2.5.3 Phase behavior of binary systems – liquid-vapor-transition

We start with a liquid mixture containing only one evaporative component, the solvent, and will derive a formula for the increase in boiling temperature in respect to that of the pure solvent. This system is formally sketched in figure 2.18.

$ YDSRU

$% OLTXLG

**Fig. 2.18: **Phase equilibrium between a binary liquid mixture
and the pure vapor of the volatile component A

At equilibrium, the chemical potentials of component A, which coexists in both the vapor and the liquid phase, have to be identical, respectively, or

ߤ_{ܣ}^{Ԣ} ൌ ߤ_{ܣ}^{ԢԢ} (Eq.2.98)

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**Basic Physical Chemistry**

**59**

**Thermodynamics**
Since we are interested in the dependence of boiling temperature on concentration of component B in
the liquid phase, we have to use the derivative, or:

݀ߤ_{ܣ}^{Ԣ} ൌ ݀ߤ_{ܣ}^{ԢԢ} (Eq.2.99)

To avoid differentiation of the expression ܴܶ ݔ_{ܣ} which is part of the chemical potential of the binary
liquid phase, we consider alternatively

݀^{ߤ}_{ܶ}^{ܣ}^{Ԣ} ൌ ݀^{ߤ}_{ܶ}^{ܣ}^{ԢԢ} (Eq.2.100)

݀^{ߤ}_{ܶ}^{ܣ}^{Ԣ}^{כ} ܴ ή ݀ ݔ_{ܣ}^{Ԣ} ൌ ݀^{ߤ}^{ܣ}_{ܶ}^{ԢԢ}^{כ} (Eq.2.101)
Note that the pressure in this case is kept constant, the only intensive variable being the boiling
temperature T and the concentration or molar fraction of the solvent in the liquid phase Hݔ_{ܣ}^{Ԣ}

To solve this differential equation and thereby obtain a quantitative relation between phase transition
temperature and solute concentration, we need the total differential of ^{ܩ}_{ܶ} which is given as:

݀ ቀ^{ܩ}_{ܶ}ቁ ൌ െ_{ܶ}^{ܪ}_{ʹ}݀ܶ ^{ܸ}_{ܶ}݀ (Eq.2102)

Noting that ߤ_{ܣ}^{Ԣ} כൌ ܩ_{ܣǡ݈݉ܽݎ}^{Ԣ} ൌ ܩ_{ܣ}^{Ԣ} כ we obtain

െ^{ܪ}_{ܶ}^{ܣ}^{Ԣ}_{ʹ}^{כ}݀ܶ ܴ ή ݀ ݔ_{ܣ}^{Ԣ} ൌ െ^{ܪ}_{ܶ}^{ܣ}^{ԢԢ כ}_{ʹ} ݀ܶ (Eq.2.103)
ቀ^{ܪ}^{ܣ}_{ܶ}^{ԢԢ כ}_{ʹ} െ^{ܪ}_{ܶ}^{ܣ}^{Ԣ}_{ʹ}^{כ}ቁ ݀ܶ ൌ^{ο}^{ܸ}_{ܶ}^{ܪ}_{ʹ}^{ܣ}݀ܶ ൌ െܴ ή ݀ ݔ_{ܣ}^{Ԣ} (Eq.2.104)
with οܸܪܣ the molar evaporation enthalpy of component A. Integration with the boundaries pure solvent

ܶ ൌ ܶܣכݔ_{ܣ}^{Ԣ} ൌ ͳ, *p* = 1 bar ) and an arbitrary mixture ܶݔ_{ܣ}^{Ԣ} , *p* = 1 bar), we finally get:

οܸܪܣ

ܴ ቀെ_{ܶ}^{ͳ}_{ܶ}^{ͳ}

ܣכቁ ൌ െ ݔ_{ܣ}^{Ԣ} (Eq.2.105)

Note that ܶܶ> ൏ ܶ൏ ܶ_{ܣ}_{ܣ}ככ, i.e. the mixture has a higher boiling temperature than the pure solvent. Also, this
effect is purely of entropic origin, since our derivation is based on the chemical potential using the formula
we previously have deduced from the mixing free enthalpy of ideal gases. Therefore, the effect also does
not depend on the chemical nature of the solute component B, but only on the relative concentration,
or better, the molar fraction of the solvent in the liquid phase ݔ_{ܣ}^{Ԣ}

This type of entropy-based effect found in binary mixtures is also called a colligative phenomenon,
further examples are lowering of the freezing temperature (see next chapter), lowering of the vapor
pressure which is the direct correspondent of the increase in boiling temperature (law of Raoult), and
finally osmotic pressure. All respective mathematical expressions are derived from the same principle,
i.e. the equality of the chemical potential of the solvent component in both coexisting phases ߤ_{ܣ}^{Ԣ} ൌ ߤ_{ܣ}^{ԢԢ}
This leads us to an alternative derivation of the increase of boiling temperature formula merely based
on this stationary phase equilibrium ߤ_{ܣ}^{Ԣ} ൌ ߤ_{ܣ}^{ԢԢ} and avoiding the explicit mathematical solution of any
differential equation:

