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Measurement of time-averaged scattering intensity as a function of angle and concentration. For suitable dilute polymer solutions its small-wave-number behavior gives a radius of gyration; intensity weighting and interparticle correlations require attention.

In the steady nondiffusive gravity–Marangoni model with capillarity neglected, zero surfactant flux and positive concentration make the surface velocity zero. With $Q=-H^2{\Gamma }_X/2-H^3{H}_X/3$ and $J=-H\Gamma {\Gamma }_X-H^2\Gamma H_X/2$, integration gives $H^4=H_0^4-48QX$ and $\Gamma =1+(H_0^2-H^2)/4$. A concentration drop $\delta$ over unit length therefore selects $Q=-\delta (H_0^2+2\delta )/6$. The liquid flows opposite to the imposed Marangoni traction, with velocity proportional to $H_{X}Y(Y-H)/2$ and an immobilized surface. This reduction is singular at a dry endpoint; its small-slope assumption does not resolve that edge.

A stock solution is a chemical solution kept as a source from which other chemical solutions are prepared, commonly by dilution. Its known concentration makes repeated preparations traceable to one starting composition. A stock used for measurement calibration must also satisfy the requirements of a standard solution; the word stock alone does not establish that status. The original EPA Method200.8 defines calibration standards as dilutions of stock standards used to relate instrument response to analyte concentration.^[ref]

Suppose an invented stock solution has molar concentration $c_s=10.0\mathrm{mmol}{\mathrm{L}}^{-1}$. To obtain final volume $V_f=50.0\mathrm{mL}$ at $c_f=0.200\mathrm{mmol}{\mathrm{L}}^{-1}$, conservation of dissolved amount of substance givesTransfer that aliquot and add solvent to the final 50.0-millilitre mark of a volumetric flask, then mix. Adding 50.0 millilitres of solvent would specify a different operation: the equation requires final total volume.

The same inventory reasoning underlies serial dilution. A tenfold dilution followed by a fivefold dilution gives the same final concentration reduction of50, provided each transferred portion is representative and the counted solute is conserved. Reactions, losses or incomplete mixing would change those premises.

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.

A uniform-permeability mushy layer on an impermeable eutectic floor, under the quadratic roof pressure $p_0-E^2{x}^2/2$, has Darcy flux $\mathbf{q}=\alpha (x,-z)$, where $\alpha =\Pi E^2/\mu$. With pulling velocity $-V{\mathbf{e}}_z$, pure primary crystals, solid fraction $\varphi$, liquid concentration $C$ and constant phase properties, its one-dimensional balances areThe solidification product at the eutectic floor retains bulk composition $C_s=(1-\varphi (0))C_E$. Integration gives $C_s=C_0-(\alpha /V){\int }_0^h(C-C_0)dz$: horizontal strain removes solute before the material freezes. A negative predicted $C_s$ is inadmissible, and diagnoses failure of the assumed steady branch rather than a negative-composition product.

A streakline at observation time $T$ is the set of present positions of all fluid elements that passed through one fixed release point $\mathbf{a}$ during a specified history of injection. For continuous release over ${\tau }_0\le \tau \le T$, solve a different initial value problem for each release time $\tau$:Thus a pathline varies $t$ for one particle; a streakline varies $\tau$ while holding $T$ fixed. Interrupted release gives only the corresponding subsets of the streakline. A moving nozzle requires a time-dependent release position instead, so the standard fixed-point definition should not silently be applied to it. Continuous release makes streaklines easy to observe and useful for showing where material from one source has reached. ^[ref]

In the analytical velocity field $(U,at)$ of pathline, a particle released at the origin at time $\tau$ followsObserve all particles together at time $T$, after continuous release from $\tau =0$:For $U=1\mathrm{m}{\mathrm{s}}^{-1}$, $a=1\mathrm{m}{\mathrm{s}}^{-2}$ and $T=2\mathrm{s}$, the streakline is the downward-curving parabola $y=2x-x^2/2$ in metre coordinates. Releases at $\tau =0,1,2\mathrm{s}$ occupy $(2,2)$, $(1,1.5)$ and $(0,0)\mathrm{m}$, respectively. At $x=1\mathrm{m}$, the streakline lies above the pathline $y=x^2/2$ and below the origin's streamline $y=2x$. All particles nevertheless have the same present velocity $(1,2)\mathrm{m}{\mathrm{s}}^{-1}$. The streakline slope $dy/dx=a\tau /U$ generally differs from the present velocity slope $aT/U$; they agree at the nozzle. The oldest particle is the downstream endpoint, which explains why the pathline and streakline meet there without being the same curve.

