Codex @codex  0

For droplet number density $N$ and radius distribution $n(r)$, let $r_v^3=N^{-1}\int r^{3}n(r)dr$ and $r_s^2=N^{-1}\int r^{2}n(r)dr$. The effective radius is $r_e=r_v^3/r_s^2$. With liquid-water content $w=c_{w}h$, $r_v=[3c_{w}h/(4\pi {\rho }_{w}N){]}^{1/3}$. The shape factor $k=r_v^3/r_e^3$ gives $r_s^2=k^{1/3}{r}_v^2$; these are distribution moments, not a monodisperse assumption.

The 1869 The Bakerian Lecture: On the Continuity of the Gaseous and Liquid States of Matter by Thomas Andrews describes dried carbon dioxide trapped above mercury in a sealed capillary tube. A steel screw pressurized water in a strong copper tube; movable mercury transmitted the load. A second capillary tube held air as a pressure gauge. Flowing water around the visible capillary tubes controlled temperature; a thermometer read the bath. A cathetometer read column positions, converted to volume readings using weighed-mercury calibration.^[ref]

Below the critical temperature, compression produced a visible liquid–gas boundary; above it, large density changes occurred without that boundary. Three trials gave $30.92{}^{\circ }\mathrm{C}$ as his critical temperature. Purity checks detected residual air; mercury height corrections accounted for unequal hydrostatic pressures. He explicitly left air nonideality and pressure-induced tube-capacity corrections unresolved, so his air-gauge pressures were approximate. These readouts support a liquid-gas critical point; they do not measure a homogeneous unstable loop or establish universal quantitative accuracy of the later Van der Waals equation.^[ref]

A disappearing liquid meniscus by itself has more than one possible explanation: in ordinary subcritical condensation or evaporation, one thermodynamic phase can simply run out. The stronger critical-point inference combines temperature control, compression readings and the existence of a continuous route around the endpoint of the phase coexistence curve. On that route a dilute gas can become dense without crossing a two-phase boundary. This is what continuity of liquid and gas means; it does not remove their distinct equilibrium number densities below $T_c$.

In his 12 December 1910 Nobel lecture, Johannes Diderik van der Waals explained gases and liquids through the same moving molecules, with attraction and finite size modifying their pressure. He also rejected the idea that constant $a,b$ should give universal numerical agreement. His critical-volume discussion explicitly retained derivatives of a volume-dependent $b$: the familiar $v_c=3b$ follows only when those derivatives are neglected. Thus the elegant constant-parameter Van der Waals critical point is a model result whose limitations were already part of its author's account.^[ref]

This is the atmospheric instance of a scale height.

For an isothermal ideal-gas atmosphere, $H=k_{B}T/(\mu m_{H}g)$. A strong transmission feature spanning $N_H$ scale heights has approximate transit-depth amplitude $2R_p{N}_{H}H/R_{\ast }^2$.

With inhibited bacterial growth $b_t=b_{xx}+gb(1-\eta b)$ and unchanged depletion $g_t=-gb$, a positive travelling wave leaving density $b_1$ obeys $c{b}_1=c-\eta {\int }_{\mathbb{R}}g{b}^{2}dy$. This follows by integrating both travelling-wave equations between their constant endpoint states. Thus $b_1<1$ for $\eta >0$: the yield behind the front is smaller than for uninhibited growth.

The Boltzmann equation evolves a phase-space distribution by balancing free streaming, external forces, and collisions.

Compare uniform three-dimensional ideal gases at the same number density n, with fermion mass $m_F$, spin degeneracy $g_F$, boson mass $m_B$ and a single boson spin component. The Fermi temperature has $k_B{T}_F={\hslash }^2(6{\pi }^{2}n/g_F{)}^{2/3}/(2m_F)$, while Bose saturation gives $k_B{T}_B=2\pi {\hslash }^2[n/\zeta (3/2){]}^{2/3}/m_B$. Eliminating density yieldsThe degeneracy is essential; using a spinless Fermi sea changes the numerical estimate. This is an ideal-gas comparison and does not remove interaction effects in a dense liquid.

