📚 B.4 Thermodynamics (HL only) | B.4 热力学（仅HL）
Welcome to the IB Physics HL revision on Thermodynamics. This section explores the fundamental laws governing energy, heat, and work, with special focus on ideal gas processes, p–V diagrams, cyclic engines, and the profound concept of entropy. Understanding these principles is essential for tackling HL problems and appreciating the limits of efficiency in real-world engines.
欢迎来到 IB 物理 HL 热力学专题复习。本章探究支配能量、热量和功的基本定律，重点讲解理想气体过程、p–V 图、循环热机以及深刻的熵概念。掌握这些原理对于解决 HL 问题、理解现实热机效率的极限至关重要。
1. First Law of Thermodynamics | 热力学第一定律
s solely on temperature, so any temperature change directly gives ΔU = (3/2)nRΔT for a monatomic gas.
热力学第一定律是能量守恒的表述：系统内能的变化 ΔU 等于系统吸收的热量 Q 减去系统对外做的功 W。IB 物理采用的符号约定为 ΔU = Q - W，其中气体膨胀时 W 为正。对理想气体而言，内能仅由温度决定，因此温度变化直接给出 ΔU = (3/2)nRΔT（单原子气体）。
ΔU = Q - W
. If a gas expands without heat input (Q = 0), its internal energy drops, causing cooling.
许多其他教材将第一定律写成 ΔU = Q + W，此时 W 表示外界对系统做的功。你在计算时必须始终使用同一种符号约定。如果气体在无热量输入（Q = 0）时膨胀，其内能减少，导致温度降低。
2. Work Done by an Expanding Gas | 气体膨胀做功
re. The total work done during a finite volume change from V₁ to V₂ is the area under the p–V curve: W = ∫ P dV. This area can be found by integration if the process path is known.
当气体反抗外部压力改变体积时，它对外做功。对于无限小的体积变化 dV，做功为 dW = P dV，其中 P 为气体的压强。在从 V₁ 到 V₂ 的有限过程中，气体所作的总功等于 p–V 曲线下的面积：W = ∫ P dV。如果已知过程路径，可通过积分求出该面积。
W = ∫ P dV
V₂/V₁). If the volume decreases, work is done ON the gas, and W becomes negative in the IB convention.
对于等压过程，功简化为 W = P ΔV。在等温膨胀中，W = nRT ln(V₂/V₁)。若体积减小，外界对气体做功，按照 IB 惯例 W 取负值。
3. p–V Diagrams and Work | p–V 图与功
p–V diagrams are the primary tool for visualising thermodynamic processes. Each point represents a state (P, V). A path between two states shows the process. The area under the path equals the work done. For a complete cycle, the enclosed area represents the net work output (if clockwise) or net work input (if counterclockwise).
p–V 图是直观展示热力学过程的主要工具。图上每一点代表一个状态（P, V），两点之间的路径表示过程。路径下方的面积等于做功的大小。对于完整的循环，闭合曲线所围的面积代表净输出功（顺时针循环）或净输入功（逆时针循环）。
ritical: an isotherm is a hyperbola (PV = constant) while an adiabat is steeper because PVγ re horizontal, isochoric lines vertical. These patterns help you quickly identify processes and compute work, heat and internal energy changes.
识别曲线形状至关重要：等温线为双曲线（PV = 常数），绝热线更陡，因为 PVγ = 常数且 γ > 1。等压线为水平线，等容线为竖直线。这些特征能帮助你快速识别过程并计算功、热量和内能变化。
4. Isothermal Process | 等温过程
y into work done by the gas. The equation of state follows Boyle’s law: P₁V₁ = P₂V₂ = nRT.
等温过程在恒定温度下进行（ΔT = 0）。对于理想气体，内能不变（ΔU = 0），因此第一定律简化为 Q = W：所有提供给气体的热量完全转化为气体对外做的功。状态方程遵循玻意耳定律：P₁V₁ = P₂V₂ = nRT。
W = nRT ln(V₂/V₁)
The work done is calculated using the logarithmic formula, where V₂/V₁ is the expansion ratio. Since the process is very slow, the system always stays in thermal equilibrium with a reservoir. On a p–V diagram, the isotherm is a hyperbola, and the area under it equals W.
做功使用对数公式计算，其中 V₂/V₁ 为膨胀比。由于过程进行得非常缓慢，系统始终与热库保持热平衡。在 p–V 图上等温线为双曲线，其下的面积就等于 W。
5. Isochoric Process | 等容过程
gas raises its internal energy and temperature. For a monatomic ideal gas, Q = (3/2)nRΔT.
等容过程保持体积恒定（ΔV = 0）。因 dV = 0 而不做功，故 W = 0。第一定律变为 Q = ΔU。所有传入气体的热量都用于提升内能和温度。对于单原子理想气体，Q = (3/2)nRΔT。
On the p– nstant. This process occurs, for example, when gas is heated in a rigid, sealed container.
在 p–V 图上，路径是竖直线。压强随温度上升而升高，遵循 P/T = 常数。例如，气体在刚性密封容器中被加热的过程就是等容过程。
6. Isobaric Process | 等压过程
or an ideal gas, the quantity ΔU + PΔV equals the change in enthalpy H. The heat capacity at constant pressure CP is larger than at constant volume because extra energy is needed to do expansion work.
