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, Peng–Robinson, 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、Peng–Robinson、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：$p_1V_1/T_1 = p_2V_2/T_2$。

Each gas has its own $R$. Values for common gases at 300 K:每種氣體各有其 $R$。常見氣體於 300 K 的數值：

$R = R_u/M$ with $R_u = 8.314$ kJ/kmol·K; $c_v = c_p - R$; $k = c_p/c_v$. Specific heats are ideal-gas values at 300 K (textbook Table A-2).$R = R_u/M$，$R_u = 8.314$ kJ/kmol·K；$c_v = c_p - R$；$k = c_p/c_v$。比熱為 300 K 的理想氣體值（教科書表 A-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：

Why temperature alone?為何僅與溫度有關？

Molecular view. Internal energy is the sum of two parts: the kinetic energy of molecular motion (translation, rotation, vibration) and the potential energy of the forces between molecules. Kinetic theory shows the average kinetic energy per molecule is set by temperature alone. The potential part depends on how far apart molecules are, which is the specific volume $v$. The ideal-gas model assumes no intermolecular forces, so that potential energy is zero at every spacing. Squeezing or expanding the gas changes nothing in $u$; only changing $T$ does.分子觀點。內能由兩部分組成：分子運動（移動、轉動、振動）的動能，以及分子間作用力的位能。由動力論可知，每個分子的平均動能僅由溫度決定。位能則取決於分子間距，即比容 $v$。理想氣體模型假設分子間無作用力，因此在任何間距下位能皆為零。壓縮或膨脹氣體都不會改變 $u$；只有改變 $T$ 才會。

Experimental view — Joule's free expansion. Joule connected a tank of compressed air to an evacuated tank, both immersed in a water bath, then opened the valve. The air expanded into the vacuum: it pushed against nothing, so $W = 0$, and the bath temperature did not change, so $Q = 0$. By the first law $\Delta u = Q - W = 0$. Yet the gas's volume and pressure changed substantially while its temperature stayed the same. Since $u$ stayed fixed while $v$ and $p$ changed, $u$ cannot depend on them — only on $T$.實驗觀點——焦耳自由膨脹。焦耳將一個裝有壓縮空氣的容器與一個抽真空的容器相連，兩者皆浸於水浴中，然後打開閥門。空氣膨脹進入真空：沒有推動任何東西，故 $W = 0$；水浴溫度不變，故 $Q = 0$。由第一定律 $\Delta u = Q - W = 0$。然而氣體的體積與壓力都大幅改變，溫度卻保持不變。既然 $v$ 與 $p$ 改變時 $u$ 不變，$u$ 就不可能是它們的函數——只能是 $T$ 的函數。

Enthalpy follows. $h = u + pv = u(T) + RT$: both terms are functions of $T$ only, so $h$ is too.焓隨之成立。$h = u + pv = u(T) + RT$：兩項皆僅為 $T$ 的函數，故 $h$ 亦然。

Caveat: Joule's water bath was too large to detect small temperature changes. Precise later experiments (Joule–Thomson) show real gases do cool slightly on expansion, because their attractive forces are not zero. The effect fades as density drops, which is exactly the regime where the ideal-gas model applies.附註：焦耳的水浴熱容量過大，無法測出微小溫度變化。後來較精密的實驗（焦耳–湯姆森）顯示，真實氣體膨脹時確實會略微降溫，因為其分子間吸引力不為零。此效應隨密度降低而消失——正是理想氣體模型適用的範圍。

Three working results follow from this fact and the definitions of the specific heats.由此事實及比熱的定義，可推得以下三個實用結果。

1 · Internal energy change1 · 內能變化

The constant-volume specific heat is defined as a partial derivative:定容比熱定義為偏導數：

For an ideal gas $u$ depends on $T$ only, so holding $v$ fixed changes nothing — the partial derivative becomes an ordinary one, and $c_v$ is itself a function of $T$ alone: $c_v(T) = du/dT$. Separate variables, $du = c_v(T)\,dT$, and integrate between two states:理想氣體的 $u$ 僅與 $T$ 有關，固定 $v$ 不影響結果——偏導數化為全導數，且 $c_v$ 本身也僅為 $T$ 的函數：$c_v(T) = du/dT$。分離變數 $du = c_v(T)\,dT$，在兩狀態間積分：

