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Cryoscopy
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Objectives:

  

1. To understand colligative properties of solutes.
2. To find the freezing point depression of a solution.
3. To determine the van ‘t Hoff factor of solutes.
4. To find the molar mass of an unknown solute.

  

Theory:

 

Cryoscopy:

 

  • Depression in Freezing Point:

 

The freezing point of a liquid substance is the temperature at which the liquid and its solid form are in equilibrium. The phenomenon that the Freezing point of a solvent will be lower when another compound is added is known as depression in freezing point, i.e., a pure solvent has a higher freezing point than a solution. This happens whenever a solute is added to a pure solvent. It is also a colligative property — dependent only on the number of particles added and not the kind. It is defined as the difference in the freezing points of the pure solvent and solution, the difference between the freezing points of the solution, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msubsup»«mi»T«/mi»«mi»f«/mi»«mo»§apos;«/mo»«/msubsup»«/math» and the pure solvent, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»T«/mi»«mi»f«/mi»«/msub»«/math».

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#9651;«/mo»«msub»«mi»T«/mi»«mi»f«/mi»«/msub»«mo»=«/mo»«msub»«mi»T«/mi»«mi»f«/mi»«/msub»«mo»-«/mo»«msubsup»«mi»T«/mi»«mi»f«/mi»«mo»§apos;«/mo»«/msubsup»«/math»

 

This is a colligative property which does not depend on the nature of solute particles but depend only on the number of those solute particles in solution. The elevation of the boiling point can be calculated by applying the assumption of the non-volatility of the solute together with the Clausius-Clapeyron relation and Raoult’s law.
 
 
Raoult's law:
 
The vapour pressure of a solution of a non-volatile solute is equal to the product of the vapour pressure of the pure solvent at that temperature and its mole fraction.
In equilibrium, the total vapour pressure , ‘«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#961;«/mi»«/math»’ of the solution is:
 
 
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#961;«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«msubsup»«mi»§#961;«/mi»«mi»A«/mi»«mo»*«/mo»«/msubsup»«msub»«mi»X«/mi»«mi»A«/mi»«/msub»«mo»§nbsp;«/mo»«mo»+«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msubsup»«mi»§#961;«/mi»«mi»B«/mi»«mo»*«/mo»«/msubsup»«msub»«mi»X«/mi»«mi»B«/mi»«/msub»«/math»
 
 
 
As the number of components in a solution increases, the individual vapour pressure decreases, i.e. the mole fraction of each component is indirectly proportional to each additional component and is individual vapour pressure for each component is:
 
 
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»§#961;«/mi»«mi»i«/mi»«/msub»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«msubsup»«mi»§#961;«/mi»«mi»i«/mi»«mo»*«/mo»«/msubsup»«msub»«mi»X«/mi»«mi»i«/mi»«/msub»«/math»
 
Where, 
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»§#961;«/mi»«mi»i«/mi»«/msub»«/math»is the partial pressure of the component i in the solution; «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msubsup»«mi»§#961;«/mi»«mi»i«/mi»«mo»*«/mo»«/msubsup»«/math» is the vapour pressure of the pure component i; and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»X«/mi»«mi»i«/mi»«/msub»«/math» is the mole fraction of the component i in the solution (in mixture).
 
 
In addition to measuring the difference (a procedure called cryoscopy) the depression in freezing point can be found using the following equation if the solute is known:

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#9651;«/mo»«msub»«mi»T«/mi»«mi»f«/mi»«/msub»«mo»=«/mo»«msub»«mi»K«/mi»«mi»f«/mi»«/msub»«mo»§nbsp;«/mo»«mo».«/mo»«mo»§nbsp;«/mo»«mi»i«/mi»«mo»§nbsp;«/mo»«mo».«/mo»«mo»§nbsp;«/mo»«mi»m«/mi»«/math»

Where,«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#9651;«/mo»«msub»«mi»T«/mi»«mi»f«/mi»«/msub»«/math» is the depression of the freezing point;
m is the molality of the solution; and i is the van ‘t Hoff factor. Molality of a solution is the ratio of  amount of substance of solute and the mass of  solvent in kg.

