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A diffusion flame sustained by the evaporation and chemical breakdown of wax delivered through a wick by capillary action.

A contact line is pinned when its position remains fixed while the surrounding drop shape or contact angle changes. Surface heterogeneity or deposited material can sustain such a state over a range of angles. Evaporation from a pinned drop then requires compensating liquid motion and can generate outward solute transport associated with the coffee-ring effect. A model with a freely receding line and fixed contact angle has a different boundary constraint and need not predict the same flow direction.

A drying front separates a positive liquid layer from a region where its thickness is zero. Its position is an unknown boundary determined with the wet-region solution. At a continuous zero-height front, a jump balance alone need not fix its speed; the vanishing-depth profile and removal law can supply that information. Source terms representing evaporation or infiltration must respect nonnegative water availability in the dry region.

Evaporation transfers molecules from a liquid surface into the surrounding vapour. It can occur below the boiling temperature, and competes with condensation from the vapour. Its net rate depends on the surrounding vapor pressure and transport conditions. Removing latent heat through evaporation can cool the remaining liquid.

The source ${\Gamma }_i$ is the mass gained by phase $i$ per mixture volume and time, through processes such as evaporation or condensation. Internal exchange conserves total mass, so its phase sources sum to zero. In an immiscible material-interface model without phase change, every ${\Gamma }_i$ is zero. The interface may still move: absence of transfer means that mass does not cross it relative to its own motion.

A membrane osmometer connects osmosis to a visible liquid level or a measured balancing pressure. In a teaching version, a solute-containing pouch is joined to a narrow vertical tube and submerged in external water. Entry increases the enclosed volume and raises the meniscus. Both the column and any stretched pouch can contribute an opposing pressure.

The University of Toronto's Osmosis (Brine) by University of Toronto Department of Physics demonstration (activity Me-012b, Description, steps 1–14) specifies soaking a dialysis membrane tube, tying one end closed and fastening the other around a thistle tube's narrow end. A stand holds the tube. Colored brine is cooled to room temperature and injected using a syringe to reduce bubbles; the pouch is completely immersed in a $1$ or $2\mathrm{L}$ beaker of distilled water. A taped marker records the initial level; subsequent meniscus height is the readout. The protocol proposes another pouch containing distilled water as a comparison, and its older instructions explicitly check seals for leaks.^[ref]

A rise is consistent with net water entry. To infer osmotic pressure quantitatively, use a retained solute and establish a stable equilibrium after accounting for dilution. If both free surfaces are at the same atmospheric pressure, the liquids have approximately equal mass densities $\rho$, and the pouch adds negligible mechanical load, the hydrostatic pressure difference at membrane height is approximately $\Delta P=\rho gh$, where $h$ is the meniscus height above the external surface and $g$ is gravitational acceleration. This is the liquid-column inference, not a pressure calibration supplied by the Toronto demonstration. For unequal mass densities, compare the two column pressures separately.

Useful additional controls are matched initial levels and geometry, constant temperature, replicate runs, and monitoring external solute to test retention. These are proposed checks, not reported Toronto results. Leaks, trapped bubbles, surface tension in the narrow tube, evaporation and membrane stretching can alter the level. In particular, a dialysis membrane may pass dissolved salt: the brine setup illustrates transient movement but cannot be assumed to retain every dissolved ion. Color improves visibility without proving salt retention. A stationary level can reflect weakening concentration differences or leakage rather than osmotic thermodynamic equilibrium; a rising level alone supplies neither $n$ nor a verified value of $nRT/V$.

