density.

This model makes the positive correction particularly clear: eliminating overlapping configurations changes the dependence of the available configuration space on volume, even though the allowed configurations all have zero interaction energy. The figure's other panels add a toy attractive shell and show how its Boltzmann factor competes with that geometry. The hard-sphere gas is a useful reference model, while a low-order virial expansion remains a dilute-gas approximation rather than a dense sphere packing formula.

A repulsive condensate next to an impenetrable planar wall has Dirichlet boundary condition $\psi (0)=0$. Writing $\psi =\sqrt{n_0}f$ reduces its stationary Gross–Pitaevskii equation to ${\ell }_h^2{f}^{\prime \prime }+(1-f^2)f=0$. The bulk conditions $f\to 1,f^{\prime }\to 0$ give ${\ell }_h^2(f^{\prime }{)}^2=(1-f^2{)}^2/2$ and hence the displayed number density. Omitting $n_0$ describes normalized number density, not dimensional number density.

The healing length of a repulsive homogeneous condensate is the length at which density-gradient kinetic energy and interaction energy balance. The displayed convention sets the dimensionless Laplacian coefficient to one in the stationary Gross–Pitaevskii equation. A hard-wall number density profile has width $\sqrt{2}{\ell }_h$; definitions without the factor $\sqrt{2}$ occur, so the equation and convention must be specified.

In the pp-I limit, helium-3 relaxation time is ${\tau }^{-1}=n_1\sqrt{2{\lambda }_{11}{\lambda }_{33}}$. At fixed proton number density, each thermonuclear reaction rate coefficient obeys $\lambda (T)/\lambda (T_0)=(T/T_0{)}^{-2/3}\exp [-\eta (T_0)((T_0/T{)}^{1/3}-1)]$. Taking their geometric mean gives the displayed relaxation ratio with $\bar{\eta }=[{\eta }_{11}(T_0)+{\eta }_{33}(T_0)]/2$. A local power-law approximation instead gives $\tau \propto T^{-(\alpha +\beta )/2}$. These are estimates rather than a complete radial model: the proton number density, composition and pp-II branch change along a real stellar composition profile.

An ideal gas is a model of a dilute gas whose particle centres move according to classical mechanics and occupy negligible volume compared with their container. Treating the particles as point masses removes the molecular-size correction; neglecting interactions between separated particles removes the intermolecular contribution to potential energy. The usual kinetic theory model retains brief elastic collisions with the walls and, optionally, with other particles: these transfer momentum while conserving the total kinetic energy of the colliding bodies. There is no appreciable interaction energy stored between collisions. These assumptions lead to the ideal gas law, $PV=N{k}_{B}T$. ^{[ref] [ref]}

The model's attraction is its universality: at the same temperature and number density, different classical ideal gases have the same pressure, even when their particle masses differ. In ideal gas pressure from wall collisions, heavier particles move more slowly at a given temperature, and the two effects cancel. For example, one mole of any ideal gas at $300\mathrm{K}$ in $0.025{\mathrm{m}}^3$ has pressure approximately $99.774\mathrm{kPa}$, as computed under the ideal gas law.

The ideal gas law describes the translational contribution to pressure; it does not specify all of the internal energy or the heat capacity. A monatomic gas has three classical translational degrees of freedom, whereas a gas of molecules can also store rotational kinetic energy and energy in vibrational normal modes. An equation of state alone therefore does not determine how much heat a chosen temperature increase requires. It also does not select a process: isothermal expansion and adiabatic expansion need different additional constraints. ^[ref]

A real gas departs from this approximation when particle size or interactions become important, especially at high number density or low temperature near a phase transition. The Van der Waals equation adds such corrections. Independently, at sufficiently low temperature or high number density, quantum statistics can become essential: even particles with negligible interactions need not obey the classical ideal gas law. The classical statistical assumption is the nondegenerate gas regime, which is distinct from neglecting interactions. ^{[ref] [ref]}

The ideal gas law is the equation of state relating the absolute pressure $P$, volume $V$, particle count $N$, and absolute thermodynamic temperature $T$ of a classical ideal gas:Here $k_B$ is the Boltzmann constant, $N_{\mathrm{A}}$ is the Avogadro constant, $R$ is the molar gas constant, and $n_{\mathrm{mol}}$ is the amount of substance in moles. The distinct symbol $n_{\mathrm{mol}}$ avoids confusing amount of substance with the number density $n=N/V$, measured in ${\mathrm{m}}^{-3}$: the same law can be written $P=n{k}_{B}T$. Use absolute pressure, relative to a vacuum, rather than an unconverted gauge pressure. ^[ref]

