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Molecular area from the adsorption slope. The supplied Gibbs adsorption equation identifies the surface excess number density $\Gamma =1/\Sigma$ throughThe logarithm is understood relative to a fixed concentration unit or reference concentration; only concentration ratios enter its differences. The pure-water point $c=0$ supplies a baseline surface tension, but cannot be included in $\ln c$.

An estimate near the later working concentration $0.055\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$ uses its two bracketing points:Taking room temperature as $298\mathrm{K}$,This corresponds to about $4.4$ adsorbed molecules per $\mathrm{n}{\mathrm{m}}^2$. The data are not perfectly linear in $\ln c$, so the inferred molecular area depends somewhat on the concentration range: a fit to all five positive-concentration points gives about $0.26\mathrm{n}{\mathrm{m}}^2$, while the higher-concentration region gives roughly $0.22\mathrm{n}{\mathrm{m}}^2$. An effective area of order $0.2$–$0.3\mathrm{n}{\mathrm{m}}^2$ is justified; a concentration-independent exact value is not.

Contact angle. Linear interpolation between $0.05$ and $0.06\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$ gives ${\gamma }_{LV}(0.055)=(48.5+45.2)/2=46.85\mathrm{mN}{\mathrm{m}}^{-1}$. For an ideal equilibrium contact angle measured through the liquid, Young's equation isThe supplied solid-vapour and solid-liquid values have difference $40\mathrm{mN}{\mathrm{m}}^{-1}$. HenceThis assumes clean homogeneous equilibrium interfaces, negligible contact-angle hysteresis, and the stated solid interfacial energies applicable to the solution.

Complete wetting. The spreading coefficient is $S={\gamma }_{SV}-{\gamma }_{SL}-{\gamma }_{LV}=40\mathrm{mN}{\mathrm{m}}^{-1}-{\gamma }_{LV}$. The onset of complete wetting occurs at $S=0$, or ${\gamma }_{LV}=40\mathrm{mN}{\mathrm{m}}^{-1}$, if the solid interfacial energies remain unchanged. The last measured value is still $42.0\mathrm{mN}{\mathrm{m}}^{-1}$, so this requires extrapolation beyond the table.

Continuing the last local slope in concentration, $(42.0-45.2)/(0.07-0.06)=-320\mathrm{mN}{\mathrm{m}}^{-1}$ per $\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$, givesContinuing a constant local Gibbs slope in $\ln c$ instead gives approximately $0.078\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$. Thus the data suggest a threshold near $0.08\mathrm{M}$ under continued-trend assumptions, rather than fixing one exactly. A surface-tension plateau caused by micellization, or a change in ${\gamma }_{SL}$, could prevent or shift this threshold; no such extra data are given.

Capillary rise. For a circular capillary of radius $a$, vertical surface-tension force $2\pi a{\gamma }_{LV}\cos \theta$ balances the weight $\rho g\pi a^{2}h$. Thus capillary action givesUsing $a=0.5\mathrm{mm}=5.0\times 10^{-4}\mathrm{m}$, $\rho =10^3\mathrm{kg}{\mathrm{m}}^{-3}$ and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$,This is the height above the reservoir level in the ideal static capillary approximation; meniscus-volume corrections and contact-angle hysteresis are neglected. The positive $\cos \theta$ gives a rise, rather than a depression.

Thin-layer chromatography spreads a thin chromatographic stationary phase on a supporting plate. A small sample spot starts on a pencil origin line. The lower edge contacts the eluent, whose level must stay below that origin; capillary action draws the chromatographic mobile phase upward. Components exchange with the chromatographic stationary phase as the wet region advances. The leading wet edge is the solvent front. Mark it after removing the plate, before solvent evaporation erases it.^[ref]

Juniata College's Science in Motion protocol grinds vegetable tissue with about $5\mathrm{mL}$ of acetone, transfers the extract, adds about $2\mathrm{mL}$ of hexane, and specifies sampling the upper layer after separation. It spots a silica gel plate with a capillary tube, drying between repeated applications; a reference pigment occupies another lane. A closed jar contains about $0.5\mathrm{cm}$ depth of 70:30 hexane–acetone eluent, below the pencil origin. The plate is removed before the front reaches the top; the front is marked immediately. Spot centres and front are measured from the origin.^[ref]

The protocol's standard provides a comparison control. It specifies no added water in this extraction and reports no measured plate positions or temperature. Its tabulated pigment values are guidance, not observations from this run.^[ref]

A useful additional control is an eluent blank, checking contamination. Compare sample and standard on the same plate, and optionally apply both together at a third origin position. Agreement under identical conditions supports a proposed identity but cannot prove it: different substances can migrate together. A second eluent giving different selectivity strengthens the comparison. One spot cannot establish purity, and a missing visible spot can reflect poor detection rather than absence. Overloading, broad spots, or an uneven solvent front also weaken the readout. These are interpretation checks, not extra observations claimed for the protocol.