📚 IB & AQA Physics: Thermodynamics Key Points Revision | IB和AQA物理：热力学考点精讲
Thermodynamics is a core topic in both IB and AQA Physics, covering everything from temperature scales to the laws that govern energy transfer and entropy. This revision guide highlights the key concepts, equations, and common pitfalls you need to master for your exams.
热力学是IB和AQA物理的核心主题，涵盖从温标到能量传递与熵定律的方方面面。本复习指南重点梳理关键概念、方程式和常见易错点，助你高效备考。
1. Temperature and Thermal Equilibrium | 温度与热平衡
Temperature measures the average random kinetic energy of particles in a substance. When two bodies at different temperatures are placed in thermal contact, energy flows from the hotter to the cooler until they reach the same temperature — this is thermal equilibrium.
温度量度物质中粒子的平均随机动能。当两个温度不同的物体热接触时，能量从高温物体流向低温物体，直到它们达到同一温度——这就是热平衡。
The Kelvin scale is the absolute thermodynamic scale. Absolute zero (0 K) is the lower limit where particles have minimum kinetic energy. The relationship between Celsius and Kelvin is:
开尔文温标是绝对热力学温标。绝对零度（0 K）是粒子动能最低的下限。摄氏温度与开尔文温度的换算关系为：
T (K) = θ (°C) + 273.15
A change of 1 °C is equal to a change of 1 K, which simplifies many calculations.
1°C的变化等于1 K的变化，这简化了许多计算。
2. Heat, Internal Energy, and Temperature | 热量、内能与温度的区别
Heat (Q) is the energy transferred between systems due to a temperature difference. It is not a property of a system but a process quantity.
热量（Q）是由于温度差而在系统间传递的能量。它不是系统的状态量，而是过程量。
Internal energy (U) is the sum of the random kinetic energy and potential energy of all particles in a system. It depends on the state of the substance (temperature and phase).
内能（U）是系统内所有粒子的随机动能和势能之和。它取决于物质的状态（温度和相态）。
Temperature is not a measure of total internal energy but of the average kinetic energy per particle. During a phase change, temperature stays constant even though internal energy increases — the added energy goes into breaking bonds, raising potential energy.
温度不是总内能的量度，而是每个粒子的平均动能。相变期间温度保持不变，但内能增加——所加的能量用于打破分子间键，升高势能。
3. Specific Heat Capacity | 比热容
The specific heat capacity (c) of a material is the energy required to raise the temperature of 1 kg of the substance by 1 K (or 1 °C).
比热容（c）是使1 kg物质温度升高1 K（或1 °C）所需的能量。
Q = mcΔT
where ΔT is the change in temperature. Heat capacity (C = mc) is the energy needed to raise the temperature of an object by 1 K.
其中ΔT是温度变化。热容（C = mc）是使物体温度升高1 K所需的能量。
Typical values: water has a high specific heat capacity (about 4200 J kg⁻¹ K⁻¹), meaning it takes a lot of energy to heat up. This is why water is used as a coolant.
典型数值：水的比热容很高（约4200 J kg⁻¹ K⁻¹），这意味着加热水需要大量能量，因此水常用作冷却剂。
Experimental determination often uses an electrical heater: energy supplied = V I t, and by measuring the temperature rise, c can be calculated, taking care to reduce heat losses.
实验测定常使用电加热器：提供的能量 = V I t，测量温升后可计算c，需注意减少热损失。
4. Latent Heat and Phase Changes | 潜热与相变
Latent heat is the energy absorbed or released during a change of state at constant temperature. Specific latent heat (L) is the energy per unit mass:
潜热是状态变化期间在恒温下吸收或释放的能量。比潜热（L）是单位质量所需的能量：
Q = mL
Specific latent heat of fusion (Lf) is for melting/freezing, and specific latent heat of vaporization (Lv) is for boiling/condensing. Lv is generally much larger than Lf because it involves completely separating particles rather than just weakening bonds.
比熔化潜热（Lf）用于熔化/凝固，比汽化潜热（Lv）用于沸腾/冷凝。Lv通常远大于Lf，因为汽化需要完全分离粒子而不仅仅是减弱结合力。
During melting or boiling, the temperature remains constant; the energy input increases the potential energy of particles while their average kinetic energy stays the same.
在熔化或沸腾过程中，温度保持不变；能量输入增加粒子的势能，而它们的平均动能不变。
5. The Ideal Gas Law | 理想气体定律
An ideal gas obeys the equation of state:
理想气体遵循状态方程：
pV = nRT
where p is pressure, V is volume, n is the number of moles, R = 8.31 J mol⁻¹ K⁻¹, and T is the absolute temperature in kelvin.
其中p是压强，V是体积，n是摩尔数，R = 8.31 J mol⁻¹ K⁻¹，T是开尔文绝对温度。
Alternatively, using the number of molecules N:
或者用分子数N表示：
pV = NkT
where k = 1.38 × 10⁻²³ J K⁻¹ is Boltzmann’s constant.
其中k = 1.38 × 10⁻²³ J K⁻¹是玻尔兹曼常数。
Key special cases: Boyle’s law (pV = constant for fixed T and n), Charles’s law (V/T = constant for fixed p and n), and Gay-Lussac’s law (p/T = constant for fixed V and n). Always use kelvin.
