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.
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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
ߤܣԢ ൌ ߤܣԢԢ (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:
݀ߤܣԢ ൌ ݀ߤܣԢԢ (Eq.2.99)
To avoid differentiation of the expression ܴܶ ݔܣ which is part of the chemical potential of the binary liquid phase, we consider alternatively
݀ߤܶܣԢ ൌ ݀ߤܶܣԢԢ (Eq.2.100)
݀ߤܶܣԢכ ܴ ή ݀ ݔܣԢ ൌ ݀ߤܣܶԢԢכ (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 HݔܣԢ
To solve this differential equation and thereby obtain a quantitative relation between phase transition temperature and solute concentration, we need the total differential of ܩܶ which is given as:
݀ ቀܩܶቁ ൌ െܶܪʹ݀ܶ ܸܶ݀ (Eq.2102)
Noting that ߤܣԢ כൌ ܩܣǡ݈݉ܽݎԢ ൌ ܩܣԢ כ we obtain
െܪܶܣԢʹכ݀ܶ ܴ ή ݀ ݔܣԢ ൌ െܪܶܣԢԢ כʹ ݀ܶ (Eq.2.103) ቀܪܣܶԢԢ כʹ െܪܶܣԢʹכቁ ݀ܶ ൌοܸܶܪʹܣ݀ܶ ൌ െܴ ή ݀ ݔܣԢ (Eq.2.104) with οܸܪܣ the molar evaporation enthalpy of component A. Integration with the boundaries pure solvent
ܶ ൌ ܶܣכݔܣԢ ൌ ͳ, p = 1 bar ) and an arbitrary mixture ܶݔܣԢ , p = 1 bar), we finally get:
οܸܪܣ
ܴ ቀെܶͳܶͳ
ܣכቁ ൌ െ ݔܣԢ (Eq.2.105)
Note that ܶܶ> ൏ ܶ൏ ܶܣܣככ, 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 ݔܣԢ
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 ߤܣԢ ൌ ߤܣԢԢ This leads us to an alternative derivation of the increase of boiling temperature formula merely based on this stationary phase equilibrium ߤܣԢ ൌ ߤܣԢԢ and avoiding the explicit mathematical solution of any differential equation:
We start with the stationary phase equilibrium given as:
ߤܣԢ כ ܴܶ ή ݔܣԢ ൌ ߤܣԢԢ כ (Eq.2.106)
and therefore, using the Gibbs-Helmholtz-expression οܸܩܣൌ οܸܪܣെ ܶοܸܵܣ
ߤܣԢԢ כ െߤܣԢ כൌ οܸܩܣ ൌ οܸܪܣെ ܶοܸܵܣൌ ܴܶ ή ݔܣԢ (Eq.2.107) withοܸܪܣ the molar evaporation enthalpy of the pure solvent, and οܸܵܣ 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
ܶ ൌ ܶܣכݔܣԢ ൌ ͳ , p = 1 bar ):
οܸܪܣെ ܶܣכή οܸܵܣൌ ܴܶ ή ͳ ൌ Ͳ, or οܸܵܣൌοܸܶܪܣ
ܣכ (Eq.2.108)
and we immediately obtain our above result
οܸܪܣ
ܴ ቀെͳܶܶͳ
ܣכቁ ൌ െ ݔܣԢ (Eq.2.109)
Note that this expression can further be simplified if we consider very dilute solutions, i.e. comparatively small values of ݔܤԢ ൌ ͳ െ ݔܣԢ and use the Taylor series expansion of the logarithm:
οܸܪܣ
ܴ ቀെͳܶܶͳ
ܣכቁ ൌ െ ሺͳ െ ݔܤԢሻ ൎ ݔܤԢ (Eq.2.110) Also the left hand side of this equation can be simplified considering that the change in boiling temperature
οܶ ൌ ܶ െ ܶܣכwill be comparatively small for such dilute systems:
οܸܪܣ
ܴ ቀെܶͳܶͳ
ܣכቁ ൌοܸܴܪܣቀܶെܶܶ ܣכ
ܣכήܶቁ ൎοܸܴܪܣቀܶοܶ
ܣכʹቁ ൌ ݔܤԢ (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
ܴ ቀܶοܶ
ܣכʹቁ ൌ ݔܤԢ ൎ݊݊ܤԢ
ܣԢ ൌ ݉ ܤԢܯܤ݉ ܣԢ
ܯܣ
ൌ݉ܯܤԢ
ܤ ήܯ݉ܣ
ܣԢ (Eq.2.112)
Or
οܶ ൌܴήܶοܣכʹήܯܣ
ܸܪܣ ή݉ܯܤԢ
ܤ ή݉ͳ
ܣԢ (Eq.2.113)
Here, ݊ܤԢ is the molar amount of solute in the mixture,݉ܤԢ the solute mass and MB 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 ܴήܶܣכʹήܯܣ
οܸܪܣ
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