This holds for any ideal-gas process — not only constant-volume ones — because $u$ cannot tell how the gas got from $T_1$ to $T_2$.此式適用於任何理想氣體過程——不限於定容——因為 $u$ 與氣體如何由 $T_1$ 到達 $T_2$ 無關。

2 · Enthalpy change2 · 焓變化

Start from the definition of enthalpy and substitute the ideal-gas equation $pv = RT$:由焓的定義出發，代入理想氣體方程式 $pv = RT$：

Both terms on the right depend on $T$ only, so $h = h(T)$. The constant-pressure specific heat $c_p \equiv (\partial h/\partial T)_p$ therefore also reduces to an ordinary derivative, $c_p(T) = dh/dT$. Separating and integrating as before:右邊兩項皆僅與 $T$ 有關，故 $h = h(T)$。因此定壓比熱 $c_p \equiv (\partial h/\partial T)_p$ 同樣化為全導數 $c_p(T) = dh/dT$。如前分離變數並積分：

3 · Relation between the specific heats3 · 比熱間的關係

Differentiate $h = u + RT$ with respect to $T$ ($R$ is constant):將 $h = u + RT$ 對 $T$ 微分（$R$ 為常數）：

Rearranged:整理得：

Both specific heats vary with temperature, but their difference is always the gas constant. Physically, $R$ is the extra energy per kelvin needed at constant pressure to do the boundary work $p\,dv = R\,dT$ as the gas expands. Dividing by $c_v$ gives the ratio $k = c_p/c_v$, so $c_v = R/(k-1)$ and $c_p = kR/(k-1)$.兩比熱皆隨溫度變化，但其差值恆為氣體常數。物理上，$R$ 是定壓加熱時每升高 1 K 額外所需、用於氣體膨脹邊界功 $p\,dv = R\,dT$ 的能量。除以 $c_v$ 得比熱比 $k = c_p/c_v$，故 $c_v = R/(k-1)$、$c_p = kR/(k-1)$。

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 (air: 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。

Because $u$ and $h$ depend on temperature alone, an ideal-gas table needs only one index column, $T$ — no pressure, no phase, no quality. Compare the steam tables, which need two properties to fix a state. A short excerpt for air:由於 $u$ 與 $h$ 僅與溫度有關，理想氣體表只需一個索引欄 $T$——不需壓力、不分相、不用乾度。相較之下，蒸汽表需要兩個性質才能定出狀態。空氣節選：

Read the change directly: $\Delta h = h(T_2) - h(T_1)$. From 300 K to 1000 K, $\Delta h = 1046.04 - 300.19 = 745.85$ kJ/kg. Constant $c_p = 1.005$ would give $1.005 \times 700 = 703.5$ kJ/kg — 6% low, because $c_p$ of air rises with temperature. That is when the table earns its place. Between rows, interpolate exactly as with the steam tables. (The full table also lists $s^\circ(T)$, used in Entropy.)直接讀取變化量：$\Delta h = h(T_2) - h(T_1)$。由 300 K 至 1000 K，$\Delta h = 1046.04 - 300.19 = 745.85$ kJ/kg。若取定值 $c_p = 1.005$，則得 $1.005 \times 700 = 703.5$ kJ/kg——偏低 6%，因為空氣的 $c_p$ 隨溫度上升。這正是需要查表的時候。介於兩列之間時，插值方法與蒸汽表完全相同。（完整表格另列 $s^\circ(T)$，用於熵一章。）

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。

The chart below shows $Z$ versus $p$ for four gases at the same temperature. At low pressure every curve starts at $Z = 1$. As pressure rises, each gas first dips if attraction wins, then climbs above 1 once molecular size dominates. H₂ is far above its critical temperature, so it rises almost immediately. CO₂ is below its critical temperature at 298 K: it dips steeply and then condenses near 6.4 MPa, where $Z$ drops abruptly to a liquid value.下圖為四種氣體在同一溫度下的 $Z$–$p$ 曲線。低壓時每條曲線都從 $Z = 1$ 出發。壓力升高時，若吸引力佔優則先下降，待分子體積主導後再升至 1 以上。H₂ 遠高於其臨界溫度，幾乎立即上升。CO₂ 在 298 K 時低於其臨界溫度：曲線急降，約於 6.4 MPa 凝結，$Z$ 驟降至液相值。