A concrete motivating experiment is the acetone–chloroform mixture in Truman State University's Liquid-Vapor Equilibrium of a Binary System, Procedure and Figure 1. A two-neck flask feeds a distillation head with a water-cooled laboratory condenser, condensate sampling arm and temperature probe. Starting with one pure liquid, the procedure adds the other component successively and waits for stable temperature with returning condensate. Small paired samples from the flask and condensate are read on an Abbé refractometer; a tabulated refractive index–composition relation gives the two compositions. Controls include three-point thermometer calibration, corrected atmospheric pressure readings, and agreement of the initial pure-liquid and condensate refractive indices with the reference value. Small withdrawals and little retained condensate reduce disturbance of composition. Sample evaporation can bias the readout. The inferred paired mole fractions, temperature and pressure describe vapor-liquid equilibrium, subject to the conversion from mass fraction to mole fraction and to representative sampling. This teaching protocol reports no measured results of a particular student run.^[ref]

How does such a measurement expose activity coefficients? Let $x_i$ and $y_i$ be the mole fractions in the liquid and gas thermodynamic phases, respectively, and let $p_i^{\ast }(T)$ be the pure-component vapor pressure. Assume the gas thermodynamic phase is an ideal gas mixture and neglect the change of the pure-liquid reference chemical potential between its saturation pressure and the experimental pressure. Equality of each component's chemical potential between the two thermodynamic phases then givesThis turns the paired composition readings into estimates of the activity coefficients. The approximation needs refinement when the gas departs appreciably from ideal gas behavior or the neglected liquid pressure effect matters.

The apparatus measures a boiling-composition series in which temperature changes as the mixture changes. Its readings therefore cannot be inserted directly into the fixed-temperature, fixed-pressure activity coefficient constraint. They motivate the more general task: test whether the measured mixture properties can arise from a common Gibbs free energy while accounting for changing state variables. A failure can reveal incompatible property fits or experimental bias; a pass supplies a consistency check, not proof that every reading is correct.

For the supplied hillslope values,The Froude number is below one, but it is not extremely small numerically; the much smaller products actually multiplying inertia and the depth-gradient term justify their leading neglect. Positive flow then satisfies $u=h^{2/3}$ andIn overland flow the source is net rainfall input after infiltration into soil, with evaporation or other local losses included when important. Lateral inflow can also contribute. Negative net input must be limited by water availability once the surface dries.

The 1864 preparations contain either crude opium or its aqueous Extractum Opii; those quantities must be distinguished from a quantity of isolated morphine. The following are the officinal opium preparations. Ratios for solids mean the mass fraction in the compounded preparation; liquid strengths specify mass per volume or the actual ingredients used.

The arithmetic uses the new avoirdupois ounce of 437.5 grains, not the old apothecaries' ounce of 480 grains. These are historical formulation proportions, not modern assays of morphine delivered after solvent extraction. Confectio Opii was omitted in 1864, and this edition's Vinum Opii contains no spices. The separate morphine hydrochloride preparations—Liquor Morphiae Hydrochloratis, Suppositoria Morphiae, Trochisci Morphiae, and Trochisci Morphiae et Ipecacuanhae—contain an isolated alkaloid salt rather than crude opium. ^[ref]

Fresh rainwater is ordinarily soft because evaporation leaves most nonvolatile dissolved salts behind; “always” is an idealization, not an exceptionless law. Water hardness is principally caused by dissolved calcium and magnesium ions. Water vapour rising from a hard-water source does not carry these ions with it as vapour. Condensation consequently supplies low-mineral water before contact with dust, surfaces or soil. Mineral particles, sea spray and a collecting surface can add salts, so actual collected rainwater need not have zero water hardness. Nor does softness establish purity or freedom from microorganisms.^[ref]

The historically appropriate quantitative principle is soap titration of water hardness. Soluble soap reacts with hardness-producing ions to form poorly soluble salts, using up soap before a persistent lather can form. For a representative carboxylate soap,Magnesium behaves similarly. Once the dissolved hardness-producing ions have substantially reacted, additional soap can sustain the specified lather. The endpoint is therefore an operational observation under standardized shaking, standing time, temperature and sample volume; it is not the first transient bubble.

Use a measured volume of sample in a stoppered vessel with enough headspace for shaking. Add a soap solution of known strength incrementally from a burette, shaking in a reproducible way, until the chosen persistent-lather endpoint appears. Determine a blank with a corresponding amount of soft or distilled water, because some soap is needed for the lather even when no hardness salts are present. Standardize the soap against a reference solution containing a known calcium amount, expressed as an equivalent mass of calcium carbonate. This standardization is essential: the apparent strength cannot safely be inferred from the mass of an incompletely characterized commercial soap.