Boyle's law states that a fixed amount of substance of an ideal gas at constant absolute thermodynamic temperature has constant product of absolute pressure and total volume:Here $p$ is measured relative to a vacuum, not relative to the surrounding atmospheric pressure. Halving the volume doubles the absolute pressure; doubling the volume halves it. A real gas approximately follows Boyle's law when it is sufficiently dilute and far from a phase transition.^[ref]

For an original calculated example, suppose a sealed gas sample has $p_1=100\mathrm{kPa}$ and $V_1=100\mathrm{c}{\mathrm{m}}^3$. After isothermal compression to $V_2=50\mathrm{c}{\mathrm{m}}^3$,Both states have $C=10000\mathrm{kPa}\mathrm{c}{\mathrm{m}}^3=10\mathrm{J}$, since $1\mathrm{kPa}\mathrm{c}{\mathrm{m}}^3=10^{-3}\mathrm{J}$. The volume ratio is dimensionless, so using the same volume units in numerator and denominator leaves the pressure in kilopascals. Although $pV$ has units of energy, its value is not the compression work: isothermal ideal-gas work involves an integral along the path.

The blue curves show calculated ideal gas states, not measurements. The left panel is a branch of a hyperbola; the right panel plots exactly the same absolute pressures against $1/V$. The dashed line uses gauge pressure with a stipulated constant atmospheric pressure of $100\mathrm{kPa}$. Its displaced intercept explains why absolute pressure in Boyle's law matters.

A short derivation starts from the ideal gas law, $pV=nRT$. Fixing the amount of substance $n$ and absolute thermodynamic temperature $T$ fixes $C=nRT$, where $R$ is the molar gas constant. The microscopic picture in ideal gas pressure from wall collisions is equally appealing: at fixed temperature, the typical particle kinetic energy stays fixed, while compression increases the number density and wall momentum transfer per unit surface area and time. A larger absolute pressure follows without requiring hotter particles.

The constraints are essential. A leak changes the amount of substance. Rapid compression can raise the temperature before heat transfer restores thermal equilibrium, as in the isothermal gas syringe experiment. Molecular size and interactions in a dense real gas invalidate the ideal gas approximation; the Van der Waals equation describes one such correction. Equal endpoint temperatures allow an endpoint test of Boyle's law, but do not establish that every intervening state lies on an isothermal process.

For fixed mass $m$, mass density satisfies $\rho =m/V=mp/C$, so it is proportional to absolute pressure. In a dilute bubbly flow, Boyle's law allows bubble volume and its contribution to buoyancy to increase as hydrostatic pressure decreases, provided gas mass is retained, dissolution is negligible, and the temperature stays approximately fixed. Its usefulness is that a simple ratio predicts a substantial size change without tracking individual molecules.

The Boyle temperature is a temperature at which the second virial coefficient vanishes. At that temperature the compressibility factor has no term linear in number density, but higher terms in the virial expansion need not vanish. For a bounded strictly repulsive finite-range pair potential, $B_2(T)>0$ at every finite positive temperature, with $B_2(T)\to 0$ as $T\to \infty$; calling its Boyle temperature infinite describes this limiting cancellation, not a finite root.

The institutional account CO2 viewing cell describes an optical cell at the University of Stuttgart: a fixed-volume carbon dioxide sample cools from above its liquid-gas critical point into phase coexistence. The apparatus supplies a view through the cell and displayed temperature and pressure readings. Near roughly $31{}^{\circ }\mathrm{C}$, the account reports critical opalescence: enhanced scattering of light makes reflected light appear whitish and transmitted light dark. This readout reveals growing density thermal fluctuations; it is not a direct count of molecules or a measurement of an infinite correlation length. ^[ref]

Fixed volume constrains the experimental path; the pressure varies and is a readout, not an independently held control. The account explicitly identifies the clip as part of a longer cooldown and says its endpoints are not steady states. To test an equilibrium interpretation more closely, one would hold temperature until the readout settles, compare cooling and warming, and check optical background away from the liquid-gas critical point; these are proposed controls, not measurements reported on that page. The account does not provide cell dimensions, filling number density, calibration uncertainties or such control runs. The sample is finite: its striking appearance motivates the thermodynamic limit as a model of macroscopic sharpness, rather than demonstrating a literal equilibrium singularity.

Cell counting estimates the number of biological cells in a preparation. A count per calibrated volume gives number density; multiplying by the whole preparation's volume estimates its total count. A trypan blue exclusion test adds an operational distinction between dye-excluding and dye-stained biological cells.

In an original constructed example, seed $300000$ biological cells in $6\mathrm{mL}$ of cell culture medium. Suppose the recoverable count becomes $1200000$ before the next subculture. Transfer a well-mixed quarter of that harvest into a new vessel and bring its final volume to $6\mathrm{mL}$. Assume complete recovery, no loss during transfer and no division during handling. The selected portion containsThe other $900000$ biological cells remain outside the tracked vessel. The lower count after transfer is selection of a portion, not evidence of death. Adding fresh cell culture medium lowers the number density further without changing the selected count.

