Clas Blomberg - Physics of life-Elsevier Science (2007)

Chapter 14. Thermodynamics formalism and examples 123
with n i the number of molecules of type “i”, and x i n i /N TOT the corresponding molecular
ratio (concentration). One might also have volume or mass concentrations:
S mix R n i ln(c i ) S mix R n i ln(c i m ) (14.12)
The difference between different expressions here is a term that can be included in the first part
of the chemical potential. It is not necessary to bother about that term in our development here.
These terms provide contributions to the concentration dependent part of the chemical
potential of a particular component with the volume concentration (the most common choice):
m i mix RT ln(c i ) (14.13)
The use of a concentration in this expression is a simplification that takes the mixing entropy
into account but not other type of interactions between various components. As ln(c) this
expression is important at very low concentrations also (when the logarithm diverges) and
then often dominates over other contributions. The expression should be modified for larger
concentrations and situations where various interactions between different kinds of molecules
are important. As discussed in the section about water, this is often the case of water solutions,
and this simple expression is, for instance, hardly a good relation even for small concentrations
of organic hydrocarbon compounds in water solution. In such cases, the concentration
in the logarithm is substituted by a more general quantity, activity a i :
m i mix RT ln(a i ) (14.14)
We will, in the further development, mainly use volume concentrations.
14B
Mixing entropy
Again, back to entropy and spreading of atoms in a room. Atoms are spread uniformly over
a room at not too fine length scales as this is the most probable way to distribute atoms.
This is an entropy effect and when one considers gases with principles like Boyle–Mariotte’s
law (that pressure is inversely proportional to volume), the pressure is indeed a result of
entropy—the entropy grows as the volume increases. A force due to an expanding gas is
mainly an entropy effect. Indeed, energy changes very little if at all while expanding.
The energy tendency is rather to decrease the volume as atoms and molecules attract each other.
We now turn to the mixing of two or more substances, which still more emphasises the
entropy concept. In the next Section 15A, there will be an example of card shuffling which
has much in common with substance mixing. If we have two kinds of molecules, there are
many more possibilities to have them mixed than having them separated, one type at one
side, and the other type at another side. There is a gain in entropy to mix molecules together,