What you'll be able to do本章學習成果
- Evaluate moving-boundary work as the area under a process curve on a p–V diagram.將p–V圖上過程曲線下的面積計算為移動邊界功。
- Compute work for polytropic, isobaric, and isothermal processes of an ideal gas.計算理想氣體的多變、等壓與等溫過程的功。
- State the closed-system energy balance and apply it, including over cycles.陳述封閉系統能量平衡並加以應用，包括循環過程。
- Define enthalpy and specific heats $c_v$, $c_p$, and use $\Delta u = c_v\Delta T$, $\Delta h = c_p\Delta T$.定義焓與比熱 $c_v$、$c_p$，並使用 $\Delta u = c_v\Delta T$、$\Delta h = c_p\Delta T$。
Key equations重要公式
Moving-boundary work移動邊界功
When a gas pushes a piston through a differential displacement, it does work $\delta W = p\,dV$. For a quasi-equilibrium process:當氣體推動活塡移動微小位移時，它做功 $\delta W = p\,dV$。對於準平衡過程：
On a p–V diagram, boundary work is the area under the process curve. It depends on the path taken, not just the end states — which is why $W$ is not a property. Over a cycle, the enclosed area is the net work.在 p–V 圖上，邊界功為過程曲線下的面積。它與所經路徑有關，而不僅取決於等狀點——這正是 $W$ 不是性質的原因。在循環過程中，封閉面積即為凈功。
Polytropic processes多變過程
Many quasi-static processes follow $pV^n = \text{const}$. Integrating gives two cases:許多準靜態過程遵循 $pV^n = \text{const}$。積分得兩種情況：
$n=0$: isobaric; $n=1$: isothermal (ideal gas); $n=k$: reversible adiabatic. For isothermal: $W = mRT\ln(V_2/V_1)$.$n=0$：等壓；$n=1$：等溫（理想氣體）；$n=k$：可逆絕熱。等溫式：$W = mRT\ln(V_2/V_1)$。
p–V work explorerp–V 功探索器
Pick a process for 1 kg of air, set the initial pressure and volume ratio, and watch the work appear as the shaded area. Compare how much work the four processes deliver for the same volume change.選擇 1 kg 空氣的過程，設定初始壓力與體積比，觀察陰影面積所代表的功。比較四種過程在相同體積變化下分別做了多少功。
Defining heat熱的定義
A foundational experiment: the adiabatic work done on a closed system between two given states is path-independent. This means adiabatic work measures a property change — the internal energy: $\Delta E = -W_{adb}$.一個基礎性實驗：對封閉系統所做的絕熱功與路徑無關。這表示絕熱功量度的是一個性質變化——內能：$\Delta E = -W_{adb}$。
For a non-adiabatic process the work differs from the adiabatic value. We define heat as that difference. Joule's paddle-wheel experiment confirmed heat and work are interchangeable forms of energy transfer.對於非絕熱過程，功與絕熱值不同。我們將該差异定義為熱。焦耳的洿輪實驗證實了熱與功是可互換的能量傳遞形式。
The closed-system energy balance封閉系統能量平衡
The change in a system's energy equals heat in minus work out:系統的能量變化等於輸入熱量減去輸出功：
Heat and work are path-dependent interactions at the boundary; energy is the property they change. Rate form: $dE/dt = \dot Q - \dot W$.熱與功是在邊界發生的路徑相關互動；能量是它們所改變的性質。速率形式：$dE/dt = \dot Q - \dot W$。
Cyclic processes循環過程
Over a complete cycle, $\oint dU = 0$ so the net heat equals the net work:在完整循環中，$\oint dU = 0$，因此凈熱等於凈功：
A work-producing cycle has net heat in; a work-consuming cycle rejects net heat. Thermal efficiency: $\eta = W_{net}/Q_{in} = 1 - Q_{out}/Q_{in}$.動力循環有凈熱輸入；消耗循環則排出凈熱。熱效率：$\eta = W_{net}/Q_{in} = 1 - Q_{out}/Q_{in}$。
Enthalpy & specific heats焓與比熱
$H = U + pV$ (per unit mass: $h = u + pv$) is called enthalpy. The specific heats:$H = U + pV$（對比値：$h = u + pv$）稱為焓。比熱：
For an ideal gas, $\Delta u = \int c_v\,dT$ and $\Delta h = \int c_p\,dT$ (or $c_v\Delta T$, $c_p\Delta T$ when constant). Linked by $c_p - c_v = R$.理想氣體：$\Delta u = \int c_v\,dT$、$\Delta h = \int c_p\,dT$（常比熱時化簡為 $c_v\Delta T$、$c_p\Delta T$）。連系：$c_p - c_v = R$。
Closed-system energy balance封閉系統能量平衡
Example範例 Isothermal compression of air空氣等溫壓縮 ›
Given: 0.40 m³ of air at 100 kPa, compressed isothermally (ideal gas, 300 K) to 0.10 m³.已知：100 kPa 的 0.40 m³ 空氣在 300 K 下等溫壓縮（理想氣體）至 0.10 m³。
Find: the work and the heat transfer.求：功與熱轉移量。
Solution. $$W=p_1V_1\ln\frac{V_2}{V_1}=100(0.40)\ln(0.25)=-55.5\text{ kJ}$$ Since $T$ is constant, $\Delta U=0$, so $Q=\Delta U+W=-55.5\text{ kJ}$ — 55.5 kJ is rejected.解： $$W=100(0.40)\ln(0.25)=-55.5\text{ kJ}$$ 因 $T$ 定，$\Delta U=0$，故 $Q=\Delta U+W=-55.5\text{ kJ}$ — 55.5 kJ 被排出。
Practice problems練習題
Work each one yourself first, then reveal the solution step by step.先自行嘗試，再逐步顯示解答。