Let $V_s$ be the sample volume in millilitres, $V$ the soap titre in millilitres and $B$ the corresponding blank. If $f$ is the standardized capacity in milligrams of calcium carbonate equivalent per millilitre of soap solution, thenFor a standard containing $m_0$ milligrams of calcium carbonate equivalent and requiring blank-corrected titre $V_0-B_0$, $f=m_0/(V_0-B_0)$. This makes the measurement quantitative without assuming an unreported soap concentration or inventing a sample titre. Historical results could instead be stated as grains of calcium carbonate equivalent per gallon, provided the gallon convention was specified.

Repeating the examination after controlled boiling, cooling, removal of the precipitate and restoration of the original volume estimates the residual or permanent hardness. The reduction estimates the boiling-removable fraction, often called temporary hardness. A representative reaction isCarbonate equilibria and magnesium chemistry make this more subtle than assuming every dissolved salt is destroyed by boiling. Very hard samples may need measured dilution; magnesium, other metal ions and contamination can disturb the lather endpoint. Thus the soap result is an experimentally calibrated hardness equivalent, not an exact identification of every original salt.

External water treatment can clean, transfer heat, provide moisture and apply mechanical pressure; the useful method depends on the object, not on a general claim that water expels disease. Historical classifications distinguished general and local bathing, cold and warm applications, affusion, showers, compresses, packing and vapour baths.

Cold applications included sponging, pouring water over the body, immersion, a local wet compress and a cold shower or douche. A renewed wet cloth removes heat by contact and evaporation; a wrapping allowed to warm and then covered is a different intervention. In 1865 these methods were used for febrile heat, local pain or swelling, and sometimes to obtain a supposed stimulating reaction in chronic debility. A cold douche also adds mechanical impact and an abrupt circulatory stimulus. Historical wet-sheet packing enclosed the body in a wet sheet with outer coverings and was either renewed to sustain cooling or left to warm. The contemporary description distinguishes these uses; the claims about stimulation and cure are historical claims.^[ref]

Tepid and warm immersion, a warm local bath or sitz bath, and warm moist fomentations were used to soothe painful or spasmodic conditions, support cleansing and soften crusts or secretions. Local applications avoid heating the whole body. The apparatus might be a bath, basin, irrigating vessel or repeatedly renewed cloth; a warm moist dressing must not become a source of sustained scalding. Historically, hot immersion and vapour baths were also used to induce sweating in rheumatic complaints and supposed retention of harmful material. Heat-induced sweating is real, but removal of a disease-specific poison by that route is a separate claim requiring evidence. Showering and douching similarly cannot be treated as dose-equivalent to quiet immersion.^[ref]

Current examples show why the distinction matters. Heatstroke requires rapid cooling and emergency care: cold-water immersion or other active cooling is used to reduce dangerous body temperature, not to provoke a later “reaction”. In contrast, routine tepid sponging is not recommended for fever in children under five. Febrile thermoregulation can provoke shivering and distress with external cooling, so the treatment of ordinary fever should not be inferred from the management of hyperthermia.^[ref][ref]

For a thermal burn, current first aid uses cool running water for 20 minutes, with the rest of the person kept warm; ice or an ice-water application is not the appropriate substitute. Cleansing or irrigation has a further purpose independent of temperature: physically removing contamination, with the fluid and technique chosen for the tissue and injury. A steam application to the face is not a harmless version of warm bathing: spilled hot water and steam cause scalds. The historical use of inhalation therefore does not validate leaning over boiling water as a general respiratory treatment. These examples specify an actual therapeutic objective without endorsing the broad claims of nineteenth-century hydropathy.^[ref][ref]

The wick supplies liquid wax by capillary action; wax vapour reacts with atmospheric oxygen, releasing the heat that keeps the supply going. The wick is a delivery system, rather than the principal fuel.

