How can be measured "surface enthalpy of water" and what is its thermodynamic relevance?
Article Hydrophobic stacking of bases and the solvent denaturation of DNA
Thermodynamically the enthalpy is a measure for an energy i.e. the heat content of a mass (specific enthalpy) or a molecular amount of substance (molar enthalpy). This quantity of energy is a function of the temperature of the substance. Also, the heat capacity (cp and cv) is characteristic for the substance's capability to store/transfer energy/heat. Cp/cv can be used to calculate the enthalpy.
What the description "surface" enthalpy should be I cannot explain.
I didn't read this paper, but I suppose it is related to the surface tension. The surface energy quantifies the energy of intermolecular forces between cohesion and adhesion, when a liquid forms a surface and this is the surface tension of liquids.
From the contour and the contact angle of a liquid drop with its surface, it is possible to determine the surface energy using the Young-Laplace equation. For constant temperature and pressure it defines the free enthalpy (Gibbs energy) and at constant temperature and volume the free energy (Helmholtz energy).
For a typical application follow this link.
http://www.fz-juelich.de/iek/iek-2/EN/Forschung/Thermochemie/Methoden/Methoden_Uebersicht/ErhitzMikrosk/ErhMikroskop_node.html
I found a clear definition of the surface enthalpy and surface free energy in Eqs. 1-3 of J. Colloid and Interface Science, 263, 2003, 554-561
h[surface] = u[surface] = f[surface] +T*s[surface]
where h is enthalpy, u is internal energy, f is free energy, s is entropy per unit area.
If one has pure surface, then
f[surface] = sigma,
s[surface] = -dsigma/dT
and h is easily calculated from surface tension data. If there is something adsorbed, things are not so simple. Check the "Surface tension and adsorption" book of Prigogine and Defay for details.
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