关键特例：玻意耳定律（定温定n时pV = 常数），查理定律（定压定n时V/T = 常数），盖-吕萨克定律（定容定n时p/T = 常数）。务必使用开尔文温度。
6. Kinetic Theory of Gases | 气体分子动理论
The kinetic theory explains macroscopic properties by the motion of microscopic particles. Key assumptions: large number of identical molecules in random motion; negligible volume of molecules; no intermolecular forces except during elastic collisions; obey Newton’s laws.
分子动理论用微观粒子的运动解释宏观性质。关键假设：大量相同的分子做随机运动；分子自身体积可忽略；除弹性碰撞外无分子间力；服从牛顿运动定律。
Pressure arises from molecular collisions with the walls. The root-mean-square speed vrms relates to the average kinetic energy:
压强来源于分子对器壁的碰撞。方均根速率vrms与平均动能相关：
(1/2) m (vrms)² = (3/2) kT
Therefore, the average translational kinetic energy of a molecule depends only on temperature. For an ideal gas, internal energy U is the sum of kinetic energies and depends only on T.
因此，分子的平均平动动能仅取决于温度。对于理想气体，内能U是所有动能之和，也只取决于T。
7. First Law of Thermodynamics | 热力学第一定律
The first law is the principle of conservation of energy applied to thermal systems:
第一定律是能量守恒原理在热力学系统中的应用：
ΔU = Q + W
where ΔU is the change in internal energy, Q is heat added to the system, and W is work done on the system. In IB and AQA physics, this sign convention is standard: Q positive if heat flows into the system; W positive if work is done on the system (compression).
其中ΔU为内能变化，Q为系统吸收的热量，W为对系统做的功。在IB和AQA物理中，此符号约定是标准：若热量传入系统，Q为正；若外界对系统做功（压缩），W为正。
If the system does work on the surroundings (expansion), W is negative. The first law unifies heat, work and internal energy into one equation.
若系统对外界做功（膨胀），则W为负。第一定律将热量、功和内能统一为一个方程。
8. Thermodynamic Processes | 热力学过程
Four specific processes are commonly tested:
常考四个典型过程：
Isothermal process: temperature constant, ΔU = 0 for an ideal gas, so Q + W = 0 → Q = -W. Heat added equals the work done by the gas (if W is work done on gas, then when gas expands, W is negative and Q positive).
等温过程：温度恒定，理想气体内能不变 ΔU = 0，因此Q + W = 0 → Q = -W。吸收的热量等于气体对外做的功（若W为对系统做的功，则气体膨胀时W为负，Q为正）。
Adiabatic process: no heat exchange Q = 0, so ΔU = W. Compression raises temperature; expansion cools the gas. p–V curve is steeper than an isothermal one.
绝热过程：无热量交换Q = 0，因此ΔU = W。压缩使温度升高；膨胀使气体冷却。p–V曲线比等温线更陡。
Isochoric (constant volume): W = 0 because ΔV = 0, so ΔU = Q. All heat changes internal energy.
等容过程：体积不变ΔV = 0，则W = 0，ΔU = Q。所有热量都改变内能。
Isobaric (constant pressure): Work done on gas is W = –pΔV. Then ΔU = Q – pΔV. Heat partly changes internal energy and partly does work.
等压过程：对气体做的功W = –pΔV，则ΔU = Q – pΔV。热量一部分改变内能，一部分对外做功。
9. p–V Diagrams and Work Done | p–V图与做功
A p–V diagram plots pressure against volume. The area under the curve represents the magnitude of work done during a process, if careful with sign conventions.
p–V图以压强对体积作图。曲线下的面积代表过程中做功的大小，需留意符号约定。
For expansion (ΔV > 0), the work done BY the gas is positive, which means work done ON the gas W is negative. The area is usually interpreted as the work done by the gas. In a cycle, the enclosed area equals the net work output.
对于膨胀（ΔV > 0），气体对外做功为正，这意味着对气体做的功W为负。通常将面积解释为气体做的功。循环过程中，封闭面积等于净输出功。
Isothermal curves are hyperbolas (p ∝ 1/V). Adiabatic curves are steeper. Constant pressure is a horizontal line; constant volume is a vertical line.
等温曲线为双曲线（p ∝ 1/V）。绝热曲线更陡。等压线是水平线段，等容线是竖直线段。
Won gas = –∫ p dV
Remember to use the same sign convention consistently in calculations.
计算时须始终使用同一符号约定。
10. Heat Engines and Efficiency | 热机与效率
A heat engine takes heat Qin from a hot reservoir, converts part of it into useful work Wout, and rejects the remaining heat Qout to a cold reservoir.
热机从高温热源吸收热量Qin，将其中一部分转化为有用功Wout，剩余热量Qout排放到低温热源。
Thermal efficiency is:
热效率为：
η = Wout / Qin = (Qin – Qout) / Qin = 1 – (Qout / Qin)
The maximum possible efficiency between two temperatures is given by a Carnot engine:
两温度间可能的最大效率由卡诺热机给出：
ηCarnot = 1 – (Tcold / Thot)
with temperatures in kelvin. Real engines always have lower efficiencies due to friction, heat losses, and irreversibilities.
温度须用开尔文。实际热机效率总是由于摩擦、热损失和不可逆因素而低于此值。
11. Second Law and Entropy | 第二定律与熵
The Second Law of Thermodynamics has several equivalent statements:
热力学第二定律有几种等价表述：
– Clausius: Heat cannot spontaneously flow from a colder body to a hotter body.
– 克劳修斯表述：热量不能自发
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