The exact SI definitions of the Avogadro constant and Boltzmann constant giveThis numerical value is exact; the applicability of the ideal gas model to a real gas is an approximation. ^{[ref] [ref]}

For $n_{\mathrm{mol}}=1\mathrm{mol}$, $T=300\mathrm{K}$, and $V=0.025{\mathrm{m}}^3$, direct substitution givesAt fixed temperature and particle count, doubling the volume to $0.050{\mathrm{m}}^3$ halves the pressure to approximately $49.887\mathrm{kPa}$: this is Boyle's law. At fixed volume and particle count, doubling the temperature from $300\mathrm{K}$ to $600\mathrm{K}$ doubles the pressure to approximately $199.547\mathrm{kPa}$. These ratios use kelvins, not readings in the degree Celsius scale: $300\mathrm{K}$ is $26.85{}^{\circ }\mathrm{C}$, and $600\mathrm{K}$ is $326.85{}^{\circ }\mathrm{C}$.

The curved slices show Boyle's law at three temperatures; the straight slice shows proportionality to absolute temperature at one fixed volume. They are relations among equilibrium states, rather than a prescribed process history. Compression could hold the temperature fixed through heat transfer, or raise it during adiabatic compression. To calculate a process's heat or work, supplement the ideal gas law with the process constraint, the first law of thermodynamics, and the relevant internal energy or heat capacity information. The graph's extrapolation toward $T=0$ is mathematical; it does not assert that a real gas remains classical or gaseous there.

For a stellar mixture with fixed mean molecular weight, the ideal gas law can be written $P=\mathcal{R}\rho T/\mu$, where $\rho$ is mass density and $\mathcal{R}$ is the stellar gas constant. It assumes negligible interaction corrections and a nondegenerate gas; excitation or ionization can change the particle count and thermodynamic response.

In his 1834 memoir on the motive power of heat, Émile Clapeyron combined gas laws into a relation between pressure, volume, and temperature while analysing heat engines (printed page 164). The equation above uses modern SI notation and absolute thermodynamic temperature; it is not a transcription of his historical temperature convention. ^[ref]

Consider a stipulated dilute collection of independent two-level systems, each with longitudinal magnetic moment $s\mu$, $s=\pm 1$, and energy $E_s=-s\mu B_0$. The prescribed vacuum/applied magnetic flux density is $B_0={\mu }_0{H}_{\mathrm{app}}$. Ignore interactions, orbital magnetic response, and the small sample-field correction. This is the magnetic version of the existing independent spin-one-half two-level system, with alignment probabilities rather than an interacting Curie–Weiss model.

For temperature $T>0$, the canonical ensemble gives the one-moment partition function $Z_1=2\cosh x$, with $x=\mu B_0/(k_{B}T)$. Taking the expected value over its two Boltzmann weights givesHere $n$ is the number density, $k_B$ the Boltzmann constant, and $\tanh$ the hyperbolic tangent. For $|x|\ll 1$, its small-argument expansion of the hyperbolic tangent gives $M\simeq {\mu }_{0}n{\mu }^2{H}_{\mathrm{app}}/(k_{B}T)$, a Curie's law response in the applied-$H$ convention. For large $|x|$, $M$ approaches the saturation magnetization $n\mu$ with sign set by the field, and its differential magnetic susceptibility falls towards zero.

Both plots use stipulated $n=10^{26}{\mathrm{m}}^{-3}$, $\mu =9.2740101\times 10^{-24}\mathrm{J}{\mathrm{T}}^{-1}$, $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, and ${\mu }_0=1.25663706\times 10^{-6}\mathrm{T}\mathrm{m}{\mathrm{A}}^{-1}$. The moment is approximately one Bohr magneton; the chosen vacuum permeability is a rounded model input. At $T=2\mathrm{K}$, $M_s=927.401\mathrm{A}{\mathrm{m}}^{-1}$ and ${\chi }_{\mathrm{model}}(0)=3.91410\times 10^{-4}$. The linear approximation overestimates $M$ by less than $3\%$ for $|x|\le 0.3$, corresponding to $|{\mu }_0{H}_{\mathrm{app}}|\le 0.89324\mathrm{T}$ or $|H_{\mathrm{app}}|\le 0.71082\mathrm{MA}{\mathrm{m}}^{-1}$. These parameters keep the modeled response weak; neglected magnetic feedback is of relative order ${\chi }_{\mathrm{model}}(0)$, rather than an explanation of a material's ordered phase.

