Equilibirum concentration in thermodynamics

[aaaa202]

Are any of you here comfortable with thermodynamic calculations for heterophase fluctuations? Then please help me understand an important bit in my book (attached)
I am doing a project on nucleation, i.e. the study of building of clusters of different atoms. In my book (attached part) it seems important to consider what is called the equilibrium distribution of clusters, though in a system with an effective supersaturation this is apparently just a theoretical distribution not a physical one.
I would like some understanding on what the idea of considering this distribution is: When does it describe the physical distribution of clusters?
And more importantly it relates the chemical potentials as per equation 7.3. Can anyone tell me how to arrive at this kind of dependence? I tried to check the reference but couldn't find the specific calculation.

 

[DrDu]

7.3 is the usual dependence of the chemical potential μ(C) at concentration C to the chemical potential μ(C0) at some standard state with concentration C0 for an ideal solution. I don't know why they mu of the standard state G, instead.

 

[aaaa202]

Where does it come from? The expression for mu_n

 

[DrDu]

Come on, that's on the first 5 pages on every book on chemical thermodynamics ...
Basically, it is due to the increase of entropy with volume.
For an ideal gas, which is analogous to an ideal solution you argue as follows:
$dA=-SdT-PdV+\mu dN$
so you get the relation $\partial \mu/\partial V=-\partial P/\partial N$.
Use P=NRT/V to get
$d\mu =-\frac{RT}{V} dV$ for constant T and N.
Integrate from $\mu_0$ to $\mu$ on the left and from $V_0$ to V to obtain
$\mu=\mu_0-RT \ln V/V_0=\mu_0+RT \ln c/c_0$.

 

[Chestermiller]

Here is some additional information about the derivation of eqn. 7.3 for an "ideal liquid solution." For a material that approximates the behavior of an "ideal liquid solution," the partial molar volume of each species is approximately equal to the molar volume of the species in its pure state at the same temperature and pressure as the solution, and the partial molar enthalpy of each species is approximately equal to the molar enthalpy of the species in its pure state. In addition, from Smith and Van Ness: "For solutions comprised of species of equal molecular volume in which all molecular interactions are the same, on can show by the methods of statistical thermodynamics that the lowest possible value of the entropy is given by an equation analogous to" the corresponding mixture equation for an ideal gas. These assumptions lead to Eqn. 7.3. In Eqn. 7.3, G(n) is supposed to be the free energy of the pure species n at the same temperature and pressure as the solution.

Chet