We start with the stationary phase equilibrium given as:

ߤ_{ܣ}^{Ԣ} כ ܴܶ ή ݔ_{ܣ}^{Ԣ} ൌ ߤ_{ܣ}^{ԢԢ} כ (Eq.2.106)

and therefore, using the Gibbs-Helmholtz-expression ο_{ܸ}ܩ_{ܣ}ൌ ο_{ܸ}ܪ_{ܣ}െ ܶο_{ܸ}ܵ_{ܣ}

ߤ_{ܣ}^{ԢԢ} כ െߤ_{ܣ}^{Ԣ} כൌ οܸܩܣ ൌ οܸܪܣെ ܶοܸܵܣൌ ܴܶ ή ݔ_{ܣ}^{Ԣ} (Eq.2.107)
withοܸܪܣ the molar evaporation enthalpy of the pure solvent, and ο_{ܸ}ܵ_{ܣ} the molar evaporation
entropy. The evaporation entropy, defined as the reversible heat uptake normalized by the evaporation
temperature, can simply be replaced if we consider this equation at the boundary case of the pure solvent

ܶ ൌ ܶܣכݔ_{ܣ}^{Ԣ} ൌ ͳ , *p* = 1 bar ):

ο_{ܸ}ܪ_{ܣ}െ ܶ_{ܣ}כή ο_{ܸ}ܵ_{ܣ}ൌ ܴܶ ή ͳ ൌ Ͳ, or οܸܵܣൌ^{ο}_{ܶ}^{ܸ}^{ܪ}^{ܣ}

ܣכ (Eq.2.108)

and we immediately obtain our above result

ο_{ܸ}ܪ_{ܣ}

ܴ ቀെ^{ͳ}_{ܶ}_{ܶ}^{ͳ}

ܣכቁ ൌ െ ݔ_{ܣ}^{Ԣ} (Eq.2.109)

Note that this expression can further be simplified if we consider very dilute solutions, i.e. comparatively
small values of ݔ_{ܤ}^{Ԣ} ൌ ͳ െ ݔ_{ܣ}^{Ԣ} and use the Taylor series expansion of the logarithm:

ο_{ܸ}ܪ_{ܣ}

ܴ ቀെ^{ͳ}_{ܶ}_{ܶ}^{ͳ}

ܣכቁ ൌ െ ሺͳ െ ݔ_{ܤ}^{Ԣ}ሻ ൎ ݔ_{ܤ}^{Ԣ} (Eq.2.110)
Also the left hand side of this equation can be simplified considering that the change in boiling temperature

οܶ ൌ ܶ െ ܶ_{ܣ}כwill be comparatively small for such dilute systems:

ο_{ܸ}ܪ_{ܣ}

ܴ ቀെ_{ܶ}^{ͳ}_{ܶ}^{ͳ}

ܣכቁ ൌ^{ο}^{ܸ}_{ܴ}^{ܪ}^{ܣ}ቀ^{ܶെܶ}_{ܶ} ^{ܣ}^{כ}

ܣכήܶቁ ൎ^{ο}^{ܸ}_{ܴ}^{ܪ}^{ܣ}ቀ_{ܶ}^{οܶ}

ܣכ^{ʹ}ቁ ൌ ݔ_{ܤ}^{Ԣ} (Eq.2.111)

**Basic Physical Chemistry**

**61**

**Thermodynamics**
Finally, we will get the following simple formula for the colligative phenomenon of boiling temperature
increase:

οܸܪܣ

ܴ ቀ_{ܶ}^{οܶ}

ܣכ^{ʹ}ቁ ൌ ݔ_{ܤ}^{Ԣ} ൎ^{݊}_{݊}^{ܤ}^{Ԣ}

ܣԢ ൌ ^{݉ ܤԢ}^{ܯܤ}_{݉ ܣԢ}

ܯܣ

ൌ^{݉}_{ܯ}^{ܤ}^{Ԣ}

ܤ ή^{ܯ}_{݉}^{ܣ}

ܣԢ (Eq.2.112)

Or

οܶ ൌ^{ܴήܶ}_{ο}^{ܣ}^{כ}^{ʹ}^{ήܯ}^{ܣ}

ܸܪܣ ή^{݉}_{ܯ}^{ܤ}^{Ԣ}

ܤ ή_{݉}^{ͳ}

ܣԢ (Eq.2.113)

Here, ݊_{ܤ}^{Ԣ} is the molar amount of solute in the mixture,݉_{ܤ}^{Ԣ} the solute mass and *M*_{B} the molar mass of
the solute, index *A* accordingly for the solvent component in the mixture. The effect οܶ is therefore, in
case of dilute solutions, directly proportional to the molal concentration of the solute *B*, and it does not
depend on the chemical nature of the solute but only on that of the solvent, with the proportionality
factor, also called ebulioscopic constant, given as ^{ܴήܶ}^{ܣ}^{כ}^{ʹ}^{ήܯ}^{ܣ}

ο_{ܸ}ܪ_{ܣ}

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