等压过程中压强保持恒定。气体做功 W = PΔV。第一定律给出 Q = ΔU + PΔV。对于理想气体，ΔU + PΔV 等于焓 H 的变化。定压热容 CP 大于定容热容，因为需要额外能量来做膨胀功。
A horizontal line on the p–V diagram represents an isobaric expansion or compression. The work is simply the rectangle’s area under that line. Common examples include heating a gas in a cylinder with a freely moving piston under constant external pressure.
p–V 图上的水平线表示等压膨胀或压缩。做功就是该线段下方矩形的面积。常见的例子有在恒定外压下、活塞可自由移动的气缸中加热气体。
7. Adiabatic Process | 绝热过程
oes work entirely at the expense of its internal energy, causing a temperature drop. For an ideal gas, the adiabatic condition is PVγ = constant, where γ = CP/CV. For a monatomic gas, γ = 5/3.
绝热过程与外界没有热交换（Q = 0）。第一定律因此给出 ΔU = -W：气体以内能减少为代价对外做功，导致温度下降。对于理想气体，绝热条件为 PVγ = 常数，其中 γ = CP/CV。单原子气体的 γ = 5/3。
PVγ = constant, TVγ-1 = constant
Adiabatic curves are steeper than isotherms because the pressure falls faster as volume increases, not only due to volume change but also due to the simultaneous temperature drop. In a rapid compression, temperature rises significantly – this principle is used in diesel engines.
绝热线比等温线更陡，因为压强随体积增大下降得更快，这不仅是体积变化的缘故，也因同时发生的温度下降。在快速压缩中，温度会急剧上升，柴油机正是利用了这一原理。
8. Cyclic Processes and Heat Engines | 循环过程与热机
A cyclic process brings the system back to its initial state, so the net change in internal energy over one complete cycle is zero (ΔUnet = 0). The net work done by the system equals the net heat input: Wnet = Qin - Qout, where Qin is heat absorbed from a hot reservoir and Qout is heat rejected to a cold reservoir. The thermal efficiency η is defined as the ratio of useful work output to heat input.
循环过程使系统回到初始状态，因此一个完整循环的净内能变化为零（ΔUnet = 0）。系统输出的净功等于净输入热量：Wnet = Qin - Qout，其中 Qin 为从高温热库吸收的热量，Qout 为排放给低温热库的废热。热效率 η 定义为有用功输出与吸热量的比值。
η = Wnet / Qin = 1 - (Qout / Qin)
On a p–V diagram, a clockwise cycle encloses a positive area representing net work output; this is a heat engine. A counterclockwise cycle represents a refrigerator or heat pump, where work is input to transfer heat from cold to hot.
在 p–V 图上，顺时针循环包围正面积，代表净输出功，即为热机。逆时针循环则代表制冷机或热泵，此时输入功将热量从低温处转移到高温处。
9. Second Law of Thermodynamics and Entropy | 热力学第二定律与熵
The second law has several equivalent statements: heat cannot spontaneously pass from a colder to a hotter body (Clausius); it is impossible to convert heat completely into work without other effect (Kelvin–Planck); and the total entropy of an isolated system never decreases. Entropy S is a state function that measures the dispersal of energy or disorder.
热力学第二定律有多种等效表述：热量不可能自发地从低温物体传到高温物体（克劳修斯表述）；不可能从单一热源吸热并将其完全转化为功而不产生其他影响（开尔文–普朗克表述）；孤立系统的总熵永不减少。熵 S 是一个态函数，量度能量的分散程度或系统的无序度。
ΔS = Qrev / T (for a reversible process)
In any irreversible process, the total entropy increases. This explains why real engines always have an efficiency lower than the ideal Carnot efficiency and why perpetual motion machines of the second kind are impossible. Entropy also provides the direction of spontaneous change.
在任何不可逆过程中，总熵都会增加。这解释了为什么真实热机的效率总是低于理想的卡诺效率，以及为什么第二类永动机是不可能实现的。熵还指明了自发变化的方向。
10. Carnot Cycle | 卡诺循环
The Carnot cycle is an idealised, reversible cycle that operates between a hot reservoir at TH and a cold reservoir at TC. It consists of four stages: isothermal expansion at TH, adiabatic expansion (cooling), isothermal compression at TC, and adiabatic compression (heating back to TH). It achieves the maximum theoretical efficiency for any heat engine operating between those two temperatures.
卡诺循环是一种理想化的可逆循环，工作在高温热库 TH 和低温热库 TC 之间。它由四个阶段组成：TH 下的等温膨胀、绝热膨胀（降温）、TC 下的等温压缩以及绝热压缩（升温回到 TH）。对于工作在两个给定温限之间的任何热机，卡诺循环能达到的理论效率最高。
ηCarnot = 1 - (TC / TH) (T in Kelvin)
This efficiency depends only on the reservoir temperatures, not on the working substance. To increase efficiency, engineers aim to raise TH or lower TC. However, practical limitations and the second law ensure that η is always less than 1. The Carnot cycle is a benchmark against which real engines, such as Otto or Diesel cycles, are compared.
该效率仅取决于热库温度，与工作物质无关。为提高效率，工程师会设法提高 TH 或降低 TC。然而，实际限制与第二定律确保效率始终小于 1。卡诺循环是评价真实热机（如奥托循环、狄塞尔循环）性能的基准。
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