Corresponding states對應狀態原理

Every gas has its own critical point, so "high pressure" means different things for different gases: 5 MPa is extreme for helium-like behavior but modest for water. The fix is to measure pressure and temperature relative to the critical point. These ratios are the reduced properties:每種氣體都有自己的臨界點，因此「高壓」對不同氣體意義不同：5 MPa 對某些氣體很高，對水卻不算什麼。解決方法是以相對於臨界點的方式量測壓力與溫度，這些比值稱為對比性質：

• Reduced temperature $T_R$ — how hot the gas is compared with its critical temperature. $T_R > 1$ means above the critical temperature, where the gas cannot be liquefied by compression alone. Large $T_R$ means molecular kinetic energy easily overwhelms intermolecular attraction.對比溫度 $T_R$——氣體溫度相對於其臨界溫度的高低。$T_R > 1$ 表示高於臨界溫度，單靠壓縮無法使其液化。$T_R$ 越大，分子動能越能壓過分子間吸引力。
• Reduced pressure $p_R$ — how compressed the gas is compared with its critical pressure. Small $p_R$ means molecules are far apart, so their own volume and mutual attraction hardly matter.對比壓力 $p_R$——氣體受壓程度相對於其臨界壓力。$p_R$ 越小，分子間距越大，分子本身體積與相互吸引幾乎無關緊要。

The principle of corresponding states: gases at the same $p_R$ and $T_R$ are equally "crowded" and equally "energetic" relative to their own molecular forces, so they have nearly the same $Z$. Plotted against reduced coordinates, the $Z$ data of many different gases collapse onto one set of curves — the generalized compressibility chart. One chart then works for any gas, given only its critical constants (Table A-1).對應狀態原理：處於相同 $p_R$ 與 $T_R$ 的氣體，相對於各自的分子力而言，「擁擠程度」與「能量高低」相同，因此 $Z$ 幾乎相同。以對比座標繪圖時，許多不同氣體的 $Z$ 數據會落在同一組曲線上——即廣義壓縮因子圖。只要知道臨界常數（表 A-1），一張圖便適用於任何氣體。

Critical constants as in Table A-1. The acentric factor $\omega$ is used by the Peng–Robinson equation below.臨界常數取自表 A-1。偏心因子 $\omega$ 用於下文的 Peng–Robinson 方程式。

Critical constants as in Table A-1. Air is a mixture, so no single $\omega$ is listed.臨界常數取自表 A-1。空氣為混合物，故不列單一 $\omega$ 值。

The acentric factor ω偏心因子 ω

Two critical constants are not quite enough to make all gases look alike. Small, spherical molecules (Ar, Kr, CH₄) obey corresponding states closely; larger or elongated molecules (CO₂, water, long hydrocarbons) drift away from it. Pitzer (1955) captured this with a third parameter, the acentric factor, defined from the saturation pressure at one specific reduced temperature:僅靠兩個臨界常數，並不足以讓所有氣體行為一致。小而球狀的分子（Ar、Kr、CH₄）很符合對應狀態原理；較大或細長的分子（CO₂、水、長鏈烴）則有偏離。Pitzer（1955）以第三個參數——偏心因子——描述此偏離，其定義取自特定對比溫度下的飽和壓力：