First, an external flame warms the wick and adjacent wax. Melting produces a small liquid reservoir, and capillary action draws it into the spaces between the wick fibres. Near the hot tip, evaporation and some pyrolysis supply gaseous fuel. Melting and evaporation are physical changes; pyrolysis and combustion change the chemical substances present. Once established, the candle flame transfers enough heat back to the wax to maintain this sequence.^[ref]

The candle flame is a diffusion flame. Its relatively dark inner region contains newly supplied fuel with limited access to oxygen. In the yellow luminous region, incomplete conversion of the hydrocarbons produces small soot particles, which become incandescent. Farther out, mixing supplies enough oxygen to consume most of the remaining fuel, carbon monoxide and soot. Thus the yellow light is largely thermal emission from hot particles, rather than a yellow gas.^[ref]

For a representative alkane component of paraffin wax, complete combustion has the chemical equationReal candle materials are mixtures, and some, such as fatty materials, already contain oxygen; the displayed chemical equation is an illustrative wax model. The principal products of complete combustion are carbon dioxide and water vapour. Convection carries the hot gases upward and brings fresh air into the flame. A cold collector can reveal condensed water; collecting the gases and passing them into limewater gives a calcium carbonate cloudiness from carbon dioxide.

Removing the oxygen supply, cooling the reaction zone, or cutting off wax delivery extinguishes the flame. The smoking wick immediately afterwards can still emit combustible vapour; without sufficient heat at the mixing zone, that vapour need not sustain a flame.

evaporation transfers water to a vapour stream and retains mainly nonvolatile solutes in a concentrate. It can give substantial volume reduction and works across many dissolved-solute compositions. Its disadvantages are heat demand, scaling and corrosion; aerosols, entrainment or volatile radioactive species can contaminate the condensate. Condensate must therefore be sampled or monitored rather than assumed releasable, while the concentrate requires radioactive waste conditioning. A demister limits droplet carryover but does not remove a genuinely volatile contaminant.

ion exchange transfers dissolved radioactive ions to an exchanger, often an ion-exchange resin. It is useful for polishing relatively dilute aqueous streams and can provide selective removal without evaporating the bulk water. Competing inactive ions consume capacity, suspended particles foul the bed, and a neutral or poorly retained species may pass through. Breakthrough measurements determine when the exchanger ceases to meet its target; exhausted resin or regeneration liquors become secondary radioactive waste. Separation performance must be established for the actual feed chemistry, not inferred from the name of the radionuclide alone.

Chemical precipitation with solid–liquid separation uses precipitation, coprecipitation and adsorption to carry contaminants into solids, followed by settling or filtration. It can treat substantial volumes and complex feeds at moderate equipment cost. Its disadvantages are a relatively bulky sludge, incomplete removal of soluble species, and sensitivity to pH, complexation and competing ions. The sludge must be dewatered and conditioned, and the clarified liquid checked for residual radioactive activity. Polishing by another method may still be needed.

In each case, measure both the treated stream and the residue so that the radioactive inventory balance can be checked. Treatment transfers or concentrates radioactive activity; only radioactive decay reduces the number of radioactive parent nuclei. These methods and the distinction between treatment and conditioning are set out in the institutional waste-management framework.^[ref]

Notation. The source uses $p_s$ for both the hypothetical dry surface pressure and the actual surface pressure after evaporation. Distinguish them as $P_d$ and $P$, respectively, and interpret the specified pressure thickness $\Delta p$ as fixed. This conserves the original hydrogen inventory and permits $P$ to vary. Reading both occurrences as one fixed number would be inconsistent with adding water vapor to a closed atmospheric column mass budget.

Let $E(T)$ denote the saturation pressure of water, let $x$ be the mole fraction of water vapor in the mixed layer, and let $M_d,M_v$ be the molar masses of hydrogen and water. Surface phase equilibrium and the ideal gas law giveBecause the layer has constant temperature, its relative humidity isThus every level strictly above the surface is subsaturated: hydrostatic balance makes $p<P$, whereas the saturation pressure stays constant. The result does not require a dilute water vapor approximation.

The Clausius-Clapeyron relation, neglecting the specific volume of liquid water and treating the water vapor as an ideal gas, givesHere $T_t,E_t$ are the triple point values, $L_v$ is the specific latent heat, and $R$ is the molar gas constant. All temperatures in these expressions are absolute. As throughout the question, this is a constant-latent heat, ideal gas model, not an extrapolation valid through the real critical point of water.