The value of van ‘t Hoff factor depends upon the number of individual ions formed in solution. i.e:
 
· i = 1 for sugar in water.
· i = 2 for sodium chloride (NaCl) in water, due to the its full dissociation into Na+ and Cl- ions.
· i = 3 for calcium chloride (CaCl2) in water, due to its full dissociation Ca2+ and 2Cl- ions.
 
The van ‘t Hoff factor is a measure of the colligative effect (the total number of particles) of the solute in solution. The value of i is usually unity for all non-electrolytes, greater than unity for electrolytes, but is less than unity for compounds that associate in solution.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»K«/mi»«mi»f«/mi»«/msub»«/math» is the molal freezing point depression constant or cryoscopic constant of the solvent, and it is depends upon the properties of the solvent. Which can be calculated as:

 

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»K«/mi»«mi»f«/mi»«/msub»«mo»=«/mo»«mi»R«/mi»«mo»§nbsp;«/mo»«mo».«/mo»«mo»§nbsp;«/mo»«msubsup»«mi»T«/mi»«mi»f«/mi»«mn»2«/mn»«/msubsup»«mo»§nbsp;«/mo»«mo».«/mo»«mo»§nbsp;«/mo»«mi»M«/mi»«mo»/«/mo»«mo»§#9651;«/mo»«msub»«mi»H«/mi»«mi»f«/mi»«/msub»«/math»

 

 Where, R is the gas constant; «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»T«/mi»«mi»f«/mi»«/msub»«/math» is the freezing temperature of the pure solvent (in K); M is the molar mass of the solvent; and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#9651;«/mo»«msub»«mi»H«/mi»«mi»f«/mi»«/msub»«/math» is the heat of fusion per mole of solvent. Note: although the above equation yields a positive value for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»K«/mi»«mi»f«/mi»«/msub»«/math», by convention «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»K«/mi»«mi»f«/mi»«/msub»«/math» is represented as a negative value, resulting in observed and calculated negative values for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#9651;«/mo»«msub»«mi»T«/mi»«mi»f«/mi»«/msub»«/math».

 

Similarly as in the case using the observed elevation in the boiling point, it is possible to calculate the molecular mass of a solute from the observed depression in the freezing point by using the equation: 

 


«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»M«/mi»«mi»B«/mi»«/msub»«mo»=«/mo»«mfrac»«mrow»«mn»1000«/mn»«mo»§nbsp;«/mo»«mo».«/mo»«mo»§nbsp;«/mo»«msub»«mi»K«/mi»«mrow»«mi»f«/mi»«mo»§nbsp;«/mo»«/mrow»«/msub»«mo».«/mo»«mo»§nbsp;«/mo»«mi»i«/mi»«mo»§nbsp;«/mo»«mo».«/mo»«mo»§nbsp;«/mo»«msub»«mi»W«/mi»«mi»B«/mi»«/msub»«/mrow»«mrow»«mo»§#9651;«/mo»«msub»«mi»T«/mi»«mrow»«mi»f«/mi»«mo»§nbsp;«/mo»«/mrow»«/msub»«mo».«/mo»«mo»§nbsp;«/mo»«msub»«mi»W«/mi»«mi»A«/mi»«/msub»«/mrow»«/mfrac»«/math»

 

Where,«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»W«/mi»«mi»B«/mi»«/msub»«/math» is the weight of the solute;«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»W«/mi»«mi»A«/mi»«/msub»«/math» is the weight of solvent; 
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#9651;«/mo»«msub»«mi»T«/mi»«mi»f«/mi»«/msub»«/math» is the elevation of the boiling point; «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»K«/mi»«mi»f«/mi»«/msub»«/math» molal freezing point depression constant; and i is the van ‘t Hoff factor.

 

 

 

 

 

 



 

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