Number Average Molecular Weight of Polymers | Calculation & Definition

Polydispersity Index (PDI) is calculated as the ratio of weight average molar mass to number average molar mass: PDI = Mw / Mn. This index indicates the breadth of the molar mass distribution in a polymer sample. A PDI value of 1 indicates a monodisperse sample where all molecules have identical molar masses, while values greater than 1 indicate a polydisperse sample with a distribution of molecular weights. Molar mass distribution is typically represented graphically, showing the relationship between the number of molecules and their molar masses.

The polydispersity index (PDI) measures the heterogeneity of a polymer sample and is calculated as the ratio of average mass molecular weight (Mw) to average number molecular weight (Mn). The formula is PDI = Mw/Mn. When PDI equals 1, the polymer is monodisperse (uniform); when PDI is greater than 1, the polymer is polydisperse (heterogeneous). For example, if Mw = 40,000 and Mn = 30,000, then PDI = 40,000/30,000 = 1.33, indicating a polydisperse sample.

Polymers consist of chains with varying molecular weights due to heterogeneity in chain length, branching, and end groups, making exact molecular weight measurement impossible; instead, four types of average molecular weights are calculated: number average molecular weight (MN̄) = Σ(n_i × M_i)/Σ(n_i), weight average molecular weight (MW̄) = Σ(n_i × M_i²)/Σ(n_i × M_i), Z-average molecular weight (MZ̄) = Σ(n_i × M_i³)/Σ(n_i × M_i²), and viscosity average molecular weight (MV̄) = [Σ(w_i × M_i^(1+α))/Σ(w_i)]^(1/α), where the polydispersity index (PDI = MW̄/MN̄) indicates the breadth of molecular weight distribution, with PDI = 1 representing monodisperse systems and values typically ranging from 1.5 to 2.5 for polydisperse polymers.

In polymer science, average molecular weights are calculated using different statistical methods: Number Average Molecular Weight (MN) = total weight / total number of chains, Weight Average Molecular Weight (MW) = Σ(wi × Mi), and Z-Average Molecular Weight (MZ) = Σ(ni × Mi³) / Σ(ni × Mi²). The Polydispersity Index (PDI or dispersity, symbol Đ) is defined as MW/MN and measures the breadth of molecular weight distribution, with values greater than 1 indicating broader distributions. Higher moments like viscosity average molecular weight (Mv) exist between MN and MW depending on the Mark-Houwink exponent (a = 0.5-0.8). Different experimental techniques measure these averages: osmotic pressure for MN, light scattering for MW, centrifugation for MZ, and intrinsic viscosity for Mv.

Polydispersity Index (PDI) measures polymer sample heterogeneity: PDI = Mw / Mn. PDI = 1 indicates monodisperse (uniform), while PDI > 1 indicates polydisperse (varying molecular weights). Molecular weight distribution is graphically represented with molecular weight on x-axis and fraction of chains on y-axis. Viscosity concepts include relative viscosity (ηr = η/η0), specific viscosity (ηsp = ηr - 1), and reduced viscosity (ηred = ηsp/c). Intrinsic viscosity [η] is the limit of reduced viscosity as concentration approaches zero. The Mark-Houwink equation [η] = K × M^a relates intrinsic viscosity to molecular weight, where K and a are constants.

This lecture covers three primary methods for determining polymer molecular weight: membrane osmometry, which uses osmotic pressure measurements to determine number-average molecular weight (MN) through the relationship Π/C = RT/MN + A2C, where A2 is the second virial coefficient; end group analysis, which calculates MN by quantifying the number of terminal functional groups per polymer chain using techniques like titration or NMR spectroscopy; and dilute solution viscometry, which measures intrinsic viscosity to determine MN. The lecture emphasizes that colligative property methods (osmometry, vapor pressure, boiling point elevation, cryoscopy) are sensitive to impurities and typically provide relative rather than absolute molecular weight values, while osmometry is theoretically absolute but practically requires careful experimental conditions.

Membrane osmometry is an experimental technique that measures the osmotic pressure of polymer solutions to determine the number average molecular weight (Mn) and chi parameter (χ) in a single plot; osmotic pressure is a colligative property arising from the thermodynamic driving force of mixing, where solvent flows through a semi-permeable membrane from pure solvent to polymer solution until equilibrium is reached when chemical potentials equalize, and this pressure can be measured by the height difference in a vertical tube and related to polymer properties through fundamental equations.

This extensive section covers experimental techniques for polymer characterization. Osmometry determines number average molar mass (Mn) using π/C = RT/Mn + B'C, where the intercept gives Mn = RT/C_intercept. Viscosity measurements determine intrinsic viscosity [η] from η/η₀ - 1 = [η] × C, with [η] obtained from the intercept of (η/η₀ - 1)/C versus C plot. The Mark-Houwink equation [η] = K × M^α then yields viscosity average molar mass. For equimolar mixtures (equal moles), Mn = (M1 + M2 + M3)/3 and Mw = (M1² + M2² + M3²)/(M1 + M2 + M3). For hemoglobin (15.5), ribonuclease (13.7), and myoglobin (17.2), Mn = 15.5 kg/mol and Mw = 15.6 kg/mol, yielding PDI = 1.006 > 1, demonstrating that Mw > Mn for any polydisperse sample.

