Welcome Number average molecular weight.

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Molecular weight:Molecular weight: The molecular weight (M.W.) of a The molecular weight (M.W.) of a molecule is the sum of the weights of all the atoms which molecule is the sum of the weights of all the atoms which make up the molecule. For example, [R-(-CHmake up the molecule. For example, [R-(-CH22-CH-CH22-)-R]-)-R]500,500, the molecular weight of this structure is 14,000.the molecular weight of this structure is 14,000.

Different types of molecular weightDifferent types of molecular weight Number average molecular weight (Number average molecular weight (MnMn)) Weight average molecular weight (Weight average molecular weight (MwMw)) Zet average molecular weight (Zet average molecular weight (MzMz)) Viscosity average molecular weight (Viscosity average molecular weight (MvMv))

The number - average molar mass MnDefinition = the sum of the products of the molar mass of each fraction multiplied by its mole fraction.

Mn = ∑XiMi ..………1.1where : Xi is the mole fraction of the molecules of molar mass Mi

Xi = Ni / ∑Ni , N = number of moles

∴Mn = ∑NiMi / ∑Ni = ∑Wi

/ ∑Wi = W/ ∑Ni ……….. 1.2Sometimes weight fraction are used in place of mole fraction.

The weight fraction Wi is defined as the mass of molecules of molar mass Mi divided by the total mass of all the molecules present.

Wi = NiMi / ∑NiMi ………..1.3

∑ ( ∑ Ni /Wi )

= Mi /∑NiMi ………….1.4

Combining equations 1.2 & 1.4 gives Mn in terms of weight fraction.

Mn = 1 / ∑Wi/Mi or ∑ Wi / Mi = 1 / Mn ……1.6

Different method of determination Number average molecular weight (Mn)

Determination of MnIn practical the number average molecular weight (Mn)

may be determined by the following method

1. Cryoscopy method i.e. Depression of freezing point

2. Ebullioscopy i.e. Elevation of boiling point3. Osmometry A. Vapor phase osmometry

B. Membrane osmometry4. End group analysis.

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◦ Freezing point of a solvent is lowered when a non-volatile solute is dissolved in it.◦ Freezing point of the solution will be depressed.◦ This phenomenon of depressing the freezing point

of a liquid by the addition of a solute is known as Cryoscopy.◦ The extent of the freezing point depends on the

number of solute molecules dissolved per unit volume of solution and its independent of the size and nature of the solute molecule.

1.So, Number average molecular wt. is measured by this technique.

2. If depression of the freezing point is denoted by ,Its related to the Number-average Molecular wt. and concentration through following equation.

3.Where, C is concentration of the solution, ρ the density of the solvent, R the universal gas constant; Tf the freezing point of the solvent, B is a constant and ΔHf is the heat of fusion of the solvent.

∆ Tf /C = (RT

f

2

/ρ∆ Hf) 1/ +M BC

∆ Tf

1. Graphical representation.2. To find out the freezing point, solvent or solution is

carefully super cooled to a temperature that is about 5C lower than the freezing point.

3. Once the freezing points of the solvent and solutions Tf different concentrations are known, ΔTf can be calculated for all these concentrations.

4. Drawback of this method is, measurement of temperatures.

5. Highly precised temperature sensors are required.6. This method can measure molecular wt. up to

30,000.

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Ebullioscopy (Elevation of boiling point)This technique is based on boiling point of solutions.Boiling point of solutions is higher than the pure

solvents. This phenomenon is used to determine the molecular wt. of polymers.

Where ΔHv and Tb are the heat of vaporization and boiling of the solvent.

∆ T b /C = (RT b

2/ρ∆ H v) 1 /M n+ BC

Continue…..

There are two types of ebulliometers. In first type, the solution of the polymer whose molecular wt. is to be determined is heated in a boiler. The solution is then brought in contact with a sensor. Another in pure solvent.In second type, both of solution and pure solvent are heated in identical containers and uniform boiling is achieved.It can also be used to determine the molecular wt. up to 30,000.

Osmometry

Osmometry is the mostly used method to determine the number average molecular weight. There are two method of osmometry they are –

Vapor phase osmometry

Membrane osmometry

These techniques are useful in different Mn ranges and depend on the change in osmotic pressure and the lowering of vapor pressure (respectively) by polymers in solution. These method are very effected to find out the molecular weight of a polymer.

Vapor phase osmometryThe determination of Mn by the use of vapor pressure osmometry operates

on the principle that the vapor pressure of a solution is lower than that of the pure solvent at the same temperature and pressure. At sufficiently low concentrations, the magnitude of the vapor pressure decrease is directly proportional to the molar concentration of solute. Monitoring vapor pressure versus a change in concentration of solute can be used in a manner similar to that of membrane osmometry.

The relationship between the vapor pressure difference of solvent and solution,

and number average molecular wt. is given as

Continue….• Vapor phase osmometry is a experimental method.

• Vapor pressure is not measured directly due to difficulties in sensitivity, but is measured indirectly by using thermistors to measure voltage changes caused by changes in temperature.

• If experimental factors are controlled very carefully, very accurate Mn numbers can be achieved. MW’s below 1000 are determinable with a precision of 0.5 %. The lower limit of MW is technically only limited by the vapor pressure of the sample, but useful lower limits are in the 250 level.

Membrane osmometry is an absolute technique which determines Mn. The solvent is separated from the polymer solution by a semipermeable membrane which is tightly held between the two chambers.

It is most widely used technique to determine number average molecular wt.

It is based on the phenomenon of osmosis.The equation of Osmotic pressure is given below- here Π is the osmotic pressure.

Membrane osmometry is also a experimental process. The instrument by which this method done contains two main components, the cell assembly and the electronics-recorder assembly. Membrane conditioning is extremely important in using this technique. Membranes are usually cellulose derivatives and shipped in alcohol or aqueous solution.

Figure : Schematic representation of a membrane osmometer.

Figure : Automatic membrane osmometer

The problems of this technique are caused mainly by the membrane. There are problems with membrane leakage and asymmetry. The main advantage of membrane osmometry is that it yields an absolute Mn for a polymer and calibration with standards is not required. The most common membranes typically can determine Mn as low as 10000-20000. Membrane osmometry also provides information about polymer-solvent interactions.

End group analysis The end group analysis is a chemical method

used for calculating the Mn of polymer samples whose molecules contain reactive functional groups at one ends of the molecules.

From knowledge of the functional group equivalent and the functionality, the molecular weight can be calculated by using the equation.

Mn = Functionality/ Functional group End functional groups are hydroxyl, carboxyl,

aldehyde, amino, ester or methyl.

It can be used reliably only for samples consisting of linear molecules with determinable end groups.

This method can be conveniently used to find out the average functionality at any polymer sample.

The method cannot be applied to branched polymers. In a linear polymer there are twice as many end of the chain and groups as polymer molecules. If having different end group, the number of detected end group is average molecular weight. End group analysis could be applied for polymerization mechanism identified . In this method molecular weight limitation up to 50,000.

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