What you'll be able to do本章學習成果
- Apply the ideal-gas equation of state and know its valid range.應用理想氣體狀態方程式並知道其適用範圍。
- Use the temperature-only dependence of $u$ and $h$, with $c_v$, $c_p$, and ideal-gas tables.利用 $u$、$h$ 僅與 $T$ 有關的特性，搭配 $c_v$、$c_p$ 與理想氣體表。
- Describe mixture composition and apply the Dalton model of partial pressures.描述混合物組成並應用分壓的道耳頓模型。
- Evaluate the compressibility factor $Z$ and judge when ideal-gas results are acceptable.計算壓縮因子 $Z$ 並判斷理想氣體結果的可用性。
- Read the generalized chart via reduced properties (corresponding states).以對比性質讀取廣義壓縮因子圖（對應狀態原理）。
Key equations重要公式
The ideal-gas equation of state理想氣體狀態方程式
The simplest — and most useful — equation of state:最簡單也最常用的狀態方程式：
$R_u = 8.314$ kJ/kmol·K is the universal gas constant; $M$ is the molar mass. Equivalent forms: $pV = mRT$, $pV = nR_uT$; between two states, $p_1V_1/T_1 = p_2V_2/T_2$.$R_u = 8.314$ kJ/kmol·K 為通用氣體常數，$M$ 為莫爾質量。等效形式：$pV = mRT$、$pV = nR_uT$；兩狀態間：$p_1V_1/T_1 = p_2V_2/T_2$。
When does it hold?何時適用？
The ideal-gas model assumes point-mass molecules with no intermolecular forces — valid at low density: low pressure and/or high temperature relative to the critical point. Air at ordinary conditions is excellent. Water vapor is ideal-gas-like below about 10 kPa (fine for air-conditioning), but not at steam-plant pressures — there the steam tables are mandatory.理想氣體模型假設分子為點質量且無分子間作用力——在低密度（低壓或相對臨界點高溫）時成立。一般狀態的空氣極為理想。水蒸氣在大約 10 kPa 以下接近理想氣體（適用於空調），但在蒸氣廠的高壓下則不適用——必須查蒸汽表。
Energy of an ideal gas理想氣體的能量
A defining feature (Joule, 1843): for an ideal gas, internal energy and enthalpy depend on temperature alone:氣體的定義特性（焦耳，1843年）：理想氣體的內能與焓僅與溫度有關：
For modest temperature ranges take specific heats constant ($\Delta u = c_v\Delta T$, $\Delta h = c_p\Delta T$). Over wide ranges, use the ideal-gas tables (e.g. air in Table A-22).對於溫度範圍不大的情況，可取比熱為常數（$\Delta u = c_v\Delta T$、$\Delta h = c_p\Delta T$）。溫度範圍寬時，應使用理想氣體表。
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:視說（平均）分子量為莫爾分率加權平均：
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：
With the Dalton model, $U$, $H$, and $S$ of the mixture are found by adding each component's contribution at the conditions it experiences. This is exactly the basis of psychrometrics, where the two components are dry air and water vapor.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的貢獻相加得到。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。
Compressibility factor壓縮因子
The deviation from ideal behavior is captured by a single dimensionless number:偷離理想行為的程度由一個無因次數字表張：
$Z = 1$ is exactly ideal. Deviations are largest near the critical point and saturation line. $Z < 1$ means attractive forces dominate; $Z > 1$ means molecular volume dominates at high pressure.$Z = 1$ 為完全理想氣體。偏差在臨界點與飽和線附近最大。$Z < 1$ 表示吸引力佔主導；$Z > 1$ 表示高壓下分子體積佔主。
Compressibility explorer壓縮因子探索器
The curves show $Z$ vs reduced pressure at several reduced temperatures. Move your state and read $Z$, the volume error, and a verdict on whether the ideal-gas model is safe. Notice the deep dip near the critical region ($T_R \approx 1$).曲線顯示 $Z$ 隨對比壓力的變化（多個對比溫度）。移動狀態點即可讀得 $Z$、體積誤差及是否適用理想氣體的判斷。在臨界區（$T_R \approx 1$）附近有一個明顯的下凹。
Generalized behavior via the Redlich–Kwong equation (corresponding-states form). Representative, not substance-exact.
Corresponding states對應狀態原理
Plotted against reduced coordinates ($p_R = p/p_{cr}$, $T_R = T/T_{cr}$), the compressibility factors of most gases collapse onto nearly one curve. This principle of corresponding states means a single generalized chart estimates $Z$ for any gas from just its critical data.以對比座標（$p_R = p/p_{cr}$、$T_R = T/T_{cr}$）繪圖，大多數氣體的壓縮因子游走於近乎同一条曲線上。對應狀態原理表明，一張廣義圖即可為任何氣體估算 $Z$，只需其臨界點資料。
Treat the ideal-gas model as "high or low" relative to the critical values. A gas is safely ideal when $p_R \ll 1$ or $T_R \gg 1$.以相對於臨界值的「高低」來判斷理想氣體模型的適用性。當 $p_R \ll 1$ 或 $T_R \gg 1$ 時，可安心使用理想氣體模型。
Other equations of state其他狀態方程式
Multi-constant equations add molecular volume and intermolecular attraction effects:多常數形式加入分子體積與分子間吸引力的效應：
- van der Waals — two constants; qualitatively right but limited accuracy.范德瓦耳斯——兩常數；定性正確但精度有限。
- Redlich–Kwong — two constants; markedly better accuracy.Redlich–Kwong——兩常數；精度明顯提升。
- Beattie–Bridgeman (5) and Benedict–Webb–Rubin (8) — accurate to high density.Beattie–Bridgeman（5個）與 Benedict–Webb–Rubin（8個）——高密度下仍精確。
- Virial equation — expands $Z$ as a power series in $1/v$.維里方程式——以 $1/v$ 的冪級數展開 $Z$。
These are developed further in Thermodynamic Property Relations where you can plot isotherms directly.這些將在熱力學性質關係式中進一步展開，可直接繪製等溫線。