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A Stirling engine repeatedly heats and cools a confined gas, whose expansion and compression drive a moving mechanical load. This makes it a particularly accessible example of a heat engine: both the repeating motion and the two regions exchanging heat can be observed.

In the published Demonstration experiments with a Stirling engine by Christopher G Deacon and others, an electrical heater warms one end of a cylinder and flowing water cools the other at $12{}^{\circ }\mathrm{C}$. A moving displacer shuttles gas through copper wool that temporarily stores and returns internal energy. A laser spot traces a closed pressure-volume path on a screen: mechanical motion controls one coordinate, pressure the other. Calibrating both scales converts the enclosed area to work per thermodynamic cycle; multiplying by rotation frequency gives gas-cycle power. At $97\mathrm{W}$ heater input, the paper reports $4.4\mathrm{W}$ gas-cycle power, while a separate Prony brake measures $3.4\mathrm{W}$ shaft output. The cold-side temperature is controlled and two higher heater settings are compared. The gas has a nonuniform temperature, and heater energy also escapes to the room: neither output/input ratio measures $W/Q_h$ for an ideal two-thermal reservoir engine. These observations demonstrate work production and losses, not a direct measurement of the bound in Carnot's theorem. The inspected apparatus, readouts and results are on pp. 180–182, Figures 1–3 and Table 1.

Stochastic thermodynamics assigns work, heat, and entropy production to individual fluctuating trajectories while recovering ordinary thermodynamics after ensemble averaging.

For dilute isolated spheres in a Newtonian solvent, $D\eta a/T=k_B/(6\pi )$. An experimental check independently measures temperature, viscosity and particle size, then obtains diffusion from a displacement variance or a scattering decay. Drift, walls, concentration effects, aggregation and a hydrodynamic radius different from a structural radius can defeat an unqualified comparison.

Take downward as positive and let gravity have magnitude $g$. For a rigid sphere of density ${\rho }_p$ in otherwise stationary fluid of density ${\rho }_f$, its weight minus buoyancy is $\frac{4}{3}\pi a^3({\rho }_p-{\rho }_f)g$. At terminal velocity, this balances the Stokes drag law:This signed downward settling velocity is negative for a buoyantly rising sphere and zero for equal densities. For fixed density contrast and dynamic viscosity, the submerged weight grows as $a^3$ while linear drag at fixed speed grows as $a$; their balance gives the settling-velocity radius scaling $U_s\propto a^2$. Meanwhile,Thus doubling the radius multiplies the predicted Stokes settling speed by four and the particle Reynolds number by eight. This makes the formula useful for separating particles by size, while also providing its own check against extrapolation to large particles.

For a computed water example, take ${\rho }_f=1000\mathrm{kg}{\mathrm{m}}^{-3}$, $\mu =1.00\times 10^{-3}\mathrm{Pa}\mathrm{s}$, ${\rho }_p=2500\mathrm{kg}{\mathrm{m}}^{-3}$, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, and $a=10.0\mu \mathrm{m}$. These are specified rounded inputs rather than a claim of exact water properties at every temperature. They giveThe diameter is $20.0\mu \mathrm{m}$, and both the net downward weight and the upward drag force are $6.16\times 10^{-11}\mathrm{N}$. At this terminal velocity, falling $1.00\mathrm{mm}$ takes $3.06\mathrm{s}$. At $a=20.0\mu \mathrm{m}$ the prediction is $1.308\mathrm{mm}{\mathrm{s}}^{-1}$ with ${\mathrm{Re}}_d=0.05232$; at $a=50.0\mu \mathrm{m}$ it is $8.175\mathrm{mm}{\mathrm{s}}^{-1}$ with ${\mathrm{Re}}_d=0.8175$, which is no longer a convincing small-Reynolds number case. The figure labels these as predictions of the Stokes settling velocity formula and shows where its inertia check fails.

The calculation describes the eventual terminal velocity, not the release transient governed by particle acceleration and possibly the Unsteady Stokes equation. Walls, hydrodynamic interactions, departures from a rigid sphere, or a different surface condition require a different resistance. For sufficiently small particles, Brownian motion can obscure individual settling trajectories; the formula still specifies the gravitational drift within its assumptions. The balancing argument already appears in article 43 of ^[ref].

