Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 64 / c

The Langmuir adsorption model assumes equivalent independent sites, a maximum of one adsorbate per site, no lateral adsorbate interactions, reversible adsorption-desorption equilibrium and non-dissociative adsorption here. Surface coverage obeys $\theta =bC/(1+bC)$ for the solution analogue.

BET adsorption allows multiple layers: occupation of one layer creates sites for further adsorption above it, and higher-layer binding is treated differently from first-layer binding, commonly using a condensation-like energy. Langmuir saturation is monolayer saturation; the BET model is not simply Langmuir with a different numerical equilibrium constant.

Invert the stated expression:Hence the line of $1/\Gamma$ against $1/C$ has slope $M=1/(b{\Gamma }_{\infty })$ and intercept $1/{\Gamma }_{\infty }$. Their ratio gives $b=(\mathrm{intercept})/(\mathrm{slope})$.

A least-squares straight-line fit in the requested reciprocal coordinates gives approximatelywhen the supplied numerical units are used. ThusThe slope and intercept units follow from the plotted reciprocal quantities. These parameters are the result of the requested reciprocal linear fit; fitting the original nonlinear isotherm with a different error model would give slightly different estimates.

The inferred saturation density is physically implausible for a single molecular layer in the printed units:That is far smaller than an atomic cross-section, let alone a butyric-acid molecule. The near-linear reciprocal plot therefore does not validate the physical monolayer interpretation. Possible issues include reported units or normalization, inaccessible/internal area not accounted for correctly, multilayer adsorption or other uptake processes. The numbers must not silently be relabelled as micromoles to obtain a more comfortable molecular area.

Electric-dipole sum-frequency generation spectroscopy is a coherent second-order optical probe. It is sensitive to appropriately oriented, symmetry-allowed molecular responses at the interface; randomly oriented populations can cancel in the orientational average, and centrosymmetric arrangements give no net electric-dipole second-order response.

Thus an ordered fraction of surface-bound carboxylate or hydrocarbon groups may dominate the measured spectrum, while disordered, differently oriented or weakly active adsorbates contribute little. The observed vibrational intensity is not a direct count of every molecule retained in an adsorption measurement. Selection rules, polarization and interference between contributions also matter.