In words: look up the substance's saturation pressure at $T = 0.7\,T_{cr}$ and divide it by $p_{cr}$. For simple spherical fluids that ratio is almost exactly $0.1$, so $\omega \approx 0$. Molecules that are less spherical or polar have lower vapor pressure at that point, and so a larger $\omega$. The name means "non-centric": it measures how far the intermolecular forces are from those between point centres. Typical values are 0.01 for methane, 0.22 for CO₂ and 0.34 for water. Peng–Robinson uses $\omega$ to tune its attraction term to each fluid.換言之：查出該物質在 $T = 0.7\,T_{cr}$ 時的飽和壓力，再除以 $p_{cr}$。簡單球狀流體的此比值幾乎恰為 $0.1$，故 $\omega \approx 0$。非球狀或極性分子在該點的蒸氣壓較低，$\omega$ 因而較大。「偏心」之名意指分子間作用力偏離點中心力的程度。典型值：甲烷 0.01、CO₂ 0.22、水 0.34。Peng–Robinson 以 $\omega$ 將吸引項調整至各流體。

Finding Z from $p_R$ and $T_R$由 $p_R$ 與 $T_R$ 求 Z

1. Look up $T_{cr}$ and $p_{cr}$ for the gas (Table A-1). Use absolute pressure and kelvin.查出該氣體的 $T_{cr}$ 與 $p_{cr}$（表 A-1）。使用絕對壓力與克氏溫度。
2. Compute $p_R = p/p_{cr}$ and $T_R = T/T_{cr}$.計算 $p_R = p/p_{cr}$ 與 $T_R = T/T_{cr}$。
3. On the chart, go up from $p_R$ on the horizontal axis to the curve for your $T_R$ (interpolate between curves if needed).在圖上由橫軸的 $p_R$ 向上找到對應 $T_R$ 的曲線（必要時在曲線間內插）。
4. Go across to the vertical axis and read $Z$. Then $v = ZRT/p$.水平移到縱軸讀取 $Z$。再由 $v = ZRT/p$ 求比容。

If $v$ is known instead of $p$, use the pseudo-reduced specific volume $v'_R = v\,p_{cr}/(RT_{cr})$; the chart's dashed $v'_R$ lines let you locate the state from $T_R$ and $v'_R$ without iterating.若已知的是 $v$ 而非 $p$，可用擬對比比容 $v'_R = v\,p_{cr}/(RT_{cr})$；圖上的 $v'_R$ 虛線可直接由 $T_R$ 與 $v'_R$ 定出狀態，無須反覆試算。

Compressibility explorer壓縮因子探索器

Try the four-step procedure here. The curves show $Z$ vs $p_R$ at several $T_R$; your chosen $T_R$ is drawn in gray and the dashed guides trace the reading. 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。

Curves generated from the Redlich–Kwong equation in reduced form — close to the generalized chart, but not substance-exact.曲線由對比形式的 Redlich–Kwong 方程式產生——與廣義圖接近，但非針對特定物質。

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 $pv = RT$。

Peng–Robinson

Proposed in 1976 for the oil and gas industry, Peng–Robinson (PR) is the most widely used cubic equation today. It keeps the two-constant structure but adds a third input, the acentric factor $\omega$, which measures how far a molecule departs from a simple sphere (argon ≈ 0, CO₂ = 0.225, water = 0.344). Its constants arePeng–Robinson（PR）方程式於 1976 年為石油與天然氣產業提出，是目前最廣泛使用的立方型方程式。它保留兩常數結構，但加入第三個輸入——偏心因子 $\omega$，用以衡量分子偏離簡單球形的程度（氬 ≈ 0、CO₂ = 0.225、水 = 0.344）。其常數為

Why it is popular: it needs only $T_{cr}$, $p_{cr}$ and $\omega$, which are tabulated for thousands of substances; it predicts liquid densities and vapor pressures noticeably better than RK; it extends naturally to mixtures; and, being cubic in $v$, it is solved quickly and reliably. It is a default model in process simulators such as Aspen HYSYS.受歡迎的原因：只需 $T_{cr}$、$p_{cr}$ 與 $\omega$，而這些數據已有數千種物質的表列值；對液體密度與蒸汽壓的預測明顯優於 RK；可自然推廣至混合物；且因為對 $v$ 是三次式，求解快速可靠。它是 Aspen HYSYS 等製程模擬軟體的預設模型之一。

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, Redlich–Kwong and Peng–Robinson 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、Redlich–Kwong 與 Peng–Robinson 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。

