CCL: translational entropy and solvation
- From: "Raphael Ribeiro"
<raphaelri/a\hotmail.com>
- Subject: CCL: translational entropy and solvation
- Date: Tue, 9 Dec 2008 17:19:14 -0500
Sent to CCL by: "Raphael Ribeiro" [raphaelri a hotmail.com]
Dear Michael,
I think you forgot that the mathematical expression that you're talking about
for translational entropy is based on the ideal gas model, the partition
function in this case is calculated using the particle in a box model.In this
case the problem of calculating the energy levels of the whole system is reduced
to calculating the energy levels of a molecule (translational,rotational and
vibrational energy levels of the molecule). Also,and more important, as you
might know in this formulation no intermolecular forces or external fields are
considered.
If you look at Landau and Lifschitz, Vol.5 Statistical Physics, you are going to
see that when he talks about the ideal Boltzman gas(pages 120-123) he derives
the equation of state of an ideal gas using a partition function that was
calculated using the translational energy levels we're talking about
(particle-in-a-box energy levels) and also the energy levels of the internal
motion of the ideal gas. This means that systems in which the translational
partition function is calculated by the particle-in-a-box model have the
equation of state of an ideal gas. That is why I can not agree that
translational partition functions are the same in the ideal gas phase and in the
solvated phase.
Again, http://biophysics.med.jhmi.edu/amzel/people/siebert/strsl_2col_bw_letter.pdf
in this article the authors talk about a statistical mechanics framework for the
estimation of translational entropy loss in associations while taking
explicitely into account the intermolecular interactions between the solute and
the solvent, it is called CM formalism.
The following quote from the paper corroborates what I said before about ideal
gas translational partition function and non-ideal gas translational partition
function, by talking about the entropy in different systems (entropy and
partition functions are directly related).
"Thus, the estimation of dStrans depend solely on the evaluation of the
integral in Eq.2 which, after integration over the momenta dp, yields:
Eq. 4
where (De Broglie wavelenght) = h/(2piMkT)^1/2 is the de Broglie wavelength of
a solute molecule.
If the N solutes were an ideal gas (i.e. without solvent nor intermolecular
interactions, (U=0), the integral in eq. 4 would be equal to V, leading to the
so called "Sackur-Tetrode"(ST) formula for translational entropy[14]:
Eq.5
This formula is sometimes used as the basis for estimating entropies of
molecular associations in solution[3,13,16]. However, the presence of a nonzero
potential U complicates considerably the evaluation of the configurational
integral in Eq.4, requiring approximate methods[1]. The goal of this paper is to
provide a framework based on cell models [2,15,17,18], to approximate
efficiently dStrans."
As you might have noticed the presence of an external potential (create by
intermolecular forces) changes the way(ideal gas phase model) we calculate
translational entropies. And that is why people make models to calculate this
component of entropy (as it was done in this paper) and understand the role of
translational component of entropy in some situations. This other article http://www.biophysj.org/cgi/content/abstract/89/4/2701 uses
a sophisticated statistical mechanic theory to analyse the role of translational
entropy of solvent molecules in the proteing folding process.
To conclude, the molecular partition function we've learned and are used to work
with is very good for ideal gases system, but not for solvated systems, as it
does not include molecular interactions and without interactions there would be
no solvated phase.
Raphael Ribeiro