双语化学辞典Chem Lexicon
本科Undergraduate 聚合反应与高分子物理 Polymerisation & polymer physics

重均分子量Weight-average molar mass

\[\bar M_w = \frac{\sum_i N_iM_i^2}{\sum_i N_iM_i} = \sum_i w_iM_i\]
z 均与黏均分子量z-average and viscosity-average
\[\bar M_z = \frac{\sum N_iM_i^3}{\sum N_iM_i^2},\qquad [\eta] = K\bar M_v^{\,a}\]

含义Meaning

按质量分数加权的平均摩尔质量，对大分子更敏感。静态光散射直接给出 \(\bar M_w\)。总有 \(\bar M_w \ge \bar M_n\)。

Molar mass averaged by weight fraction, emphasising large chains; static light scattering measures it directly. Always \(\bar M_w \ge \bar M_n\).

符号说明Symbols

符号Symbol含义 Meaning单位 Unit
\(w_i\)质量分数weight fraction1

推导Derivation

  1. 以质量分数 \(w_i = N_iM_i/\sum N_jM_j\) 为权重对 \(M_i\) 取平均。

    Weight each \(M_i\) by its mass fraction.

  2. 光散射强度正比于 \(\sum N_iM_i^2\)，归一化后即 \(\bar M_w\)（齐姆图外推）。

    Scattered intensity ∝ ΣN_iM_i², giving \(\bar M_w\) (Zimm plot).

  3. 由柯西–施瓦茨不等式 \(\left(\sum N_iM_i\right)^2 \le \sum N_i\sum N_iM_i^2\) 得 \(\bar M_w \ge \bar M_n\)。

    Cauchy–Schwarz gives \(\bar M_w \ge \bar M_n\).

适用条件与局限Assumptions & limitations

  • 少量超高分子量组分（凝胶、聚集体）会显著抬高 \(\bar M_w\)。

    A little very-high-mass material (gel, aggregates) inflates \(\bar M_w\).

例题Worked example

同质量的 10 与 100 kg/mol 链：M_w = 55 kg/mol，而 M_n = 18.2 kg/mol。

Equal masses of 10 and 100 kg/mol: M_w = 55, M_n = 18.2 kg/mol.