偶极矩 Dipole moment — 公式推导、符号与局限

\[\boldsymbol\mu = \sum_i q_i\mathbf r_i,\qquad \mu = q\,d\ (\text{两点电荷}),\qquad 1\ \mathrm D = 3.336\times10^{-30}\ \mathrm{C\,m}\]\[\boldsymbol\mu = \sum_i q_i\mathbf r_i,\qquad \mu = q\,d\ (\text{two point charges}),\qquad 1\ \mathrm D = 3.336\times10^{-30}\ \mathrm{C\,m}\]

含义Meaning

衡量分子中正负电荷中心分离程度的矢量，方向（化学惯例）从正指向负。分子偶极矩是各键矩的矢量和，所以对称分子（CO₂、CCl₄、BF₃）为零。

A vector measuring the separation of positive and negative charge centres. The molecular dipole is the vector sum of bond dipoles, so symmetric molecules (CO₂, CCl₄, BF₃) have zero dipole.

符号说明Symbols

符号Symbol	含义 Meaning	单位 Unit
\(\mu\)	偶极矩dipole moment	D, C·m
\(q\)	电荷charge	C
\(d\)	电荷间距separation	m

推导Derivation

电荷分布的一阶矩即偶极矩；总电荷为零时与原点选择无关。
The first moment of the charge distribution; origin-independent for neutral species.
键矩矢量相加：H₂O 中两个 O–H 键矩夹角 104.5°，合成 \(\mu = 2\mu_\text{OH}\cos(104.5°/2)\)。
Add bond moments vectorially: for water \(\mu = 2\mu_\text{OH}\cos 52.25°\).
测量：德拜方程 \(P_m = \frac{N_A}{3\varepsilon_0}\left(\alpha + \frac{\mu^2}{3k_\mathrm BT}\right)\)，由摩尔极化随温度的变化分离出 μ。
Debye: molar polarisation versus 1/T separates μ from polarisability.

适用条件与局限Assumptions & limitations

键矩可加性只是近似（孤对、诱导效应）；点电荷模型只是定性。
Bond-moment additivity is approximate.

例题Worked example

HCl：μ = 1.08 D，键长 127 pm；若完全离子化 μ = e × 127 pm = 6.1 D，所以离子性约 18%。

HCl: 1.08 D vs 6.1 D if fully ionic → ~18 % ionic.