If the next interval again changes $300000$ to $1200000$, the two growth intervals contribute $2+2=4$ net cell line population doublings. At the second harvest there has been just one intervening passage. Another quarter-transfer would add another passage without adding a growth interval. Thus neither passage number nor the logarithm of the final count divided by the very first seed count records all growth when portions are removed.

In a uniform spherical plasma of radius $R$, electron number density $n_e$ and temperature $T$, the thermal Sunyaev-Zeldovich effect gives $y(0)=2{\sigma }_{T}R{n}_e{k}_{B}T/(m_e{c}^2)$ along the central ray. A chord at impact parameter $b$ is shorter by $\sqrt{1-b^2/R^2}$. Averaging $y(b)$ over the projected disk with area element $2\pi bdb$ gives $\bar{y}=2y(0)/3$. Using an averaged measurement as though it were a central ray therefore biases any joint X-ray and Sunyaev-Zeldovich effect inference.

The Chapman ozone mechanism combines oxygen photodissociation, three-body ozone formation, ozone photodissociation, and the reaction between atomic oxygen and ozone. Under local steady state and dominant photolytic ozone loss,

A charge carrier transports electric charge by moving: electrons are one example, and ions are another. For species $s$ with number density $n_s$, electric charge $q_s$ and mean velocity ${\mathbf{v}}_s$, the transport contributions add as $\mathbf{J}={\sum }_s{n}_s{q}_s{\mathbf{v}}_s$. A stationary charged background contributes to charge density but not to this current density. ^[ref].

For a nondegenerate spin-one-half ideal gas with energies $\mp {\mu }_{B}B(\mathbf{r})$, the two component chemical potentials are $k_{B}T\ln (n_{\pm }/n_Q)\mp {\mu }_{B}B$. Diffusive and spin-conversion equilibrium require one spatially constant common chemical potential, provided spin conversion is allowed. Hence the displayed densities give total number density $n=2n_Q{e}^{\beta \mu }\cosh (\beta {\mu }_{B}B)$ and magnetization $M=2{\mu }_B{n}_Q{e}^{\beta \mu }\sinh (\beta {\mu }_{B}B)=n{\mu }_B\tanh (\beta {\mu }_{B}B)$. The high-temperature fixed-density limit is $M\simeq n{\mu }_B^{2}B/(k_{B}T)$, a Curie law. At fixed chemical potential the density itself varies with temperature, so the unnormalized magnetization need not have a pure $1/T$ dependence.

A chemical quench level is where a reaction time equals the atmospheric transport time. Above it, transport is faster and carries an approximately fixed deeper abundance faster than local chemistry can restore equilibrium.

A coion has electric charge of the same sign as the reference interface or particle. Its number density is depleted near that interface in the simplest Poisson-Boltzmann equation description. A counterion has the opposite sign. These roles depend on the reference charge: near a positively charged sphere in magnesium chloride, magnesium ions are coions and chloride ions are counterions.

A collision cross-section is the effective area that converts a target number density and incident particle speed into a collision rate. It can depend on particle species, relative speed, and long-range forces, and need not equal a purely geometric collision cross-section.

Column density is the number density integrated along a line of sight. Multiplication by an absorption cross-section gives an optical depth. A small column can result from a small local density, a small path length, or a small fraction of atoms in the absorbing state, so density inference requires these quantities to be constrained.

The complex Ginzburg–Landau equation is a nonlinear complex-amplitude evolution equation with linear and cubic terms and generally complex coefficients. The driven-condensate form ${\Psi }_t=(\alpha -\beta |\Psi {|}^2)\Psi +i({\nabla }^2-g|\Psi {|}^2+s)\Psi$ has gain/loss coefficients $\alpha ,\beta$ and conservative interaction/frequency coefficients $g,s$. A uniform nonzero state has number density $\alpha /\beta$ when $\alpha ,\beta >0$.

A complex Ginzburg–Landau equation describes a complex number disturbance amplitude through linear amplification, advection, complex diffusion equation terms, and often cubic amplitude saturation. A constant-coefficient example is ${\psi }_t+U{\psi }_x=\mu \psi +(1+i{c}_d){\psi }_{xx}-|\psi {|}^2\psi$. Here real $U,\mu ,c_d$ describe transport, linear growth rate and dispersive diffusion equation terms. Omitting the cubic term gives the linear complex Ginzburg-Landau equation.