Editing Colligative properties (section)

== Boiling point and freezing point ==

Addition of solute to form a solution stabilizes the solvent in the liquid phase, and lowers the solvent's [[chemical potential]] so that solvent molecules have less tendency to move to the gas or solid phases. As a result, liquid solutions slightly above the solvent boiling point at a given pressure become stable, which means that the boiling point increases. Similarly, liquid solutions slightly below the solvent freezing point become stable meaning that the freezing point decreases. Both the [[boiling point elevation]] and the [[freezing point depression]] are proportional to the lowering of vapor pressure in a dilute solution.

These properties are colligative in systems where the solute is essentially confined to the liquid phase. Boiling point elevation (like vapor pressure lowering) is colligative for non-volatile solutes where the solute presence in the gas phase is negligible. Freezing point depression is colligative for most solutes since very few solutes dissolve appreciably in solid solvents.

=== Boiling point elevation (ebullioscopy) ===
{{Main|Boiling point elevation}}
The [[boiling point]] of a liquid at a given external pressure is the temperature (<math>T_{\rm b}</math>) at which the vapor pressure of the liquid equals the external pressure. The ''normal boiling point'' is the boiling point at a pressure equal to 1 [[atmosphere (unit)|atm]].

The boiling point of a pure solvent is increased by the addition of a non-volatile solute, and the elevation can be measured by [[ebullioscopy]]. It is found that 
    
:<math>\Delta T_{\rm b} = T_{\rm b,\text{solution}} - T_{\rm b,\text{pure solvent}} = i\cdot K_b \cdot m </math><ref name=":0">{{Cite book|last=Tro|first=Nivaldo J.|title=Chemistry; Structure and Properties|publisher=[[Pearson Education]]|year=2018|isbn=978-0-134-52822-9|edition=2nd|pages=563–566|type=Textbook.}}</ref>

Here ''i'' is the [[van 't Hoff factor]] as above, ''K''<sub>b</sub> is the [[ebullioscopic constant]] of the solvent (equal to 0.512&nbsp;°C kg/mol for water), and ''m'' is the [[molality]] of the solution.

The boiling point is the temperature at which there is equilibrium between liquid and gas phases.  At the boiling point, the number of gas molecules condensing to liquid equals the number of liquid molecules evaporating to gas. Adding a solute dilutes the concentration of the liquid molecules and reduces the rate of evaporation. To compensate for this and re-attain equilibrium, the boiling point occurs at a higher temperature.

If the solution is assumed to be an [[ideal solution]], ''K''<sub>b</sub> can be evaluated from the [[thermodynamic]] condition for liquid-vapor equilibrium. At the boiling point, the [[chemical potential]] μ<sub>A</sub> of the solvent in the solution phase equals the chemical potential in the pure vapor phase above the solution.

:<math>\mu _A(T_b)  = \mu_A^{\star}(T_b)  + RT\ln x_A\  = \mu_A^{\star}(g, 1 \,\mathrm{atm}),</math>

The asterisks indicate pure phases. This leads to the result <math>K_b = RMT_b^2/\Delta H_{\mathrm{vap}}</math>, where R is the [[molar gas constant]], M is the solvent [[molar mass]] and Δ''H''<sub>vap</sub> is the solvent molar [[enthalpy of vaporization]].<ref name=Engel>T. Engel and P. Reid, Physical Chemistry (Pearson Benjamin Cummings 2006) p.204-5</ref>

=== Freezing point depression (cryoscopy)===
{{Main|Freezing point depression}}
The freezing point (<math>T_{\rm f}</math>) of a pure solvent is lowered by the addition of a solute which is insoluble in the solid solvent, and the measurement of this difference is called ''cryoscopy''. It is found that 
    
:<math>\Delta T_{\rm f} = T_{\rm f,\text{solution}} - T_{\rm f,\text{pure solvent}} = - i\cdot K_f \cdot m </math><ref name=":0" /> (which can also be written as <math>\Delta T_{\rm f} = T_{\rm f,\text{pure solvent}} - T_{\rm f,\text{solution}} = i\cdot K_f \cdot m </math>)

Here ''K<sub>f</sub>'' is the [[cryoscopic constant]] (equal to 1.86&nbsp;°C kg/mol for the freezing point of water), ''i'' is the van 't Hoff factor, and ''m'' the molality (in mol/kg). This predicts the melting of ice by [[road salt]].

In the liquid solution, the solvent is diluted by the addition of a solute, so that fewer molecules are available to freeze. Re-establishment of equilibrium is achieved at a lower temperature at which the rate of freezing becomes equal to the rate of liquefying. At the lower freezing point, the vapor pressure of the liquid is equal to the vapor pressure of the corresponding solid, and the chemical potentials of the two phases are equal as well. The equality of chemical potentials permits the evaluation of the cryoscopic constant as <math>K_f = RMT_f^2/\Delta_{\mathrm{fus}}H</math>,  where Δ<sub>fus</sub>''H'' is the solvent molar [[enthalpy of fusion]].<ref name=Engel/>