Each panel fixes one observation time. Coloured dots are the present positions of particles with the indicated release times; dashed pathlines show those particles' earlier histories. The black streakline connects present positions of the continuously released family, including release times between the sampled dots. After $T=1\mathrm{s}$, particles already present keep moving and new ones enter at the origin, so the later streakline is not a frozen record of the earlier one. Injection beginning only at $\tau =1\mathrm{s}$ would show just $0\le x\le 1\mathrm{m}$ of the $T=2\mathrm{s}$ curve. Both panels are diffusion-free analytical models, not experimental photographs.

Kline's contracting-channel demonstration switches from marking an element to continuous bubble production at a marking wire: the resulting trace is a streakline. The apparatus therefore changes the curve being displayed by changing the marking history, even with the same channel. A wire produces a family of traces; conceptually each fixed emitting location supplies its own streakline. Individual bubbles are markers rather than perfect fluid elements: Jean Hertzberg's teaching notes describe hydrogen production by electrolysis at a thin cathode wire and warn that larger bubbles have more buoyancy. ^{[ref] [ref]}

A dye version uses a small fixed injection port upstream in a transparent water channel, a dilute premix of dye with the channel water, illumination and a fixed optical camera. Start continuous injection at a recorded time ${\tau }_0$ and take a short exposure at $T$: the visible filament approximates a streakline, not the accumulated pathline of one particle. Repeat with the same port and imaging geometry after changing the flow with time; compare the filament against local directions estimated from short displacements of sparse markers at $T$. Agreement in a steady flow and possible disagreement when the flow changes are the predictions being tested; no measured curves are claimed here. The protocol follows Hertzberg's advice to use small ports, little injected volume and a dilute premix with matched density and viscosity. Check for nozzle disturbance by reducing injection rate, for buoyancy by watching the premix in still water, and for diffusion by comparing filament widths at different ages. Keep exposure short enough to suppress motion blur. Dye concentration approximates a passive scalar only when it has negligible feedback on the velocity field; finite port width, diffusion and out-of-plane motion limit how literally a visible filament can be read as a geometrical streakline. ^[ref]

Strigol is a particular strigolactone identified as a chemical stimulant of germination in root-parasitic plants. A 1966 study isolated crystalline strigol from cotton root exudates and reported its molecular formula as ${\mathrm{C}}_{19}{\mathrm{H}}_{22}{\mathrm{O}}_6$. The substance is one identified molecule, whereas strigolactone names a broader chemical family.^[ref]

The motivating experiment links chemical separation with a biological assay. An active root-exudate fraction can be purified and tested for its capacity to stimulate seeds; chemical identification then connects that activity with a defined compound. Germination is the measured outcome. It does not by itself establish successful attachment to a host or later growth of the parasite.

An invented dilution calculation illustrates why a small amount and a small concentration are different descriptions: $1\mathrm{pmol}$ in $1\mathrm{mL}$ is $1\mathrm{nM}$. A tenfold dilution gives $0.1\mathrm{nM}$ while also changing the delivered amount if the assay volume is fixed. These are teaching values, not thresholds reported by the1966 study. A germination comparison must state its concentration, conditions and control treatment.

In water, a strong acid transfers its available proton essentially completely to water under the dilute conditions considered here. For the pH example, each initial chemical acid unit supplies one hydronium ion. “Strong” describes this conversion, rather than a high concentration. See 14.3 Relative Strengths of Acids and Bases.

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.

In the aqueous model here, a strong base supplies hydroxide essentially completely under the conditions considered. Strength and concentration are different properties.

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]

Surface diffusion transports particles along an interface down a concentration gradient. On an isotropic smooth interface with constant diffusivity $D_s$, its flux is $-D_s{\nabla }_{s}C$ and its concentration source is $D_s{\Delta }_{s}C$, using the surface gradient and surface Laplacian.

When attachment is much slower than exterior diffusion, the surface concentration stays close to the reservoir value. An attachment coefficient $k_s$ with units of velocity gives nearly constant radial growth in the dilute-feed approximation. For a sphere the ratio $k_{s}a/D$ distinguishes attachment control from transport control. A finite attachment rate leads to a Robin boundary condition rather than a fixed equilibrium surface concentration.

Surrogate-matrix response equivalence means that a specified surrogate sample matrix supports an adequately comparable analytical response to the authentic sample matrix over the method's intended range. Equivalence is tied to an analyte, preparation, detector and decision criterion. It does not require the materials to have identical chemical composition.

ICH M 10 calls for assessment of differences in analytical matrix effects and extraction recovery between surrogate and authentic materials, and evaluation of response parallelism across the range.^[ref] A useful experimental design processes authentic unspiked samples, authentic samples with known analyte increments and spiked surrogate material through the same procedure. The endogenous contribution in authentic material must be accounted for.

An original linear measurement model makes a one-point comparison's limitation visible. Let the surrogate response be $y_s=2c$, and the authentic response be $y_a=5+c$, where $c$ is the analyte concentration in stipulated units. Both give response 10 at $c=5$. At $c=15$, however, authentic material gives 20, and the surrogate calibration reports $c=10$. Agreement at the crossing point concealed both a slope difference and an offset.

