Editing Raoult's law (section)

==Thermodynamic considerations==
Raoult's law was first observed empirically and led [[François-Marie Raoult]]<ref name = F-MRaoult1 /><ref name = F-MRaoult2 /> to postulate that the vapor pressure above an ideal mixture of liquids is equal to the sum of the vapor pressures of each component multiplied by its mole fraction.<ref name = K-CChao>{{cite book|title = Albright's Chemical Engineering Handbook|editor-first = Lyle F.|editor-last = Albright|publisher = [[CRC Press]]|year = 2008|isbn = 9780824753627|chapter-url = https://books.google.com/books?id=HYB3Udjx_FYC&pg=PA325|chapter = Thermodynamics of Fluid Phase and Chemical Equilibria|pages = 255–392|first1 = Kwang-Chu|last1 = Chao|authorlink1 = Kwang-Chu Chao|first2 = David S.|last2 = Corti|first3 = Richard G.|last3 = Mallinson}}</ref>{{rp|325}} Taking compliance with Raoult's Law as a defining characteristic of ideality in a  solution, it is possible to [[Ideal solution#Formal definition|deduce]] that the [[chemical potential]] of each component of the liquid is given by
: <math>\mu _i  = \mu_i^\star  + RT \ln x_i,</math>
where <math>\mu_i^\star</math> is the chemical potential in the pure state and <math>x_i</math> is the mole fraction of component <math>i</math> in the ideal solution.  From this equation, other thermodynamic properties of an ideal solution may be determined.  If the assumption that the vapor follows the ideal gas law is added, Raoult's law may be derived as follows.

If the system is ideal, then, at [[chemical equilibrium|equilibrium]], the chemical potential of each component <math>i</math> must be the same in the liquid and gas states. That is,

: <math>\mu_{i, \text{liq}} = \mu_{i, \text{vap}}.</math>

Substituting the formula for chemical potential gives
: <math>\mu_{i, \text{liq}}^\star + RT \ln x_i = \mu_{i, \text{vap}}^\ominus + RT \ln\frac{f_i}{p^\ominus},</math>
as the gas-phase mole fraction depends on its [[fugacity]], <math>f_i</math>, as a fraction of the pressure in the reference state, <math>p^\ominus</math>.

The corresponding equation when the system consists purely of component <math>i</math> in equilibrium with its vapor is
: <math>\mu_{i, \text{liq}}^\star = \mu_{i, \text{vap}}^\ominus + RT \ln\frac{f_i^\star}{p^\ominus}.</math>

Subtracting these equations and re-arranging leads to the result<ref name = K-CChao />{{rp|326}}
: <math>f_i = x_i f_i^\star.</math>

For the ideal gas, pressure and fugacity are equal, so introducing simple [[pressure]]s to this result yields Raoult's law:
: <math>p_i = x_i p_i^\star.</math>

===Ideal mixing===
An ideal solution would follow Raoult's law, but most solutions deviate from ideality. Interactions between gas molecules are typically quite small, especially if the vapor pressures are low. However, the interactions in a liquid are very strong. For a solution to be ideal, the interactions between unlike molecules must be of the same magnitude as those between like molecules.<ref>Rock, Peter A. ''Chemical Thermodynamics'' (MacMillan 1969), p. 261. {{ISBN|1891389327}}.</ref> This approximation is only true when the different species are almost chemically identical. One can see that from considering the [[Gibbs free energy of mixing|Gibbs free energy change of mixing]]:

: <math>\Delta_\text{mix} G = nRT (x_1 \ln x_1 + x_2 \ln x_2).</math>

This is always negative, so mixing is spontaneous. However, the expression is, apart from a factor <math>-T</math>, equal to the entropy of mixing. This leaves no room at all for an enthalpy effect and implies that <math>\Delta_\text{mix} H</math> must be equal to zero, and this can only be true if the interactions between the molecules are indifferent.

It can be shown using the [[Gibbs–Duhem equation]] that if Raoult's law holds over the entire concentration range <math>x \in [0,\ 1]</math> in a binary solution then, for the second component, the same must also hold.

If deviations from the ideal are not too large, Raoult's law is still valid in a narrow concentration range when approaching <math>x \to 1</math> for the majority phase (the ''solvent''). The solute also shows a linear limiting law, but with a different coefficient. This relationship is known as [[Henry's law]].

The presence of these limited linear regimes has been experimentally verified in a great number of cases, though large deviations occur in a variety of cases.  Consequently, both its pedagogical value and utility have been questioned at the introductory college level.<ref name="Hawkes 1995">{{cite journal|last = Hawkes|first = Stephen J.|year = 1995|title = Raoult's Law Is a Deception|journal = [[J. Chem. Educ.]]|volume = 72|issue = 3|pages = 204–205|doi = 10.1021/ed072p204| s2cid=95146940 |doi-access = free| bibcode=1995JChEd..72..204H }}</ref> In a perfectly ideal system, where ideal liquid and ideal vapor are assumed, a very useful equation emerges if Raoult's law is combined with [[Dalton's Law]]:

: <math>x_i = \frac{y_i p_\text{total}}{p_i^\star},</math> <!-- Please do not completely delete from article. Very valuable equation. -->

where <math>x_i</math> is the [[mole fraction]] of component <math>i</math> in the ''solution'', and <math>y_i</math> is its [[mole fraction]] in the ''gas phase''. This equation shows that, for an ideal solution where each pure component has a different vapor pressure, the gas phase is enriched in the component with the higher vapor pressure when pure, and the solution is enriched in the component with the lower pure vapor pressure. This phenomenon is the basis for [[distillation]].

===Non-ideal mixing===
In elementary applications, Raoult's law is generally valid when the liquid phase is either nearly pure or a mixture of similar substances.<ref>{{cite book |last1=Felder |first1=Richard M. |last2=Rousseau |first2=Ronald W. |last3=Bullard |first3=Lisa G.|date= 2004-12-15|title=Elementary Principles of Chemical Processes |publisher=Wiley |page=293 |isbn=978-0471687573 }}</ref> Raoult's law may be adapted to non-ideal solutions by incorporating two factors that account for the interactions between molecules of different substances.  The first factor is a correction for gas non-ideality, or deviations from the [[ideal-gas law]]. It is called the [[fugacity coefficient]] (<math>\phi_{p,i}</math>). The second, the [[activity coefficient]] <math>\gamma_i</math>, is a correction for interactions in the liquid phase between the different molecules.<ref name = K-CChao />{{rp|326}}

This modified or extended Raoult's law is then written as<ref name = "Che Thermo">
{{Citation
  | last1 = Smith| first1 = J. M.
  | last2 = Van Ness| first2 = H. C.
  | last3 = Abbott| first3 = M. M.
  | title = Introduction to Chemical Engineering Thermodynamics
  | place = New York
  | publisher = McGraw-Hill
  | year = 2005
  | pages = 545
  | edition = seventh
  | isbn = 0-07-310445-0}}
</ref>
: <math>y_i \phi_{p,i} p = x_i \gamma_i p_i^\star.</math>