平均摩尔质量 Average molar mass of a mixture — 公式推导、符号与局限

\[\bar M = \frac{m_{\text{总}}}{n_{\text{总}}} = \sum_i x_i M_i\]\[\bar M = \frac{m_{\text{total}}}{n_{\text{total}}} = \sum_i x_i M_i\]

由气体密度或相对密度from gas density or relative density

\[\bar M = \rho\, V_m = D\cdot M_{\text{参比}}\]\[\bar M = \rho\, V_m = D\cdot M_{\text{ref}}\]

含义Meaning

混合物的总质量除以总物质的量，等于各组分摩尔质量按摩尔分数的加权平均。

Total mass over total amount — the mole-fraction-weighted mean of the component molar masses.

符号Symbol	含义 Meaning	单位 Unit
\(\bar M\)	平均摩尔质量average molar mass	g·mol⁻¹
\(x_i\)	组分 i 的摩尔分数（对气体也是体积分数）mole fraction of i (= volume fraction for gases)	1
\(M_i\)	组分 i 的摩尔质量molar mass of i	g·mol⁻¹

\(m_{\text{总}} = \sum n_i M_i\)，\(n_{\text{总}} = \sum n_i\)。
\(m_{\text{tot}} = \sum n_i M_i\), \(n_{\text{tot}} = \sum n_i\).
相除并令 \(x_i = n_i/n_{\text{总}}\)：
Divide and set \(x_i = n_i/n_{\text{tot}}\):
\[\bar M = \frac{\sum n_i M_i}{\sum n_i} = \sum x_i M_i\]
二组分时 \(\bar M = M_1 + x_2(M_2 - M_1)\)，随组成线性变化，用“十字交叉法”可快速求组成。
For two components \(\bar M = M_1 + x_2(M_2 - M_1)\) — linear in composition, the basis of the ‘cross method’.

加权必须用摩尔分数，不能用质量分数。
Weight by mole fraction, not mass fraction.
对发生反应或缔合的气体（如 NO₂ ⇌ N₂O₄），\(\bar M\) 随平衡移动而变化，可据此求平衡组成。
For reacting or associating gases (NO₂ ⇌ N₂O₄) \(\bar M\) shifts with the equilibrium — useful for finding equilibrium composition.

空气约含 78% N₂、21% O₂、1% Ar（体积）：M̄ = 0.78×28 + 0.21×32 + 0.01×40 = 29 g/mol。

Air (78 % N₂, 21 % O₂, 1 % Ar by volume): M̄ ≈ 29 g/mol.