Thermodynamics Key Points for IB & CIE Physics | IB/CIE 物理热力学考点精讲

📚 Thermodynamics Key Points for IB & CIE Physics | IB/CIE 物理热力学考点精讲

Thermodynamics is a central topic in both IB and CIE A-Level Physics. It brings together concepts of heat, work, internal energy and the fundamental laws that govern energy transfer. This revision guide highlights the key definitions, equations, process types, engine cycles and entropy ideas you are likely to meet in exam questions.

热力学是 IB 与 CIE A-Level 物理的核心模块，将热量、功、内能与能量传递的基本规律串联在一起。本考点精讲梳理关键定义、公式、过程类型、热机循环以及熵等重要概念，助你轻松应对考试。

1. Temperature and Thermal Equilibrium | 温度与热平衡

Temperature measures the average random kinetic energy of particles in a substance. The SI unit is the kelvin (K), and conversion with the Celsius scale is given by T (K) = θ (°C) + 273.15. Two bodies are in thermal equilibrium when they have the same temperature and there is no net flow of thermal energy between them.

温度衡量物质中粒子无规则运动的平均动能。国际单位是开尔文（K），与摄氏温标的换算关系为 T (K) = θ (°C) + 273.15。当两个物体温度相同且彼此没有净热量流动时，它们处于热平衡。

Thermometers rely on a physical property that varies with temperature, such as the volume of a liquid or the resistance of a metal. Fixed points, e.g. the triple point of water (273.16 K) or the ice point (0 °C), are used to calibrate temperature scales.

温度计利用随温度变化的物理属性（如液体体积或金属电阻）进行测温。固定点（例如水的三相点 273.16 K 或冰点 0 °C）用于标定温标。

T (K) = θ (°C) + 273.15

2. Internal Energy and Heat | 内能与热量

Internal energy U is the sum of the random kinetic energy and the potential energy of all particles in a system. It depends only on the state of the system (temperature, phase) and not on how that state was reached.

内能 U 是系统内所有粒子无规则动能与势能的总和。它只取决于系统的状态（温度、物相），与如何达到该状态无关。

Heat Q is energy transferred due to a temperature difference. It is not a property of a system; you cannot say a body ‘contains’ heat. Specific heat capacity c relates heat to temperature change: Q = mcΔθ. Latent heat L is the energy needed to change phase without a temperature change: Q = mL.

热量 Q 是因温差而传递的能量，它不是系统的属性，不能说物体“含有”热量。比热容 c 将热量与温度变化联系起来：Q = mcΔθ。潜热 L 是改变物相而不改变温度所需的能量：Q = mL。

Q = mcΔθ Q = mL

3. First Law of Thermodynamics | 热力学第一定律

The first law is a statement of energy conservation: the change in internal energy of a system equals the heat added to the system plus the work done on the system. In this guide we adopt the sign convention

第一定律是能量守恒的表述：系统内能的变化等于系统吸收的热量加上外界对系统所做的功。本文采用符号约定

ΔU = Q + W

where W is positive when work is done on the system (compression) and negative when the system does work on the surroundings (expansion). Some textbooks use ΔU = Q − W with W being work done by the system; always check the given convention in exam papers.

其中当外界对系统做功（压缩）时 W 为正，当系统对外做功（膨胀）时 W 为负。有些教材使用 ΔU = Q − W，此时 W 代表系统对外做的功；考试时务必留意题目给出的约定。

For an ideal gas, internal energy depends only on temperature. An isothermal process therefore has ΔU = 0, and Q = −W = positive area under a pV curve (when the gas expands). In an adiabatic process Q = 0, so ΔU = W; the gas cools when it expands adiabatically.

对于理想气体，内能仅与温度有关。因此等温过程中 ΔU = 0，有 Q = −W（膨胀时等于 pV 曲线下面积）。绝热过程中 Q = 0，因此 ΔU = W；绝热膨胀时气体温度降低。

4. Thermodynamic Processes | 热力学过程

Four special processes are commonly examined:

常考的四种特殊过程：

Isobaric – constant pressure. Work done on gas is W = −P ΔV. Heat transferred changes both internal energy and does work.

等压过程 – 压强恒定。外界对气体做功 W = −P ΔV。传递的热量同时改变内能并对外做功。
Isochoric (isovolumetric) – constant volume. W = 0, so ΔU = Q. All heat changes internal energy.

