Number-average vs weight-average molecular weight; polydispersity index

4.6.5 · Chemistry › Polymers

HUM KYA average kar rahe hain?

In masses ka average lene ke do honest tarike hain.

HAR AVERAGE ko scratch se kaise banayein

1. Number-average molecular weight ${\bar{M}}_n$

Derivation (first principles):

Sample ka total mass = har type ki (number × mass) ka sum: $\text{Total mass}={\sum }_i{N}_i{M}_i$

Molecules ki total sankhya: $\text{Total number}={\sum }_i{N}_i$

Per molecule average mass = total mass ÷ total number:

Yeh bas ordinary arithmetic mean hai jo kitne molecules hain us hisaab se weighted hai.

2. Weight-average molecular weight ${\bar{M}}_w$

Derivation (first principles):

Species $i$ ka weight fraction (total mass mein uska hissa): $w_i=\frac{N_i{M}_i}{{\sum }_j{N}_j{M}_j}$

Mass-fraction se weighted average mass: ${\bar{M}}_w={\sum }_i{w}_i{M}_i=\frac{{\sum }_i{N}_i{M}_i\cdot M_i}{{\sum }_j{N}_j{M}_j}$

Polydispersity Index (PDI)

Worked Examples

Forecast-then-Verify

Common Mistakes (Steel-manned)

Active Recall

Connections

• Addition vs Condensation Polymerization — condensation Flory follow karta hai: PDI $=1+p\to 2$ high conversion par.
• Gel Permeation Chromatography — size ke hisaab se separate karta hai, poori distribution aur dono averages extract kar sakta hai.
• Light Scattering and Osmometry — light scattering → ${\bar{M}}_w$; osmotic pressure (colligative) → ${\bar{M}}_n$.
• Colligative Properties — particles ki number par depend karte hain, isliye ${\bar{M}}_n$ dete hain.
• Degree of Polymerization — ${\bar{X}}_n={\bar{M}}_n/M_0$ (monomer mass).
• Variance and Standard Deviation — ${\bar{M}}_w/{\bar{M}}_n-1$ distribution ke relative variance se related hai.

Concept Map