含义Meaning
逐步（缩）聚合中数均聚合度与反应程度的关系。要得到高分子量，必须使反应程度极接近 1（p = 0.99 才 \(\bar X_n\) = 100），并严格控制等当量配比。
Relates number-average degree of polymerisation to conversion in step growth. High polymer needs p very close to 1 (p = 0.99 gives only X̄n = 100) and exact stoichiometry.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\bar X_n\) | 数均聚合度number-average degree of polymerisation | 1 |
| \(p\) | 官能团反应程度extent of reaction of functional groups | 1 |
| \(r\) | 两种官能团的摩尔比（≤ 1）molar ratio of functional groups (≤ 1) | 1 |
| \(f_\text{av}\) | 平均官能度average functionality | 1 |
推导Derivation
起始 N₀ 个单体（等当量双官能），每形成一个键分子总数减 1。
Start with N₀ bifunctional monomers; each bond formed reduces the molecule count by one.
反应程度 p 时剩余分子数 \(N = N_0(1 - p)\)。
At conversion p, \(N = N_0(1-p)\) molecules remain.
\(\bar X_n = N_0/N = 1/(1 - p)\)。
\(\bar X_n = N_0/N = 1/(1-p)\).
非等当量时剩余端基由过量组分决定，推得含 r 的形式；p → 1 时 \(\bar X_n \to (1+r)/(1-r)\)，1% 过量即把上限限制在 201。
With imbalance r the limit is (1+r)/(1−r): a 1 % excess caps X̄n at 201.
适用条件与局限Assumptions & limitations
假设官能团等活性（与链长无关，弗洛里假设）、无环化和副反应。
Assumes equal reactivity (Flory), no cyclisation or side reactions.
凝胶点公式高估 p_c；弗洛里–斯托克迈尔统计理论更准确。
The gel-point estimate is too high; Flory–Stockmayer statistics are better.
例题Worked example
尼龙-66：p = 0.995 → X̄n = 200（约 100 个重复单元，M_n ≈ 22.6 kg/mol）。
Nylon-6,6: p = 0.995 → X̄n = 200 (~100 repeat units, M_n ≈ 22.6 kg/mol).