等容过程 – 体积恒定。W = 0，因此 ΔU = Q，热量全部转化为内能变化。
Isothermal – constant temperature. ΔU = 0, so Q = −W. For an ideal gas, pV = constant and the work done is obtained from the area under the hyperbolic curve.

等温过程 – 温度恒定。ΔU = 0，因此 Q = −W。对于理想气体，pV = 常量，做功由双曲线下面积给出。
Adiabatic – no heat exchange (Q = 0). The first law reduces to ΔU = W. The relation pV^γ = constant holds, where γ = C_P/C_V.

绝热过程 – 无热交换（Q = 0）。第一定律简化为 ΔU = W，且满足 pV^γ = 常量，其中 γ = C_P/C_V。

5. PV Diagrams and Work Done | PV 图与做功

A pressure–volume graph is an essential tool. The area under a process curve on a pV diagram represents the magnitude of the work done by the gas on the surroundings. Under the ΔU = Q + W sign convention, the work done on the gas is the negative of this area.

压强–体积图是重要的分析工具。pV 图上过程曲线下的面积表示气体对外界做功的大小。在 ΔU = Q + W 的约定下，外界对气体做的功等于该面积的负值。

A clockwise cyclic process on a pV diagram represents a heat engine; the net work output per cycle equals the area enclosed by the loop. An anticlockwise cycle indicates a refrigerator or heat pump, where net work is done on the system.

pV 图上的顺时针循环代表热机，每循环净输出功等于环路所围面积。逆时针循环表示制冷机或热泵，此时外界对系统做净功。

The shape of the curve distinguishes processes: a horizontal line is isobaric, a vertical line is isochoric, a hyperbola (p ∝ 1/V) is isothermal, and a steeper curve is adiabatic (p ∝ 1/V^γ).

曲线的形状可区分过程：水平线为等压，竖直线为等容，双曲线（p ∝ 1/V）为等温，更陡的曲线（p ∝ 1/V^γ）为绝热。

6. Molar Specific Heat Capacities | 摩尔热容

For an ideal gas, two principal molar heat capacities are defined: C_V (constant volume) and C_P (constant pressure). They are linked by Mayer’s relation:

对于理想气体，定义了两个主要摩尔热容：C_V（等容）和 C_P（等压），它们通过迈耶公式关联：

C_P = C_V + R

where R = 8.31 J mol⁻¹ K⁻¹ is the molar gas constant. The value of C_V depends on the degrees of freedom f of the gas particles: C_V = (f/2)R. For a monatomic gas f = 3, C_V ≈ 12.5 J mol⁻¹ K⁻¹; for a diatomic gas at moderate temperatures f = 5, so C_V ≈ 20.8 J mol⁻¹ K⁻¹.

其中 R = 8.31 J mol⁻¹ K⁻¹ 是摩尔气体常数。C_V 的值取决于气体粒子运动的自由度 f：C_V = (f/2)R。单原子气体 f = 3，C_V ≈ 12.5 J mol⁻¹ K⁻¹；常温下双原子气体 f = 5，C_V ≈ 20.8 J mol⁻¹ K⁻¹。

The adiabatic index γ = C_P/C_V is used in adiabatic relations: TV^γ−1 = constant, pV^γ = constant.

绝热指数 γ = C_P/C_V 用于绝热关系：TV^γ−1 = 常量，pV^γ = 常量。

7. Ideal Gas Law and Kinetic Theory | 理想气体定律与分子运动论

The equation of state for an ideal gas links the macroscopic variables:

理想气体的状态方程将宏观量联系起来：

pV = nRT = NkT

where n is the number of moles, N is the number of molecules and k = 1.38 × 10⁻²³ J K⁻¹ is Boltzmann’s constant.

其中 n 为摩尔数，N 为分子总数，k = 1.38 × 10⁻²³ J K⁻¹ 为玻尔兹曼常数。

Kinetic theory models gas molecules as tiny, elastic spheres in random motion. The pressure exerted by an ideal gas can be expressed as p = (1/3) ρ <c²>, where ρ is density and <c²> is the mean square speed. This leads to the fundamental relation between average translational kinetic energy and temperature:

分子运动论将气体分子视为随机运动的小弹性球。理想气体产生的压强可表示为 p = (1/3) ρ <c²>，其中 ρ 为密度，<c²> 为均方速率。由此推出平均平动动能与温度的基本关系：

(1/2) m <c²> = (3/2) kT

The Maxwell−Boltzmann distribution describes the spread of molecular speeds. The root-mean-square speed c_rms = √(<c²>) = √(3kT/m) increases with temperature and decreases with molecular mass.