Successful membrane osmometry requires precise temperature control, sufficiently dilute solutions (<20 g/L), and appropriate membrane selection based on solvent type (cellulose for organics, glass for corrosives). Membrane porosity determines cutoff and equilibration time. The method has molecular weight limits: upper limit ~10⁶ g/mol due to very low osmotic pressures for high MN, and lower limit determined by membrane cutoff (smaller MN chains pass through and don't contribute). Vapor pressure osmometry uses two thermistors in a sealed chamber saturated with solvent vapor—one containing pure solvent, the other polymer solution. Lower solvent chemical potential in solution causes vapor condensation, raising solution thermistor temperature. Using Clausius-Clapeyron equation, ΔT/C = (K × ΔH_vap)/(MN × R × T²). Unlike membrane osmometry, VPO requires calibration with known polymers because steady-state rather than true equilibrium is reached. It is limited to MN ~10⁴ g/mol due to small temperature differences. Both techniques are complementary: membrane osmometry for moderate to high MN, VPO for lower MN ranges.

This section covers polymer molar mass concepts and measurement techniques. Due to random polymerization, real samples contain chains of varying lengths. The number average molar mass (Mn) is calculated as the sum of (molar mass × number of molecules) divided by total molecules. The weight average molar mass (Mw) is calculated as the sum of (molar mass² × number of molecules) divided by the sum of (molar mass × number of molecules). The polydispersity index (PDI = Mw/Mn) indicates distribution breadth: PDI = 1 means monodisperse, PDI > 1 means polydisperse. Two experimental methods determine molar mass: membrane osmometry measures Mn via osmotic pressure (π = RT/Mn × c), while light scattering measures Mw by analyzing scattered light intensity at different angles using the Zimm equation.

Molecular weight fundamentally characterizes polymer properties. Higher molecular weight increases melting/boiling temperatures, glass transition temperature, impact resistance, strength, toughness, and viscosity. Due to chain entanglement and Van der Waals interactions, a tenfold chain length increase can produce over 1,000-fold viscosity increase. Polymers exhibit molecular weight distributions rather than uniform chains. Number average molecular weight equals total polymer weight divided by total molecules (near distribution peak). Weight average molecular weight shifts toward higher values because heavier chains contribute proportionally more to weight. The degree of polymerization equals number average molecular weight divided by monomer molecular weight.

Glass transition temperature (Tg) is when amorphous polymers transition from hard glassy to soft rubbery state. Melting temperature (Tm) is when crystalline polymers transition from solid to liquid. Tg is gradual, Tm is sharp. Plasticizers increase chain mobility, lowering Tg. Molecular weight affects properties: higher molecular weight increases strength and toughness. Polymers have different chain lengths, so molecular weight is an average. Number-average (Mn) and weight-average (Mw) molecular weights describe distribution breadth.

Polymers exhibit diverse properties based on their molecular structure, with molecular weight and stereochemistry playing critical roles in determining material behavior; number-average molecular weight (Mn) represents the conventional average of all polymer chains, while weight-average molecular weight (Mw) gives more weight to longer chains, and the ratio Mw/Mn (polydispersity index) indicates the breadth of molecular weight distribution, with higher molecular weights generally increasing polymer toughness and thermal stability through greater chain entanglement.

Physical properties determine polymer behavior: (1) Degree of polymerization is the number of repeating units per chain; molecular weight equals degree of polymerization multiplied by repeating unit weight; since polymer molecules vary in size, averages are used: number average molecular weight (Mn = Σ(ni×mi)/Σnj) and weight average molecular weight (Mw = Σ(ni×mi²)/Σ(nj×mj)); (2) Crystallinity describes chain arrangement - lamellar crystalline structures have regular folded chains while amorphous regions have irregular arrangements; percentage crystallinity is calculated using densities; (3) Thermal properties include melting point (first-order transition for crystalline regions) and glass transition temperature (second-order transition for amorphous regions); Tg increases with molecular weight up to ~20,000, affected by intermolecular forces, chain stiffness, cross-linking, pendant groups, plasticizers, and molecular weight.

Polymer samples have distributions of chain lengths, with the average called molecular weight. Longer chains are more tangled and resistant to stress, while shorter chains slip past each other easily. As molecular weight increases, tensile strength and impact resistance increase initially then plateau, while melt viscosity increases continuously. Higher molecular weight polymers are more expensive due to greater stiffness and deformation resistance. The distribution shape (normal or weighted) affects processing ease and mechanical properties.