What you'll be able to do本章學習成果
- Convert between a molar analysis and a gravimetric (mass) analysis of a gas mixture.在氣體混合物的莫爾分析與質量分析之間互相轉換。
- Compute the apparent molecular weight and the mixture gas constant $R = R_u/M$.計算視平均分子量與混合物氣體常數 $R = R_u/M$。
- Apply the Dalton model and partial pressures to an ideal-gas mixture.將道耳頓模型與分壓應用於理想氣體混合物。
- Evaluate mixture $U$, $H$, and $S$ by summing component contributions.以各成分貢獻求和的方式計算混合物的 $U$、$H$ 與 $S$。
Key equations重要公式
Describing mixture composition描述混合物組成
Most working gases are mixtures. For component $i$:大多數工作氣體均為混合物。對於成分 $i$：
Composition is given as mass fraction and mole fraction, each summing to unity:組成以質量分率與莫爾分率表示，各自總和為 1：
The apparent (average) molecular weight is the mole-fraction average:視平均分子量為莫爾分率加權平均：
With $M$ in hand the mixture behaves as a single ideal gas of gas constant $R = R_u/M$ — which is exactly where air's familiar $R = 0.287$ kJ/kg·K comes from ($M = 28.97$ kg/kmol).求得 $M$ 之後，混合物即可視為氣體常數 $R = R_u/M$ 的單一理想氣體——空氣常用的 $R = 0.287$ kJ/kg·K 正是這樣來的（$M = 28.97$ kg/kmol）。
Worked example範例 Molar analysis → mass fractions莫爾分析轉質量分率 ›
A gas mixture is 50% N₂, 35% CO₂, 15% O₂ by mole. Find (a) the apparent molecular weight and (b) the mass-fraction analysis.
(a) Using rounded molecular weights: $M = 0.50(28) + 0.35(44) + 0.15(32) = 34.2\;\tfrac{\text{kg}}{\text{kmol}}$.
(b) Base it on 1 kmol of mixture, so $n_i = y_i$ and $m_i = n_i M_i$:
| Component | nᵢ | Mᵢ | mᵢ (kg) | mfᵢ |
|---|---|---|---|---|
| N₂ | 0.50 | 28 | 14.0 | 40.9% |
| CO₂ | 0.35 | 44 | 15.4 | 45.0% |
| O₂ | 0.15 | 32 | 4.8 | 14.0% |
| Total | 1.00 | 34.2 | 34.2 | 100% |
The heavier CO₂ carries a larger mass share than its mole share.較重的 CO₂ 占有的質量分率遠大於其莫爾分率。
The Dalton model道耳頓模型
When the mixture and each component behave as ideal gases, the Dalton model treats each component as if it alone filled volume $V$ at temperature $T$:當混合物及各成分都行為如理想氣體時，道耳頓模型將每一成分視為單獨充滿體積 $V$ 且溫度為 $T$：
Each component exerts a partial pressure $p_i$ equal to its mole fraction times the total, and partial pressures sum to the total:每一成分施加的分壓 $p_i$ 等於其莫爾分率乘以總壓，各分壓相加等於總壓：
The model works because ideal-gas molecules do not interact: each component is blind to the others and fills the container as though it were alone. That assumption is also the model's limit — at high pressure, where molecules do feel each other, partial pressures no longer add cleanly.此模型成立的原因在於理想氣體分子彼此不作用：每一成分「看不見」其他成分，如同獨自充滿容器。這個假設同時也是模型的極限——在高壓下分子彼此有感，分壓便不再單純相加。
Mixture U, H, and S混合物的 U、H 與 S
With the Dalton model, $U$, $H$, and $S$ of the mixture are found by adding each component's contribution at the conditions it experiences — temperature $T$ and its own partial pressure $p_i$:在道耳頓模型中，混合物的 $U$、$H$ 與 $S$ 由各成分在其各自條件下（溫度 $T$ 與自身分壓 $p_i$）的貢獻相加得到：
Note the asymmetry: $u$ and $h$ of an ideal gas depend on temperature alone, so the composition enters only through the mass fractions. Entropy is different — it depends on pressure, and each component sees its partial pressure, not the total. That single detail is what produces the entropy of mixing.注意此處的不對稱性：理想氣體的 $u$ 與 $h$ 僅與溫度有關，組成只透過質量分率進入計算。熵則不同——它與壓力有關，而各成分所感受的是自身的分壓而非總壓。正是這一點造就了混合熵。
This is exactly the basis of psychrometrics, where the two components are dry air and water vapor, and the vapor's partial pressure sets the dew point.這正是濕空氣學的基礎，其中兩個成分分別為乾空氣與水蒸氣，而水蒸氣的分壓決定露點。