麦克斯韦−玻尔兹曼分布描述了分子速率的分布。方均根速率 c_rms = √(<c²>) = √(3kT/m) 随温度升高而增大，随分子质量增大而减小。

8. Heat Engines and Efficiency | 热机与效率

A heat engine absorbs heat Q_h from a hot reservoir, converts part of it into useful work W_out, and rejects the remainder Q_c to a cold reservoir. The thermal efficiency is

热机从高温热源吸收热量 Q_h，将其中一部分转化为有用功 W_out，其余热量 Q_c 排向低温热源。热效率定义为

η = W_out / Q_h = 1 − Q_c/Q_h

By the first law, for a complete cycle the net change in internal energy is zero, so net work equals the net heat absorbed: W_out = Q_h − Q_c.

根据第一定律，完成一个循环后内能净变化为零，因此净功等于净吸热量：W_out = Q_h − Q_c。

Real engines always operate irreversibly and have efficiencies lower than the theoretical limit. A pV loop that encloses a larger area with the same heat input gives higher efficiency.

实际热机总是不可逆运行，效率低于理论极限。在相同吸热量下，pV 回路包围的面积越大，效率越高。

9. Carnot Cycle and Maximum Efficiency | 卡诺循环与最大效率

The Carnot cycle is a theoretical, reversible cycle that provides the upper limit of efficiency for any heat engine operating between two reservoirs at absolute temperatures T_h and T_c. It consists of two isothermal and two adiabatic processes.

卡诺循环是一种理想的可逆循环，给出了在高温热源 T_h 和低温热源 T_c 之间运行的任何热机效率的上限。它由两个等温过程和两个绝热过程组成。

η_Carnot = 1 − T_c/T_h

This formula emphasises that to raise efficiency, the hot reservoir should be as hot as possible and the cold reservoir as cold as possible. The temperatures must be expressed in kelvin.

该公式表明，要提高效率，高温热源应尽可能高，低温热源应尽可能低。温度必须使用开尔文。

No real engine can exceed the Carnot efficiency. When T_c = 0 K, the efficiency would reach 1, but the third law of thermodynamics prevents reaching absolute zero.

任何真实热机的效率都不可能超过卡诺效率。当 T_c = 0 K 时效率将达到 1，但热力学第三定律指出无法达到绝对零度。

10. Second Law and Entropy | 第二定律与熵

The second law of thermodynamics has several equivalent formulations. One states that heat cannot spontaneously flow from a colder to a hotter body. Another, the Clausius statement, says no cyclic process can transfer heat from a cold reservoir to a hot reservoir without an input of work.

热力学第二定律有多种等价表述。一种说法是热量不能自发地从低温物体流向高温物体。克劳修斯表述为：没有外界做功的情况下，循环过程无法将热量从低温热源传递到高温热源。

Entropy S is a measure of the disorder of a system. The change in entropy when a reversible process adds heat Q at constant temperature T is

熵 S 是系统无序度的量度。在可逆过程中，恒定温度 T 下加入热量 Q 引起的熵变为

ΔS = Q_rev / T

For an isolated system, the total entropy change always increases for irreversible processes and remains constant for reversible ones: ΔS_total ≥ 0. This is the entropy statement of the second law.

对于孤立系统，不可逆过程的总熵总是增加，可逆过程总熵保持不变：ΔS_total ≥ 0。这便是第二定律的熵表述。

In thermal physics problems, you may be asked to calculate entropy changes for melting (Q = mL) or heating (Q = mcΔT, requiring integration because temperature changes). Entropy helps explain why certain processes, such as the free expansion of a gas, are irreversible despite conserving energy.

在热学问题中，你可能需要计算熔化（Q = mL）或加热（Q = mcΔT，因温度变化需积分）过程的熵变。熵有助于解释某些过程（如气体的自由膨胀）虽然能量守恒却是不可逆的。

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Thermodynamics Key Points for IB & CIE Physics | IB/CIE 物理热力学考点精讲