PAPER No.6: PHYSICAL CHEMISTRY-II (Statistical

Transcription

1 Subject PHYSICAL Paper No and Title Module No and Title Module Tag 6, PHYSICAL -II (Statistical 34, Method for determining molar mass - I CHE_P6_M34

2 Table of Contents 1. Learning Outcomes 2. Introduction 3. Measurement of Molar Mass and Size 4. Colligative Property Measurement 5. Summary 4.1 Number Average Molar Mass 4.2 Osmosis and Osmotic Pressure 4.3Expression of Osmotic Pressure for a Polymer Solution

3 1. Learning Outcomes After studying this module you shall be able to: Learn about the osmometry technique for determination of Molar Mass of polymers. Understand the importance of second virial coefficient in case of polymer soultions. Mathematically prove that osmosis technique gives Number Average Molar Mass of a polymer. 2. INTRODUCTION Macromolecules, also known as polymers, are formed by the covalent linkages between many repeating small molecules known as monomers. In the preparation of the polymer from monomer units, the polymerization reactions proceed through different extents of reaction. Thus, this results in polydispersity in molar masses. As a result, macromolecules do not have a unique molar mass. Depending upon the method by which the macromolecule is produced, same molecule may have different molar mass. Hence, for macromolecules the concept of average of molar masses is used. In order to describe the distribution of molar masses, the following averages are commonly used: 1. Number Average Molar Mass (M ) n : = N im i M n i N i i (1) Where: N i is the number of molecules each having molar mass M i 2. Mass Average Molar Mass: M w = i w i M i (2) Where : w i = Mass fraction M i = Molar mass 3. Z average molar mass: = N i M z 3 i M i N i i M i 2 (3) 4. Viscosity Average molar mass: = ( N a+1 i im i M v ) i N i M i 1 a (4)

4 3. MEASUREMENT OF MOLAR MASS AND SIZE The molar mass of polymers can be determined by a variety of physical and chemical methods such as by functional group analysis, by measuring the colligative properties, by light scattering method, by ultracentrifugation, by viscosity measurement of dilute solutions. All these methods, except the viscosity method, are in principle, absolute: Molar masses can be calculated without reference to calibration by some other method. However, dilute solution viscosity measurement method does not involve direct estimation of molar mass. Its value lies in the simplicity of the technique and the fact that its results can be related empirically to molar masses for many systems. Thus, dilute solution viscosity method yields Viscosity Average Molar Mass which is not an absolute value but a relative mass based on prior calibration with known molar mass for the same polymer-solvent-temperature conditions. With the exception of some types of end-group analysis, all molar-mass methods require solubility of the polymer, and all involve extrapolation to infinite dilution or operation in a θ (theta) solvent in which ideal-solution behavior is attained. Table 1 lists the different molar masses described above along with the experimental methods to determine the various average molar mass. Average Definition Methods M n i N i M i Osmotic pressure and other i N i colligative properties, End M w M z M v 4. Colligative Property Measurement w i M i i group analysis Light Scattering, Sedimentation velocity 3 i N i M i Sedimentation equilibrium 2 i N i M i 1 a Intrinsic viscosity ( N a+1 i im i ) i N i M i Table 1: Description of average molar masses. The relations between the colligative properties and molar masses, for infinitely dilute solutions, lay upon the fact that the activity of the solute becomes equal to its mole fraction as the solution concentration becomes sufficiently small. The activity of the solvent must equal to its mole fraction under these conditions, and follows that the depression of the activity of the solvent by a solute is equal to mole fraction of the solute. The colligative property

5 methods used are: lowering of vapour pressure, elevation of boiling point (ebulliometry), depression in freezing-point (cryoscopy) and the osmotic pressure measurement. A colligative property measures the number of particles in solution. Therefore, colligative property measurement of a polymer yields number average molar mass. 4.1 Number Average molar mass Number average molar mass can be calculated by using a dilute solution of a polymer through any of the methods from ebulliometry, cryoscopy and osmometry. Direct measurement of lowering of vapour pressure for dilute polymer solutions, is not precise and gives uncertain results. However, Vapour-phase osmometry, is an indirect method based Clapeyron equation, using which the vapour pressure lowering of a polymer solution, at equilibrium, can be related to a temperature difference that is comparable to or of the same order of magnitude as those observed in cryoscopy and ebulliomtry. These methods require calibration with low molar mass standards and they may produce reliable results for polymer with molar masses < 30,000. The working equations for ebulliometric, cryoscopic and osmometric measurements are as follows: T 2 b RT 1 lim x 0 c H f M n (5) T 2 f RT 1 lim x 0 c H f M n (6) RT lim x 0 c M n (7) Where: T b = elevation in boiling-point T f = depression in freezing-point = osmotic pressure = density of the solvent c = solute concentration (in g/ml or g/cm 3 ) and Hf are the enthalpies of vaporization and fusion, respectively, of the solvent per H v gram The number-average molar mass M n has been introduced in equations (5), (6) and (7) in order to make theses equations applicable to poly-disperse solutes.

