道尔顿分压定律 Dalton's law of partial pressures — 公式推导、符号与局限

\[p_{\text{总}} = \sum_i p_i, \qquad p_i = x_i\, p_{\text{总}}\]\[p_{\text{total}} = \sum_i p_i, \qquad p_i = x_i\, p_{\text{total}}\]

含义Meaning

混合理想气体的总压等于各组分分压之和；某组分的分压等于它单独占据整个容器时的压强。

The total pressure of an ideal-gas mixture is the sum of the partial pressures; each is the pressure that gas would exert alone.

符号Symbol	含义 Meaning	单位 Unit
\(p_i\)	组分 i 的分压partial pressure of i	Pa
\(x_i\)	摩尔分数（= 体积分数）mole fraction (= volume fraction)	1

理想气体分子之间无作用，每种气体各自满足 \(p_iV = n_iRT\)。
Ideal molecules do not interact, so each gas obeys \(p_iV = n_iRT\).
相加：\(\left(\sum p_i\right)V = \left(\sum n_i\right)RT = n_{\text{总}}RT\)，所以 \(p_{\text{总}} = \sum p_i\)。
Sum: \(\left(\sum p_i\right)V = n_{\text{tot}}RT\), so \(p_{\text{tot}} = \sum p_i\).
两式相除：\(p_i/p_{\text{总}} = n_i/n_{\text{总}} = x_i\)。
Divide: \(p_i/p_{\text{tot}} = n_i/n_{\text{tot}} = x_i\).

只对理想气体混合物严格成立；高压下分子间作用使其偏离（应用逸度）。
Exact only for ideal mixtures; high pressure needs fugacities.
排水集气时要扣除水蒸气分压：\(p_{\text{气}} = p_{\text{大气}} - p(\mathrm{H_2O})\)。
Gas collected over water: subtract the water vapour pressure.

空气中 O₂ 的摩尔分数 0.21，在 101 kPa 下 p(O₂) = 21 kPa；在 3000 m 高原（约 70 kPa）只有 15 kPa。

x(O₂) = 0.21: p(O₂) = 21 kPa at sea level but only 15 kPa at 3000 m (≈70 kPa).