I made a stupid sign mistake in one of my video lectures and have spent part of the weekend sorting things out in my mind. So here is a note to self before while it is still fresh in my mind.
The non-polar free energy of solvation can be written as
\Delta G_{\text{np-solv}} = \gamma_{\text{np}} SASA
where where SASA is the solvent accessible surface area. The argument is that one of the main contributions to \Delta G_{\text{np-solv}} is the energy required to make the molecular cavity in the solvent, which, for macroscopic objects, is a function of the surface tension of the liquid \gamma and the surface area of the cavity.
\Delta G_{\text{np-solv}} \propto \gamma SASA
\gamma is positive for water so \Delta G_{\text{np-solv}} is positive in water.
So far, so good. But this simple picture fails when considering the solvation entropy
\Delta S_{\text{np-solv}} = - \left( \frac{\partial \gamma_{\text{np}}}{\partial T} \right) SASA
For water the bulk surface tension decreases with increasing temperature as you would expect, which suggests that \Delta S_{\text{np-solv}} is positive when in fact it is observed to be negative.
So if \gamma_{\text{np}} has anything to do with \gamma this would imply that the surface tension associated with molecular-sized cavities increase with temperature. It is not clear why that would be so and this, in part, has led Graziano to argue that, effectively, \gamma_{\text{np}} has nothing to do with \gamma but is a strictly empirical parameter.
A little more detail that ultimately doesn't shed any more light
\gamma_{\text{np}} is also positive but not equal to \gamma. One reason is that \Delta G_{\text{np-solv}} also contains contributions from repulsion and dispersion interactions with the solute. However, if one computes \Delta G_{\text{np-solv}} from hard sphere simulations the corresponding \gamma_{\text{np}} values still does not match the \gamma value for bulk water.
Tolman has argued that the surface tension depends on the curvature of the surface and suggested the following approximation
\gamma (R) = \gamma \left( 1 - \frac{2\delta}{R} \right)
where R is the cavity radius and \delta is a parameter called the Tolman length. When R < 2\delta
\frac{\partial \gamma (R)}{\partial T} = \left( \frac{\partial \gamma}{\partial T} \right) \left( 1 - \frac{2\delta}{R} \right)
will indeed be positive, but only when \Delta G_{\text{np-solv}} is negative. What is observed is positive \Delta G_{\text{np-solv}} and \Delta S_{\text{np-solv}}.
This work is licensed under a Creative Commons Attribution 4.0
The non-polar free energy of solvation can be written as
\Delta G_{\text{np-solv}} = \gamma_{\text{np}} SASA
where where SASA is the solvent accessible surface area. The argument is that one of the main contributions to \Delta G_{\text{np-solv}} is the energy required to make the molecular cavity in the solvent, which, for macroscopic objects, is a function of the surface tension of the liquid \gamma and the surface area of the cavity.
\Delta G_{\text{np-solv}} \propto \gamma SASA
\gamma is positive for water so \Delta G_{\text{np-solv}} is positive in water.
So far, so good. But this simple picture fails when considering the solvation entropy
\Delta S_{\text{np-solv}} = - \left( \frac{\partial \gamma_{\text{np}}}{\partial T} \right) SASA
For water the bulk surface tension decreases with increasing temperature as you would expect, which suggests that \Delta S_{\text{np-solv}} is positive when in fact it is observed to be negative.
So if \gamma_{\text{np}} has anything to do with \gamma this would imply that the surface tension associated with molecular-sized cavities increase with temperature. It is not clear why that would be so and this, in part, has led Graziano to argue that, effectively, \gamma_{\text{np}} has nothing to do with \gamma but is a strictly empirical parameter.
A little more detail that ultimately doesn't shed any more light
\gamma_{\text{np}} is also positive but not equal to \gamma. One reason is that \Delta G_{\text{np-solv}} also contains contributions from repulsion and dispersion interactions with the solute. However, if one computes \Delta G_{\text{np-solv}} from hard sphere simulations the corresponding \gamma_{\text{np}} values still does not match the \gamma value for bulk water.
Tolman has argued that the surface tension depends on the curvature of the surface and suggested the following approximation
\gamma (R) = \gamma \left( 1 - \frac{2\delta}{R} \right)
where R is the cavity radius and \delta is a parameter called the Tolman length. When R < 2\delta
\frac{\partial \gamma (R)}{\partial T} = \left( \frac{\partial \gamma}{\partial T} \right) \left( 1 - \frac{2\delta}{R} \right)
will indeed be positive, but only when \Delta G_{\text{np-solv}} is negative. What is observed is positive \Delta G_{\text{np-solv}} and \Delta S_{\text{np-solv}}.
Ashbaugh has pointed out that a temperature-dependent \delta solves this problem but Graziano fired back that since there is no analytical form for \delta, \frac{\partial \delta}{\partial T} is just another temperature dependent parameter, and you might as well use \frac{\partial \gamma_{\text{np}}}{\partial T} as a parameter (I am paraphrasing here).
This work is licensed under a Creative Commons Attribution 4.0