6 Majorly, osmotic pressure measurements are used for studying macromolecules because osmotic changes are larger than the changes in other colligative properties. This is due to the fact that even a very small concentration of the solution produces a fairly large magnitude of osmotic pressure while at this small concentration, the values of other colligative properties such as boiling point elevation and freezing point depression are too small to be experimentally determined. Thus, osmometry technique is more useful and thus, is more widely used than other colligative techniques for polymer systems. 4.2 Osmosis and Osmotic pressure When a solution and pure solvent are separated with the help of a semi-permeable membrane, diffusion of solvent molecules take place from the side of pure solvent to solution side. This flow of solvent molecules, from a region of their higher concentration (i.e. pure solvent) to a region of their lower concentration (i.e solution) is known as Osmosis. As a result of osmosis, the meniscus of the solution tube rises, whereas that of the pure solvent falls, until equilibrium is reached, when the meniscus of the solution tube does not rise further. At this stage, the excessive hydrostatic pressure created on the solution side exactly balances the tendency of the solvent molecules to pass through the membrane (fig1). This excessive hydrostatic pressure is known as the osmotic pressure of the solution and is represented by the symbol Π. Thus, osmotic pressure arises because of the equilibrium between the solvent in solution and the pure liquid solvent. Fig1: The phenomenon of osmosis and generation of osmotic pressure The osmotic pressure of a solution depends on its concentration i.e larger the concentration larger will be the osmotic pressure. The relationship of osmotic pressure with the concentration of the solution can be derived thermodynamically. In this module we will not consider the entire derivation; we will only use the following results: ΠV l,m = RT ln x 1 (8) Where:

7 V l,m = molar volume of the solvent x 1 = amount fraction of solvent in the solution We will simplify equation (8) for a dilute solution using the following approximations: 1. Since the solution is dilute x 2 (amount fraction of the solute) 1, therefore the term ln x 1 can be approximated as: ln x 1 = ln(1 x 2 ) x 2 (9) 2.The amount fraction of solute is given by: x 2= n2 n1+n2 = n 2 n1 (10) 3.For a solution, the total volume V can be written in terms of the partial molar volumes following the additivity rule: V = n 1 V 1 + n 2 V 2 Where : V1 = partial molar volume of the solvent V2 = Partial volume of the solute. Thus, for a dilute solution, we make the following assumptions: i) Partial molar volume of the solvent in the solution is same as that of the pure solvent i.e.: V1 = V l,m ii) The factor n2v2 << n1v1 since n2<<n1 and can be neglected. Thus, we have V n 1 V 1 = n 1 V l,m (11) Subsituting equations 9, 10 and 11 in equation 8 we get: Or or or V n 1 Π = RT( x 2 ) = RTx 2 V n 1 Π = RT n 2 n 1 VΠ = n 2 RT

8 Where: Π = ( n 2 ) RT = crt (12) V Π = Osmotic pressure c = molar concentration of solute in the solvent R = Gas constant T = Temperature Equation 12 is known as the van t Hoff equation for determining osmotic pressure of the solution. This equation is exactly identical to the ideal gas equation i.e pv = nrt (13) Except that Π replaces the gas pressure p. This resemblance indicates that the solute molecules are dispersed in the solvent in the similar way as the gas molecules are dispersed in the empty space. Thus, the solute is analogous to the gas molecules and the solvent is analogous to the empty space between the gas molecules. 4.3 Expression of Osmotic pressure for a polymer solution We have already studied that osmotic pressure measurement is a method primarily used in the determination of the molar masses of substances which have either very low solubility or very high molar masses, such as proteins or polymers. The van t hoff equation (12) is valid either only at infinite dilution or to a very dilute solution where the solute-solute interaction is negligible. Thus, this equation might not be applicable to a substance of high molar mass at concentrations which are employed for experimental methods. Moreover a polymer solution does not behave as an ideal solution because there is a large difference in the molar volumes of polymeric solute and that of low molar mass solvent. Therefore we take this fact into consideration and apply following conditions to equation (8). i.e. Firstly, we perform the expansion of logarithm in expression (9): ln x 1 = ln(1 x 2 ) = (x 2 + x x ) (14) And thus, expression 8 can be expressed as: ΠV l,m = RT (x 2 + B x 2 + C x ) (15)