For a nonzero horizontal Fourier mode $k$, solve $(D^2-k^2{)}^{2}w=R{k}^2\theta$, $w=w^{\prime \prime }=0$, to obtain $w={\mathcal{W}}_R\theta$. Under homogeneous temperature Dirichlet boundary conditions, the sine modes diagonalize this self-adjoint operator, with eigenvalues $R{k}^2/(k^2+n^2{\pi }^2{)}^2$. The temperature linear operator is ${\mathcal{L}}_R=D^2-k^2+{\mathcal{W}}_R$. At a marginal mode $g=\sin (n\pi z)$ it has ${\mathcal{W}}_{R}g=(k^2+n^2{\pi }^2)g$.

An incompressible Boussinesq fluid with uniform magnetic field, uniform vertical rotation and uniform temperature gradient supports waves combining magnetic tension, Coriolis force and buoyancy. Put $K=|\mathbf{k}|$, $h^2=K^2-k_z^2$, $a^2=(\mathbf{B}\cdot \mathbf{k}{)}^2/({\mu }_0\rho )$ and $H=g\alpha \beta$, with ${\partial }_z{T}_0=-\beta$. Eliminating pressure and the temperature and magnetic perturbations gives the displayed dispersion relation for nonzero frequencies. Stable thermal stratification has $H<0$. Clearing frequency denominators requires checking any resulting zero roots in the original equations; a stationary magnetostatic-buoyancy balance can exist independently of the propagating branches.

Proposed climate intervention that adds material to the stratosphere to increase reflection of sunlight. Sulfur-dioxide injection relies on conversion into a sufficiently persistent sulfate particle population. A global-mean shortwave forcing target does not uniquely specify regional temperature, rainfall or ozone responses.

Conversion of relatively unreactive chlorine reservoirs into forms that produce active radicals in sunlight. Heterogeneous reactions on appropriate particle surfaces can turn hydrogen chloride and chlorine nitrate into photolabile species. Available surface composition, temperature, reservoir supply and subsequent illumination all matter; adding arbitrary particles does not guarantee activation.

For a fixed-temperature layer with stress-free boundary conditions, a vertical Fourier sine series has $a^2={\lambda }^2+n^2{\pi }^2$, where $\lambda$ is horizontal wavenumber. The coupled vertical velocity and temperature amplitudes satisfy the displayed quadratic for their growth rate $\sigma$. Its roots are real for positive Rayleigh number, while both roots have negative real parts for negative Rayleigh number. Setting $\sigma =0$ gives the free-slip convection neutral curve.

For density of states proportional to ${ϵ}^a$, $a>-1$, and positive temperature, set $q=\mu /(kT)$, $n(x)=(1+e^{x-q}{)}^{-1}$, $b=a+1$ and $M_j={\int }_0^{\infty }{x}^{a+j}n(1-n)dx$. Differentiation and integration by parts giveThe inequality is strict Cauchy-Schwarz inequality, since $x$ is not constant under this positive weight. Thus fixed entropy per particle forces fixed $\mu /T$. This is the injectivity step behind the adiabatic scaling of an ideal Fermi gas.

Consider an original ideal model: $V_A=10.00\mathrm{mL}$ of hydrochloric acid at $c_A=0.100\mathrm{mol}{\mathrm{L}}^{-1}$, titrated with sodium hydroxide at $c_B=0.100\mathrm{mol}{\mathrm{L}}^{-1}$. Treat these numbers as exact model inputs. Assume complete dissociation of an ionic compound, complete strong acid ionization, additive volumes, uniform mixing, and activity coefficients equal to one. Use stipulated $K_w=10^{-14}$, approximately appropriate near $25{}^{\circ }\mathrm{C}$. These are calculated pH values, not measurements of these materials.

The chemical equation for neutralization isInitial acid inventory is $n_A=1.000\mathrm{mmol}$, so the equivalence point is $v=10.00\mathrm{mL}$. Added water does not change this inventory. But the final volume $V_A+v$, including the titrant, determines all concentrations.