For a well-mixed ideal gas mixture, its mean molar mass and water vapor mass fraction areBy hydrostatic balance, the mixed layer has total mass per unit area $\Delta p/g$, of which $w_v\Delta p/g$ is water vapor. The remaining atmosphere contains no water vapor. Consequently conservation of hydrogen mass givesEquivalently the requested quadratic equation isFor each $T_s$, calculate $E$, take the positive root of this quadratic equation, and retain it only if $0\le x<1$ and $P>\Delta p$. The last condition leaves a nonempty dry layer. In particular the dry constituent's surface partial pressure is not generally $P_d$: $P_d$ measures its column weight, and the hydrogen and water vapor have different molar masses.

At fixed ground temperature, warming the cloud increases its downward radiant flux $e\sigma T_c^4$, giving a positive contribution to the ground's radiative heating. That is the immediate radiative effect. A self-consistent final answer must also remove the convective heat transfer used to pin the original cloud to the dry adiabatic process.

For pure radiative equilibrium of both bodies, with no net nonradiative energy exchange, their two balances areFor $e>0$ this givesThe $e\to 0^{+}$ limit gives the selected limiting cloud temperature; at exactly $e=0$, only $T_g=T_0$ is fixed radiatively. The maximum greenhouse effect at fixed $Q$ occurs at $e=1$:The pinned dry-adiabatic process case instead has maximum factor $1/q=(p_s/p_c{)}^{\kappa }$. It exceeds $2^{1/4}$ for a high, cold cloud with $q<2^{-1/4}$, equals it at the level found in part b, and is smaller for a low, warm cloud.

In particular, for $q^4<1/2$ the final purely radiative ground is cooler than the pinned solution, even though a warmer cloud increases downward radiation at fixed $T_g$. To see why these statements are compatible, let $H$ be the net upward nonradiative energy transfer from the ground to the cloud in the pinned state. Its conservation of energy equation requiresFor the cold cloud, $H<0$: maintaining it on the assumed dry adiabatic process requires removing its net radiative heating and returning energy elsewhere. Setting $H=0$ changes that energy redistribution. It is therefore incorrect to infer the final ground temperature by changing only the downward radiant flux. Indeed the final cloud is warmer than the pinned one for $q^4<1/2$ and $0<e<1$, while the final ground is cooler. At $e=1$ the cloud temperature is $T_0$ in both whole-column equilibria.

Fixed cloud properties are an artificial constraint. Warming a saturated cloud increases the saturation pressure, tending to evaporate its cloud droplets; the released water vapor, changing latent heat budget and altered buoyancy can change its altitude and emissivity. Cooling favours condensation and potentially atmospheric precipitation. A warm perturbation that survives evaporation can rise by convection, but neither a unique vertical displacement nor survival of the cloud follows without moisture supply and an environmental temperature profile. Finally, pure radiative equilibrium of the two bodies does not require the intervening transparent gas to remain on the original dry adiabatic process; imposing that extra constraint is generally inconsistent except at the matching level in part b.

Let $1<S<S_c$ be the ambient saturation ratio. There are two equilibrium radii: $r_{-}<r_c<r_{+}$. Condensation occurs if $S>S_{\mathrm{eq}}(r)$; evaporation occurs if $S<S_{\mathrm{eq}}(r)$. Therefore droplets with $r<r_{-}$ grow towards $r_{-}$, while those with $r_{-}<r<r_c$ shrink towards $r_{-}$. The smaller equilibrium is stable.

Above $r_c$, droplets with $r_c<r<r_{+}$ still undergo evaporation and ultimately approach $r_{-}$. Droplets with $r>r_{+}$ grow, and the larger equilibrium $r_{+}$ is unstable. Merely exceeding $r_c$ is therefore not sufficient for continued growth at subcritical ambient atmospheric supersaturation. For example, if $(S-1)/(S_c-1)=0.6$, the two intersections have approximately $r_{-}/r_c=0.676$ and $r_{+}/r_c=2.349$; a droplet of radius $1.5r_c$ undergoes evaporation. If instead $S>S_c$, there is no equilibrium barrier in this approximation and droplets activate into growing cloud droplets.