9 Where : B = second virial coefficient C = Third virial coefficient If the concentration if solution is expressed in mass of solute per unit volume of solution, then: Also, (16) n 2 = m 2 M 2 and n 1 = V (17) V l,m Using equation 17 in equation 11 we have: (18) Substituting equation 18 in equation 15 we get: The higher terms are insignificant. Thus retaining only the first two terms on the right hand side, we get: (19)

10 Equation (20), predicts that for a non-ideal solution, the plot of Π/c 2 RT and c 2 is linear: The slope obtained from this graph is equal to the second virial coefficient B and its intercept is equal to the reciprocal of the molar mass M2 of the solute. (20) Fig 2: plot of Π/c 2 (reduced osmotic pressure) versus c 2 We can also plot Π/c 2 (reduced osmotic pressure) versus c 2 (Fig 3). From this graph, which when extrapolated to c 2 = 0 gives: Slope= B RT Intercept= RT/M2 i.e lim C2 0 ( Π c 2 ) = RT/M Discussion of Second Virial Coefficient (B ). The virial coefficients are determined empirically for a given solute-solvent system. The second virial coefficient i.e B represents the interaction of a single solute particle with the solvent while the higher order virial coefficients are associated with correspondingly larger number of solute particle clusters interacting with the solvent. Thus, second virial coefficient is a measure of inter-particle interactions and hence, is temperature dependent.

11 Fig 3: Plot of reduced osmotic pressure Π/c 2 versus c 2 at different temperatures The temperature at which, for a given polymer-solvent pair, the polymer exists in an unperturbed state is termed as theta (θ) temperature. This is also the temperature at which the second virial coefficient becomes zero. In Fig 2, T2 is θ temperature. This temperature is also the critical solution temperature for a polymer of infinite molar mass. Since B is temperature dependent therefore, for each polymer-solvent pair, there exists a unique temperature at which B = 0. At theta temperature, there exists no interactions between different polymer chains; the polymer solution behaves as an ideal solution at this temperature. Above theta temperature, i.e T1 (Fig 2) there is positive deviation and a positive value of B indicates that the polymer is insoluble in the solvent, while below theta temperature, there is negative deviation (T3) (Fig 2) and the polymer chain segments will attract one other and eventually phase separation will occur. Moreover, a negative value of B is indicative of a good solvent (i.e. the polymer will be highly soluble in the solvent due to favourable polymer-solvent intermolecular interactions) Mathematical proof: Osmotic pressure gives number average molar mass. The osmotic pressure measurement leads to number-average molar mass because osmotic pressure is a colligative property which depends upon the number of molecules of the polymer and not on their masses. All the molecules, whether heavy or light, make equal contribution. Mathematically: Since osmotic pressure (Π) consists additively of the partial osmotic pressures Π 1, Π 2, Π 3, for the molecules of different degrees of polymerization 1,2,3, etc. Thus from equation 12:

12 Π = Π 1 + Π 2 + Π = c 1RT M 1 + c 2RT M 2 + c 3RT M The same holds for the total concentration: c = c1+ c2 +c3 +. Thus, the osmotic-average molar mass can be written as: M osmotic = (c 1 +c 2 +c 3 + )RT Π 1+ Π 2+ Π 3+ = (c 1 +c 2 +c 3 + )RT RT( c 1 M1 + c 2 M2 + c 3 M3 + ) = i c i i c i i M i = M n Hence the expression proves that osmotic pressure yields number average molar mass (M n). 5. SUMMARY In this module, we have learnt: Osmometry technique for determination of molar mass of polymers. Osmotic pressure of a solution is given by the van t hoff equation i.e. Π = ( n 2 ) RT = crt V Polymer solution is a non ideal solution. Thus osmotic pressure for a polymer solution is given by: Where: B = second virial coefficient. M 2= Molar mass of the polymer. Osmometry technique provides number average molar mass i.e M n M n = i N im i i N i