Keep self-ionization of water even at the equivalence point. With $c^{\circ }=1\mathrm{mol}{\mathrm{L}}^{-1}$, define the dimensionless acid excess $d$, hydronium level $h$, and hydroxide level $o$ byThe sodium cation and chloride anion inventories fix the electric charge balance $h-o=d$, while chemical equilibrium gives $ho=K_w$. Eliminating $o$ yields a quadratic equation $h^2-dh-K_w=0$ with exactly one positive root. For stable calculation on either side, writeThe second expression follows by multiplying by $s-d$; it avoids subtracting two nearly equal large quantities on the alkaline side. A logarithm acts on the dimensionless $h$, as required by the pH definition.

At $v=5.00\mathrm{mL}$, the excess is $0.500\mathrm{mmol}$ in $15.00\mathrm{mL}$, giving $d=1/30$ and pH $1.477121\dots$. At $v=15.00\mathrm{mL}$, excess hydroxide is $0.500\mathrm{mmol}$ in $25.00\mathrm{mL}$, giving $d=-1/50$ and pH $12.301030\dots$. Neglecting dilution would give the wrong values even far from the equivalence point.

Exactly at $v=10.00\mathrm{mL}$, $d=0$ and $h=o=\sqrt{K_w}=10^{-7}$, giving pH $7$ in this model. At $9.999\mathrm{mL}$ and $10.001\mathrm{mL}$, the same smooth expression gives pH $5.300835\dots$ and $8.699121\dots$. The familiar excess-only formulas would predict zero hydronium or zero hydroxide at the equivalence point; retaining self-ionization of water removes that failure. A steep curve is still continuous.

The sensitivity is especially revealing. Differentiating $h-K_w/h=d$ giveswhen $c_A=c_B$. At this model's equivalence point, the derivative is about $1.086\times 10^4$ pH units per millilitre. This is a local derivative, not a linear prediction over a whole drop. Real endpoint accuracy also depends on dosing, mixing, measurement uncertainty, temperature, extra reacting chemical species, and nonideal thermodynamic activities.

A strong electrolyte yields essentially all ions from its dissolved substance in the specified solvent and conditions. Strength does not specify concentration.

For dissolved sodium chloride in dilute water, one dissolved formula unit supplies one sodium cation and one chloride anion. For hydrogen chloride in water, ions instead arise by proton transfer to the solvent, giving hydronium and chloride. Thus separating existing ions by dissociation of an ionic compound and producing ions from neutral molecules by ionization can both give a strong electrolyte.^[ref]

The denominator in “essentially all” is the dissolved inventory. Solubility asks how much enters the chemical solution; electrolyte strength asks what happens to the part that enters. A sparingly soluble ionic compound can therefore have an essentially fully dissociated dissolved portion. Undissolved solid does not count as neutral dissolved chemical species, and low electrical conductivity alone does not establish a weak electrolyte.^[ref]

The diagram's weak electrolyte has the smaller converted fraction but more ions in the same volume: ten charged symbols versus four. Even at equal total ion concentrations, different ion drift velocities can give different electrical conductivities. Strength is useful for predicting dissolved chemical species in chemical equations; concentration and transport properties are additionally needed to predict electric current.

An inspected motivating experiment is Harvard Natural Sciences Lecture Demonstrations' Conductivity of Solutions by Harvard Natural Sciences Lecture Demonstrations, Setting it Up. Separate labeled beakers hold aqueous sodium chloride, sucrose, acetic acid and hydrochloric acid samples specified at $0.10\mathrm{mol}{\mathrm{L}}^{-1}$, with about $150\mathrm{mL}$ in $250\mathrm{mL}$ beakers; distilled water supplies a blank. Metal electrodes enclosed in tubing connect to a lamp. The readout is lamp brightness; the page reports that an ionic chemical solution lights the lamp. A separate deionized-water rinse station limits carryover between samples.^[ref]