At $B_0=0$, cooling increases the zero-field slope as $1/T$. At fixed $B_0\ne 0$, cooling instead gives ${\chi }_{\mathrm{model}}\sim 4{\mu }_{0}n{\mu }^2{e}^{-2\mu |B_0|/(k_{B}T)}/(k_{B}T)$: a saturated moment can hardly change further. The order of the zero-field and zero-temperature limits matters. This independent model has no finite-temperature collective phase transition; the diverging zero-field slope alone does not establish one.

For one ionic species of number density $n_i$, the ion-sphere radius is $a_i=(3/(4\pi n_i){)}^{1/3}$. The ionic Coulomb coupling parameter compares the interaction energy from Coulomb's law at that separation with thermal energy $k_{B}T$. At fixed composition, $\Gamma \propto {\rho }^{1/3}/T$. Values near one mark significant corrections to the ideal ion gas; strong coupling can lead to an ordered solid. Screening, mixtures and quantum motion modify detailed transitions, so this estimate is not a universal crystallization boundary. It is separate from the electron degeneracy criterion based on the electron Fermi temperature.

Kinetic theory describes many-particle systems through a distribution on phase space, retaining both position and velocity. In the number-density convention, $f(t,\mathbf{x},\mathbf{v})d^{3}x{d}^{3}v$ is the expected number of particles in a small position–velocity region. For identical particles of mass $m$, its integrals give the number density $n$ and bulk velocity $\mathbf{u}$:This $f$ is not normalized to one: its integral over the whole phase space gives the total expected particle number. Transport moves particles along trajectories, while collision or diffusion terms change the distribution. The Boltzmann equation, Hamiltonian Liouville equation and Fokker-Planck equation describe different physical assumptions, rather than interchangeable collision laws.

An ideal gas is a motivating example: the pressure on a wall comes from microscopic momentum transfers, even when the bulk velocity is zero. The Maxwell-Boltzmann velocity distribution separates this random molecular motion from bulk motion; see ideal gas pressure from wall collisions. The attractive feature of kinetic theory is that a distribution can describe two opposing populations where one average velocity hides almost everything. It also tells us when averaging into a continuum approximation loses essential information: compare the intermolecular mean free path with the apparatus scale using the Knudsen number. ^[ref]

The Knudsen number is the dimensionless ratio of an intermolecular mean free path ${\ell }_{\mathrm{mfp}}$ to a specified characteristic length $L$ of the apparatus or flow. The mean free path measures the distance travelled between gas–gas collisions; $L$ measures the distance over which boundaries or macroscopic conditions matter. Wall encounters must not be counted as gas–gas collisions in this comparison: doing so would hide the very wall-dominated behavior we want to detect. The same gas can therefore have a small Knudsen number in a large vessel and a large Knudsen number in a tiny opening.

For a dilute hard-sphere gas of identical molecular diameter $d$ in thermal equilibrium, use the conventionHere $n$ is the number density, $p$ the pressure, $T$ the absolute temperature, and $k_B$ the Boltzmann constant; the second equality uses the ideal gas law. The collision cross-section is $\pi d^2$, and the $\sqrt{2}$ accounts for both partners moving with independent equilibrium velocities. Mixtures and velocity-dependent collision cross-sections require a stated molecular model and mean free path convention; the displayed expression is not an exact formula for every real gas. ^{[ref] [ref]}

As a numerical toy example, choose $T=300\mathrm{K}$ and $d=0.37\mathrm{nm}$, without claiming that this is a measured molecular diameter for a particular gas. The formula gives ${\ell }_{\mathrm{mfp}}\simeq 68.1\mathrm{nm}$ at $p=10^5\mathrm{Pa}$. It yields $\mathrm{Kn}\simeq 6.81\times 10^{-6}$ for a $1\mathrm{cm}$ vessel but $\mathrm{Kn}\simeq 0.681$ for a $100\mathrm{nm}$ gap. At $p=1\mathrm{Pa}$ the same toy gas has ${\ell }_{\mathrm{mfp}}\simeq 6.81\mathrm{mm}$, comparable to the centimetre vessel. Reducing either pressure or apparatus size can thus expose molecular transport.

Choose $L$ for the question being asked: a tube diameter compares the mean free path with wall separation, while tube length describes a different geometric scale. For a positive macroscopic field $q$, a local gradient scale can instead be $L_q=|q|/|\nabla q|$, where its gradient is nonzero. A large vessel can contain a thin region with a large local Knudsen number. Small geometric Knudsen number alone does not guarantee a slowly varying near-equilibrium flow. The time, velocity and space scalings must also be stated when the Knudsen number appears as a coefficient in a Boltzmann equation. ^[ref]