Köhler theory explains activation of cloud condensation nuclei, rather than rapid production of atmospheric precipitation. Diffusion-controlled droplet growth is limited by diffusion of water vapor and removal of released latent heat; approximately $d(r^2)/dt$ is proportional to the ambient atmospheric supersaturation. It slows in radius and tends to narrow a droplet size distribution. The available water vapor, changing temperature and shared depletion of atmospheric supersaturation also invalidate unlimited growth at prescribed $S$. Formation of rain-sized drops normally requires collision-coalescence, while colder clouds can additionally use cloud ice nucleation versus ice growth.

Stretch $z=\delta Z$ and write $f=\delta h$, $w=\delta W$, with horizontal velocity and time left order one. The longitudinal viscous term is ${\delta }^{-2}{u}_{ZZ}$, so set $P={\delta }^{2}p$. For $Re{\delta }^2\ll 1$, the leading bulk equations are $P_Z=0$, $u_{ZZ}=P_x$ and $u_x+W_Z=0$.

The normal-stress condition gives $P=-{\delta }^3\hat{C}{h}_{xx}$ at leading order, and the tangential condition gives $u_Z(h)=0$. HenceNo slip at the bed, $u(0)=0$, yields $u=P_x(Z^2/2-hZ)$. Averaging through the depth givesThe kinematic condition becomes $h_t+u(h)h_x-W(h)=-\widehat{ϵ}/\delta$ at leading order. Integration of continuity from the impermeable bed then gives

A nonzero contact speed or evaporative flux is singular in the strictly no-slip sharp-contact-line model with mobility proportional to $h^3$. A finite nonzero limiting $\overline{u}$ would require $h_{xxx}=3\overline{u}/(C{h}^2)$, which is singular for a linearly vanishing height and generally incompatible with a prescribed finite slope at the exact line. Thus a dynamic finite-contact-angle condition needs microscopic regularization, such as slip or a precursor film. In the slow-drying limit a fixed contact angle can instead be used for the leading quasistatic outer profile; it is not an exact regular finite-rate contact-line solution.

At leading quasistatic order $h_{xxx}=0$, so the symmetric profile satisfying the imposed outer angle isIts cross-sectional volume is $V={\int }_{-s}^{s}hdx=2\theta s^2/3$. The moving endpoints contribute no boundary-height term, and the end flux vanishes, giving $V_t=-2ϵs$. ConsequentlyThese are leading slow-time results up to extinction; the approximation need not remain uniform at the final vanishing scale.

To recover the small velocity, differentiate the quasistatic profile using $s_t=-3ϵ/(2\theta )$:Symmetry makes the flux zero at the centre, so $h\overline{u}=-ϵx(s^2-x^2)/(4s^2)$. Division by the height givesThe flow is inward on both sides, suggesting transport of suspended matter toward a central deposit rather than a perimeter ring. Coffee commonly leaves the opposite, ring-like pattern. A possible missing ingredient is contact-line pinning: holding the perimeter fixed makes evaporation require outward replenishment, and stronger evaporation near the edge reinforces that transport. Receding constant-angle and pinned contact lines therefore have qualitatively different deposit mechanisms.

For comparison, the same uniform-evaporation parabolic model with fixed $s$ gives ${\theta }_t=-3ϵ/s$ and $\overline{u}=ϵx/(s\theta )$, an outward velocity. This comparison isolates the boundary assumption even before adding edge-enhanced evaporation.

The receding solution is the quasistatic evaporation of a receding parabolic drop.

Separation or transformation of contaminants in liquid waste. evaporation, ion exchange and chemical precipitation concentrate radionuclides into residues that need further management.

Entry of surface water into soil pore space. The infiltration rate depends on water availability, soil permeability, capillary suction and the existing moisture profile. In a shallow overland-flow model it is a sink from the surface layer, distinct from evaporation into the atmosphere. An imposed infiltration capacity cannot remove more liquid than is locally available.

Sweating supports thermoregulation through evaporation of water, which carries away latent heat.