Chapter 8. Measurement of molecular
               weight and size
               8.1. End-group analysis
               8.2. Colligative property measurement 
               8.3. Osmometry
               8.4. Gel-permeation chromatography 
               8.5. Ultracentrifugation
               8.6. Light-scattering methods
               8.7. Solution viscosity and molecular size      
          
	
         
 
                                 
            Molecular weights of polymers can be determined by chemical or
            physical methods of functional-group analysis. They are measurement
            of the colligative properties, light scattering, or ultracentrifugation;
            or by measurement of dilute-solution viscosity.
            Molecular weights can be calculated without reference to callibration
            by another method.
            Dilute-solution viscosity is not a direct measure of molecular weight 
            and empirically related to molecular weight for many systems. 
            All methods require the solubility of polymer, involve extrapolation to
            infinite dilution, and operate in a Θ solvent in which have ideal-solution
            behaviour.
            
	
               
 
                                         
             Typical polymers consist of mixtures of many molecular species and
             molecular weight methods always provide average values.
             The sum of all molecular species of the number of moles Ni of each
             species:    ∞
                        ∑Ni
                         i=1
             The total weight of the sample, w, = the sum of the weights of each 
             molecular species 
                              ∞       ∞
                         w=∑wi=∑NiMi
                              i=1     i=1
             The number-average molecular weight, M      molecular weight as 
                                                      n
             weight of sample per mole
                                        ∞
                                        ∑NiMi
                         M = w = i=1
                           n    ∞         ∞
                               ∑Ni       ∑Ni
                                i=1      i=1
            
	
                 
 
                                          
           8.1. End-group analysis
           The end-group analysis needs the information about the number of 
           determinable groups per molecule. This method is not suitable for 
           high molecular weight because the fraction of end groups becomes
           too small to measured with precision (>25,000).
           Condensation polymers
           End-group analysis in condensation polymers usually involves 
           chemical methods of analysis for functional groups.
           Examples:
           • Carboxyl groups in PE and in polyamides are titrated with base in an
             alcoholic or phenolic solvent.
           • Amino groups in PA are titrated with acid 
           • Hydroxyl groups is reacted with a titratable reagent