# Confusing Chemical potential of mixtures

I feel that there are very few textbook that treat the chemical potential of mixtures in an understandable clear way, which is why I wanted to ask here about certain things? Although I do not have a single question, but 5 of them, I think that it is merely impossible to separate them thematically, which is why I hope to meet an expert here.

If we have an ideal gas then if we know the chemical potential for a particular (p,T) we have:

$$\mu(p_2,T) \approx \mu(p_1,T) + \left(\frac{\partial \mu}{\partial p} \right)_T(p_1,T) (p_2-p_1) = \mu(p_1,T) + \frac{V}{N}(p_2-p_1).$$

Similarly for changes with respect to temperature we have $$\mu(p,T_2) \approx \mu(p,T_1) + \left(\frac{\partial \mu}{\partial T} \right)_p(p,T_1) (T_2-T_1) = \mu(p,T_1) - \frac{S}{N}(T_2-T_1).$$

Now, if we have either a gas or a liquid (Question 1: is this true, does this law hold for both fluids?) $A$ and put another material $B$ into it, then we have by Raoult's law:

$$\mu_{A,\text{ with mixed B}}(p,T) = \mu_A(p,T) + RT ln( \frac{N_A}{N_A + N_B} )$$

Question 2: Now, is the chemical potential of the total fluid given by $A$ and $B$ then the sum $\mu_{A+B} = \mu_{A,\text{ with mixed B}}(p,T) + \mu_{B,\text{ with mixed A}}(p,T)$?

Let's imagine that $A$ and $B$ would be gases, would this mean that

Question 3 : $$\mu_{A,\text{ with mixed B}}(p,T) = \mu_A(p,T) + RT ln( \frac{p_A}{p_A + p_B} ) ?$$

Question 4 : And what if $A$ and $B$ would be two two miscible liquids could we still use the vapour pressures to calculate the chemical potential of the liquids mixture?

$$\mu_{A,\text{ with mixed B}}(p,T) = \mu_A(p,T) + RT ln( \frac{p_A}{p_A + p_B} ) ?$$

Now, imagine that I have a liquid $A$ and salt $B$ in it. Also, I have a gas layer $C$ of $A$ above the liquid. What exactly is now in equilibrium $\mu_{A+B} = \mu_{C}$ or $\mu_{A,\text{ with mixed B}} = \mu_{C}$?

Question 5: My last question would be: Imagine that you have a liquid(let's say water) and some gaseous water phase above (according to the vapour pressure). Now, you add oxygen to this. What would this mean for the chemical potentials of the liquid/gas water?

• Shouldn't this be in chemisty? – ʇolɐǝz ǝɥʇ qoq Feb 12 '15 at 0:49
• I think chemical potential is also considered in Statistical Physics – Xin Wang Feb 12 '15 at 14:52
• What is "chemical potential of the total fluid"? I suspect that there is no such thing. – Steve Byrnes Feb 12 '15 at 23:27
• @SteveB well, the question is: if I have to liquids that are miscible and I put them together, then I get a liquid. How do I calculate the chemical potential of this liquid? – Xin Wang Feb 12 '15 at 23:56
• I don't think there is any sensible or useful way to define chemical potential for a liquid mixture. So I would say it has no chemical potential. Lots of things do not have a chemical potential: The color orange, beauty, etc. :-D Chemical potential is a useful concept to the extent that something maintains its identity in different environments, and is usually (or always) conserved, loosely speaking. So it's a very useful concept for atoms or molecules, but not really for a liquid mixture, since the composition can easily and smoothly change. – Steve Byrnes Feb 13 '15 at 16:06