Plot isotherms for CO₂, water or nitrogen and compare the ideal gas with the van der Waals, Redlich–Kwong and Peng–Robinson equations. Switch the substance with the buttons at the top of the panel; the temperature range and axes adjust to bracket each fluid's critical point.繪製 CO₂、水或氮的等溫線，比較理想氣體、范德瓦耳斯、Redlich–Kwong 與 Peng–Robinson 方程式。以面板上方按鈕切換物質；溫度範圍與座標軸會自動調整，涵蓋各流體的臨界點。

What to look for.觀察重點。

• Above $T_{cr}$ ($T_R > 1$): every curve is smooth and slopes downward. Pressure falls steadily as volume grows, and there is no phase change.高於 $T_{cr}$（$T_R > 1$）：所有曲線平滑且單調下降——體積增大時壓力穩定下降，不發生相變。
• Below $T_{cr}$: the cubic equations develop the van der Waals loop, a wiggle with a rising section where $\partial p/\partial v > 0$. That section is physically impossible. A real fluid follows a flat line at the saturation pressure instead, placed so the areas above and below it are equal (Maxwell's equal-area rule).低於 $T_{cr}$：立方型方程式出現范德瓦耳斯迴圈——一段 $\partial p/\partial v > 0$ 的上升區，物理上不可能。真實流體改沿飽和壓力下的水平線變化，其位置使線上下兩塊面積相等（馬克士威等面積法則）。
• Large $v$ (right side): all models merge with the ideal gas. Molecules are far apart, so neither their size nor their attraction matters.大 $v$（右側）：所有模型皆趨近理想氣體——分子相距甚遠，其體積與吸引力皆可忽略。
• Small $v$ (left side): the curves shoot up as $v \to b$, because the molecules' own volume leaves no room left to compress. The models disagree most here.小 $v$（左側）：當 $v \to b$ 時曲線急遽上升——分子本身的體積使其無法再被壓縮。各模型在此差異最大。
• Compare fluids. N₂ ($\omega \approx 0.04$) is nearly spherical, so the three cubics agree closely. Water ($\omega = 0.34$) is strongly polar, the models spread apart, and van der Waals is worst. This is why steam calculations use tables rather than a cubic equation.比較流體。N₂（$\omega \approx 0.04$）接近球形，三種立方型方程式結果相近。水（$\omega = 0.34$）極性強，各模型差異擴大，范德瓦耳斯最差——這正是蒸汽計算使用蒸汽表而非立方型方程式的原因。

The results panel reports each model's pressure at a fixed reference volume (about twice the critical volume) and the ideal-gas error relative to Peng–Robinson. Watch how that error grows as $T$ falls toward $T_{cr}$. Now slide $T$ upward. Near $T_R \approx 2$–$2.5$, the Boyle temperature, attraction and molecular size cancel: the error passes through zero and all four curves nearly coincide. At still higher $T$, molecular size dominates, $Z$ rises slightly above 1, and the real pressure exceeds the ideal one. The ideal gas becomes exact only as $v \to \infty$.結果面板列出各模型在固定參考比容（約為臨界比容兩倍）下的壓力，以及理想氣體相對於 Peng–Robinson 的誤差。觀察 $T$ 降向 $T_{cr}$ 時誤差如何增大。再將 $T$ 往上調：在 $T_R \approx 2$–$2.5$（波以耳溫度）附近，吸引力與分子體積效應相互抵消，誤差通過零，四條曲線幾乎重合。溫度更高時分子體積主導，$Z$ 略大於 1，真實壓力高於理想值；理想氣體僅在 $v \to \infty$ 時才完全成立。

Multi-constant equations & choosing an EOS多常數方程式與狀態方程式的選擇

More constants buy more accuracy over a wider range, at the cost of needing more fluid-specific data. The table summarizes the equations in this chapter and where each is typically used:常數越多，可涵蓋的範圍越廣、精度越高，但需要更多該流體的數據。下表整理本章各方程式及其常見用途：

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。

Exercises習題

Work the example first — it shows the full reasoning chain. Then try E1–E5 before opening the answers. Critical constants are from the table above (Table A-1).先完成範例——它展示完整的推理流程。再嘗試 E1–E5，之後才打開解答。臨界常數取自上表（表 A-1）。