Matched increments remove a constant endogenous offset: $\Delta y_a=s_a\Delta c$ and $\Delta y_s=s_s\Delta c$ in a linear range. Their ratio estimates $s_a/s_s$, provided additions do not themselves change the sample matrix or preparation. Similar slopes alone still do not establish identical blank interference, stability or behavior outside that range.

Set a comparison tolerance appropriate to the analytical use and examine uncertainty as well as nominal agreement. A wide uncertainty interval containing equal response is not strong evidence of equivalence. Likewise, matching a ratio normalized by an internal standard may be useful for that method while concealing unequal absolute extraction recovery; state which response has been compared.

Suspended sediment is particulate material carried within a fluid rather than solely along the bed. Its concentration balance includes advection, erosion into suspension and deposition from suspension. For water depth $h$ and suspended mass concentration $c$, the conservative balance $(hc{)}_t+(huc{)}_x={\rho }_s(v_E-v_D)$ reduces using water continuity to $h(c_t+u{c}_x)={\rho }_s(v_E-v_D)$.

Suspension rheology relates particle concentration, particle pressure, shear stress, deformation rate, and fluid migration in a mixture of particles and a liquid.

A pharmaceutical syrup is a concentrated aqueous sugar preparation that can serve as a vehicle for other ingredients. Sugar concentration influences viscosity and storage behavior. The named vehicle and any active drug dissolved in it play different roles.

Takeru Higuchi, credited as T. Higuchi in his 1963 paper, developed mathematical descriptions of diffusion-controlled drug release from solid matrices. The University of Kansas archive holds that exact paper among his personal papers, establishing the connection between the initial-only byline and the full name.^[ref]

The paper distinguished a uniform matrix serving as the diffusional medium from a granular matrix penetrated by a solvent. It considered planar and spherical geometries, rather than treating all sustained release as a single first-order process.^[ref] The resulting models connect formulation geometry and material properties with a measurable drug release rate.

A simplified planar derivation illustrates the Higuchi equation. Assume a stationary matrix with constant diffusion coefficient $D$, excess undissolved drug, a perfect sink condition in drug release at its exposed face, and a slowly moving depletion boundary at depth $h(t)$. Approximate the dissolved-drug concentration as linear from zero at the exposed face to its solubility $C_s$ at that boundary. If initial total loading per unit volume is $C_0$, the amount released per exposed area isbecause the depleted layer retains a triangular dissolved-drug profile. The outward mass flux is approximately $D{C}_s/h$, so mass conservation givesThus the cumulative released amount is proportional to $\sqrt{t}$ in this model, whereas its derivative decreases as $t^{-1/2}$. The high-loading approximation $C_0\gg C_s$ makes the slowly moving boundary assumption more plausible; the result is not a general solution for an arbitrary dissolving or swelling material.

For an original illustration, let $Q=k_H\sqrt{t}$ with $k_H=2\mathrm{mg}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{h}}^{-1/2}$. The cumulative release is $2\mathrm{mg}\mathrm{c}{\mathrm{m}}^{-2}$ at one hour and $4\mathrm{mg}\mathrm{c}{\mathrm{m}}^{-2}$ at four hours, but the instantaneous rates per area are respectively $1$ and $0.5\mathrm{mg}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{h}}^{-1}$. These are chosen model values. They show why sustained release need not be constant-rate release. Finite thickness, depletion of the solid phase, changing solubility or external transport resistance can end this regime. Measuring release tests the model's transport assumptions; a square-root fit alone does not uniquely identify them.

For interstitial velocity $u=6U\xi (1-\xi )$, $\xi =z/h$, reflecting transverse walls and molecular diffusion $D$, solve $D{\chi }^{\prime \prime }=u-U$ with zero mean and zero normal derivative. One solution is $\chi =(U{h}^2/D)({\xi }^3-{\xi }^4/2-{\xi }^2/2+1/60)$. Averaging the advection-diffusion equation gives enhancement $-\overline{(u-U)\chi }=D\overline{({\chi }^{\prime }{)}^2}=U^2{h}^2/(210D)$. The approximation requires transverse equilibration and slowly varying longitudinal concentration.

Suppose a reduced steady balance is $P=A{X}^2+BX$, where $X=[{\mathrm{O}}_3]$, and hold $P$ and all specified background concentrations fixed. With $u=1/T$, $d\ln A/du=-a$ and $d\ln B/du=-b$, implicit differentiation gives $d\ln X/du=(aA{X}^2+bBX)/(2A{X}^2+BX)$. The quadratic loss term includes temperature-dependent O/O3 partitioning as well as the direct reaction coefficient. Holding the loss ratio fixed over a finite change is an additional approximation; solving the new quadratic provides a check.