For a controlled interpretation, keep electrode spacing and immersed area fixed, compare at the same temperature, and recheck the blank after rinsing. These are additional controls: the page supplies no numerical electrode geometry, temperature or calibration. Brightness supports the presence of mobile ions; it cannot alone measure the converted fraction because concentration, ion mobility and circuit response also matter. Its classification of insoluble compounds as weak should not replace the dissolved-inventory distinction above.^[ref]

A regime where optical-depth-one transmission occurs in the far wings of a strongly absorbing line. With a uniformly mixed absorber, $\Delta ={\Delta }_{s}p/p_s$ and $C=qS{p}_s/(2\pi g{\Delta }_s)\gg 1$, the radiative-wing optical depth is $\tau \simeq qS{\Delta }_s{p}^2/[2\pi g{p}_s(\nu -{\nu }_0{)}^2]$. This condition does not make pressure broadening valid in the arbitrarily rarefied upper atmosphere.

A strong, nonradiative shock in a monatomic ideal gas gives $p_{\mathrm{d}}=3{\rho }_{\mathrm{u}}{v}_{\mathrm{u}}^2/4$ and ${\rho }_{\mathrm{d}}=4{\rho }_{\mathrm{u}}$. The ideal gas law therefore gives the temperature in the title, with $\mu$ the mean particle mass. If the upstream speed is a Newtonian free-fall speed, $k_{\mathrm{B}}{T}_{\mathrm{d}}=3\mu GM/(8R)$. Mass alone does not determine this temperature without a radius and composition; electron-ion disequilibrium and radiation can also make the emergent spectrum differ from the ideal gas temperature.

Real atoms have multiple internal quantum states. Optical pumping and spatially varying electromagnetic polarization can remove kinetic energy by additional mechanisms, producing sub-Doppler cooling below the ideal Doppler cooling limit. In polarization-gradient cooling, state-dependent energy shifts from the Stark effect and delayed optical pumping make atoms repeatedly climb potential energy hills. This mechanism needs the internal-state structure omitted by the independent-beam two-level system above. ^[ref].

William D. Phillips's team tested unexpectedly cold sodium molasses with time-of-flight atom detection. A probe above the cloud initially saw no atoms; deliberately heating them verified that the detector worked. Moving the probe below the cloud produced arrival signals consistent with temperatures below the Doppler cooling limit. Independent release/recapture checks used sharply apertured beams and included gravity; uncertain cloud and probe geometry still limited the inferred temperature. These controls tested the thermometer as well as the cooling theory. ^[ref].

For a locally linear ice temperature with positive gradient $G$ into the solid, a corrugated sublimation front receives a perturbation in heat flux proportional to its horizontal wavenumber $\alpha$. The Gibbs-Thomson effect suppresses short waves, giving the displayed quasi-static growth rate. It requires $V/(\kappa \alpha )\ll 1$ and $|\sigma |/(\kappa {\alpha }^2)\ll 1$, as well as a perturbation depth short enough for the linear background approximation. Unstable modes have $0<\alpha <\sqrt{G/\Gamma }$.

The sudden-decay approximation converts a dominant cold relic's rest-mass density into radiation at $H\simeq \Gamma$, neglecting expansion during the conversion. Its pre-decay radiation temperature obeys $T_{-}^3\simeq 3M_{\mathrm{Pl}}^2{\Gamma }^2/(c_1{g}_{\star }Ym)$, while the reheating temperature is set by the post-decay radiation energy density. Constant relativistic counts and prompt thermalization are additional assumptions.

A sun valve automatically restricts a gas light's supply in daylight and allows it again as illumination falls. In Gustaf Dalén's design, a blackened rod and three polished rods respond differently to sunlight. Their relative thermal expansion operates the gas valve through a membrane; a pilot flame permits relighting. The museum dates this design to 1907.^[ref]

The useful comparison is between two thermal responses. Both kinds of rod expand when warmed; a dark surface can absorb more incident radiant power, so its temperature rises more. Comparing their extensions can reject a common ambient-temperature change while responding to illumination. The mechanism converts light into heat, heat into a difference of length, and that difference into a change of gas flow. Differential thermal model of a sun valve makes those assumptions quantitative.