Equations of State狀態方程式

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。
• 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）。
• Use cubic and multi-constant equations of state — van der Waals, Redlich–Kwong, virial — and explain what each term represents physically.運用立方型與多常數狀態方程式——范德瓦耳斯、Redlich–Kwong、維里——並說明各項的物理意義。

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：$p_1V_1/T_1 = p_2V_2/T_2$。

Mixtures of ideal gases — air, combustion products, moist air — follow the same equation with an apparent molar mass; that case has its own chapter, Ideal-Gas Mixtures.理想氣體混合物——空氣、燃燒產物、濕空氣——以視平均分子量代入同一方程式即可；該主題另見理想氣體混合物一章。

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。

Compressibility factor壓縮因子

The deviation from ideal behavior is captured by a single dimensionless number:偏離理想行為的程度由一個無因次數字表示：

$Z = 1$ 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幾乎落在同一條曲線上。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。

Cubic equations of state立方型狀態方程式

An accurate $p$–$v$–$T$ relationship is the foundation for every other property. Beyond the ideal gas, two-constant cubic equations add the molecular effects the ideal model ignores — finite molecular volume and intermolecular attraction:精確的 $p$–$v$–$T$ 關係式是所有其他性質的基礎。在理想氣體之外，兩常數立方型方程式加入了理想模型所忽略的分子效應——有限分子體積與分子間吸引力：

The constants $a$ and $b$ come from the critical-point data. Read the structure physically: subtracting $b$ from $v$ says the molecules cannot occupy all the volume, which raises pressure; subtracting the $a$ term says attraction pulls molecules back from the wall, which lowers it. Set $a = b = 0$ and you recover $pv = RT$.常數 $a$ 與 $b$ 由臨界點資料決定。從物理上解讀其結構：自 $v$ 減去 $b$，代表分子無法佔滿全部體積，使壓力升高；減去含 $a$ 的項，代表吸引力把分子從器壁拉回，使壓力降低。令 $a = b = 0$ 即還原為 $pv = RT$。

Equation-of-state p–v plotter狀態方程式 p–v 繪圖器

Plot isotherms for CO₂ and compare the ideal gas with the van der Waals and Redlich–Kwong equations. Above the critical temperature the curves are smooth and monotonic; drop below $T_c = 304$ K and the cubic equations develop the famous van der Waals loop — the equation's attempt to describe the two-phase region. Toggle each model and watch where the ideal gas diverges.繪製 CO₂ 的等溫線，比較理想氣體、范德瓦耳斯方程式與 Redlich–Kwong 方程式。在臨界溫度以上，曲線平滑單調；低於 $T_c = 304$ K 時，立方型方程式會出現著名的范德瓦耳斯迴圈——方程式試圖描述兩相區的結果。切換各模型，觀察理想氣體在哪裡開始偏離。

Multi-constant equations多常數方程式

More constants buy more accuracy over a wider range:常數越多，可涵蓋的範圍越廣、精度越高：

• 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 $Z$。

Any of these can be fed into the machinery of Thermodynamic Property Relations, which turns $p$–$v$–$T$ data into the $u$, $h$, and $s$ values printed in the tables.任一方程式皆可送入熱力學性質關係式的推導機制，將 $p$–$v$–$T$ 資料轉換為性質表中所列的 $u$、$h$、$s$ 數值。