= Solution
\b[Molecular <area> from the <adsorption> slope.] The supplied <Gibbs adsorption equation> identifies the <surface excess> <number density> $\Gamma=1/\Sigma$ through
$$
\Gamma=-\frac1{k_BT}\frac{d\gamma}{d\ln c},\qquad
\Sigma=-\frac{k_BT}{d\gamma/d\ln c}.
$$
The 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 $0.055\,\mathrm{mol\,dm^{-3}}$ uses its two bracketing points:
$$
\frac{d\gamma}{d\ln c}\simeq
\frac{(45.2-48.5)\times10^{-3}}{\ln(0.06/0.05)}
=-1.810\times10^{-2}\,\mathrm{N\,m^{-1}}.
$$
Taking room <temperature> as $298\,\mathrm K$,
$$
\boxed{\Sigma\simeq\frac{4.11\times10^{-21}\,\mathrm J}
{1.810\times10^{-2}\,\mathrm{J\,m^{-2}}}
=2.27\times10^{-19}\,\mathrm{m^2}\simeq0.23\,\mathrm{nm^2}.}
$$
This corresponds to about $4.4$ adsorbed molecules per $\mathrm{nm^2}$. 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 $0.26\,\mathrm{nm^2}$, while the higher-concentration region gives roughly $0.22\,\mathrm{nm^2}$. An effective <area> of order \b[$0.2$–$0.3\,\mathrm{nm^2}$] is justified; a concentration-independent exact value is not.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2017-q3a-surface-tension.png]
{title=Surface-tension interpolation and extrapolated wetting threshold, alongside the local Gibbs adsorption slope}
\b[<Contact angle>.] Linear interpolation between $0.05$ and $0.06\,\mathrm{mol\,dm^{-3}}$ gives $\gamma_{LV}(0.055)=(48.5+45.2)/2=46.85\,\mathrm{mN\,m^{-1}}$. For an ideal equilibrium <contact angle> measured through the liquid, <Young's equation> is
$$
\gamma_{SV}-\gamma_{SL}=\gamma_{LV}\cos\theta.
$$
The supplied solid-vapour and solid-liquid values have difference $40\,\mathrm{mN\,m^{-1}}$. Hence
$$
\boxed{\theta=\cos^{-1}\left(\frac{300-260}{46.85}\right)
=31.4^\circ.}
$$
This assumes clean homogeneous equilibrium interfaces, negligible contact-angle hysteresis, and the stated solid interfacial energies applicable to the solution.
\b[Complete wetting.] The <spreading coefficient> is $S=\gamma_{SV}-\gamma_{SL}-\gamma_{LV}=40\,\mathrm{mN\,m^{-1}}-\gamma_{LV}$. The onset of <complete wetting> occurs at $S=0$, or $\gamma_{LV}=40\,\mathrm{mN\,m^{-1}}$, if the solid interfacial energies remain unchanged. The last measured value is still $42.0\,\mathrm{mN\,m^{-1}}$, so this requires extrapolation beyond the table.
Continuing the last local slope in concentration, $(42.0-45.2)/(0.07-0.06)=-320\,\mathrm{mN\,m^{-1}}$ per $\mathrm{mol\,dm^{-3}}$, gives
$$
\boxed{c_{\rm wet}\simeq0.07+\frac{42.0-40.0}{320}
=0.0763\,\mathrm{mol\,dm^{-3}}.}
$$
Continuing a constant local Gibbs slope in $\ln c$ instead gives approximately $0.078\,\mathrm{mol\,dm^{-3}}$. Thus the data suggest a threshold near \b[$0.08\,\mathrm M$ 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.
\b[Capillary rise.] For a circular capillary of radius $a$, vertical surface-tension <force> $2\pi a\gamma_{LV}\cos\theta$ balances the weight $\rho g\pi a^2h$. Thus <capillary action> gives
$$
h=\frac{2\gamma_{LV}\cos\theta}{\rho ga}
=\frac{2(\gamma_{SV}-\gamma_{SL})}{\rho ga}.
$$
Using $a=0.5\,\mathrm{mm}=5.0\times10^{-4}\,\mathrm m$, $\rho=10^3\,\mathrm{kg\,m^{-3}}$ and $g=9.81\,\mathrm{m\,s^{-2}}$,
$$
\boxed{h=\frac{2(0.040)}{(10^3)(9.81)(5.0\times10^{-4})}
=1.63\times10^{-2}\,\mathrm m\simeq16\,\mathrm{mm}.}
$$
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.
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