📚 Thermal Physics | 热物理
Thermal physics is the study of temperature, heat, and how energy is transferred and transformed within matter. Whether we are looking at the expansion of a railway track on a summer day or the microscopic collisions of gas molecules in a cylinder, the principles of thermal physics underpin countless natural and engineered systems. This article covers the key concepts for CIE A‑Level Physics: temperature scales, heat capacity, latent heat, the ideal gas model, kinetic theory, and the laws of thermodynamics.
热物理是研究温度、热量以及能量在物质中如何传递和转化的学科。无论是夏日铁轨的膨胀，还是气缸内气体分子的微观碰撞，热物理原理都支撑着无数自然与工程系统。本文涵盖了CIE A‑Level物理的核心概念：温标、热容、潜热、理想气体模型、分子动理论以及热力学定律。
1. Temperature and Thermal Equilibrium | 温度和热平衡
Temperature is a physical quantity that indicates how hot or cold an object is. It determines the direction of net energy transfer by heating: thermal energy flows spontaneously from a region of higher temperature to a region of lower temperature until thermal equilibrium is reached. At thermal equilibrium, two bodies in contact have the same temperature and there is no net heat flow between them.
温度是表示物体冷热程度的物理量。它决定了由加热引起的净能量传递方向：热量自发地从高温区域流向低温区域，直至达到热平衡。处于热平衡时，相互接触的两个物体温度相同，彼此之间没有净热流。
2. Temperature Scales and Absolute Zero | 温标和绝对零度
The Celsius scale is defined such that the melting point of pure ice is 0 °C and the boiling point of pure water is 100 °C at standard atmospheric pressure. The thermodynamic (Kelvin) scale is independent of any material property. Its unit, the kelvin (K), is defined by setting the triple point of water to 273.16 K. The lowest possible temperature is absolute zero, 0 K, corresponding to −273.15 °C, where the kinetic energy of particles is at a minimum. Temperature in kelvin is related to Celsius by:
T / K = θ / °C + 273.15
摄氏温标规定，在标准大气压下纯冰的熔点为0 °C，纯水的沸点为100 °C。热力学（开尔文）温标不依赖于任何物质的性质。其单位开尔文（K）通过设定水的三相点为273.16 K来定义。可能的最低温度为绝对零度0 K，相当于−273.15 °C，此时粒子的动能达到最小值。开尔文温度与摄氏温度的关系为：
T / K = θ / °C + 273.15
3. Heat, Internal Energy, and Work | 热量、内能和功
Internal energy is the sum of the random kinetic and potential energies of all particles in a system. Heat is energy transferred due to a temperature difference alone; it is not a property stored in an object. Work, in the thermodynamic sense, is energy transferred mechanically, for instance by compressing or expanding a gas. Both heating and doing work change the internal energy of a system.
内能是系统中所有粒子无规则运动的动能与势能之和。热量是仅由于温度差而传递的能量，不是储存在物体内的属性。热力学意义上的功是通过机械方式传递的能量，例如压缩或膨胀气体。加热和做功都能够改变系统的内能。
4. Specific Heat Capacity and Heat Capacity | 比热容和热容
The heat capacity C of a body is the energy required to raise its temperature by 1 K: C = Q / ΔT. Specific heat capacity c is the heat capacity per unit mass:
Q = m c Δθ
where m is the mass and Δθ is the change in Celsius temperature (numerically equal to ΔT). The energy supplied depends on mass, the material, and the temperature rise. Water, for example, has a very high specific heat capacity (about 4200 J kg⁻¹ K⁻¹), which makes it an excellent coolant and thermal store.
物体的热容C是使其温度升高1 K所需的能量：C = Q / ΔT。比热容c是单位质量的热容：
Q = m c Δθ
其中m为质量，Δθ为摄氏温度的变化（数值上等于ΔT）。所需能量取决于质量、材料以及温升。例如水的比热容非常高（约4200 J kg⁻¹ K⁻¹），这使它成为优良的冷却剂和储热介质。
5. Specific Latent Heat | 比潜热
When a substance changes state at constant temperature, energy is absorbed or released without a change in temperature. The specific latent heat of fusion Lf is the energy per unit mass needed to change a solid to a liquid at its melting point. The specific latent heat of vaporisation Lv applies to the liquid–gas change. The energy transferred is:
Q = m L
where L is the appropriate latent heat. During boiling, for instance, the added energy breaks intermolecular bonds rather than increasing kinetic energy, which is why the temperature remains constant until the phase change is complete.
物质在温度不变的情况下发生状态变化时，会吸收或释放能量而不伴随温度变化。熔解比潜热Lf是指单位质量的固体在熔点变为液体所需的能量，汽化比潜热Lv则适用于液‑气相变。传递的能量为：
Q = m L
其中L为相应的潜热。例如在沸腾过程中，加入的能量主要用于破坏分子间键，而不是增加动能，因此在整个相变完成之前温度保持不变。
6. Ideal Gas Equation of State | 理想气体状态方程
An ideal gas obeys the three historical gas laws exactly. Boyle’s law (p ∝ 1/V at constant T), Charles’s law (V ∝ T at constant p), and the pressure law (p ∝ T at constant V) are combined in the equation of state:
pV = nRT
where p is pressure, V is volume, n is the number of moles, R is the molar gas constant (8.31 J mol⁻¹ K⁻¹), and T is the absolute temperature. For a fixed mass of gas, pV/T is constant. The equation also appears in molecular form:
pV = NkT
where N is the number of molecules and k is the Boltzmann constant (1.38 × 10⁻²³ J K⁻¹).
理想气体严格遵循三条历史气体定律。波义耳定律（恒温下p ∝ 1/V）、查理定律（恒压下V ∝ T）和压力定律（恒容下p ∝ T）被综合为状态方程：
pV = nRT
其中p为压强，V为体积，n为摩尔数，R为摩尔气体常数（8.31 J mol⁻¹ K⁻¹），T为绝对温度。对于一定质量的气体，pV/T为常量。该方程也可以用分子数量表示：
pV = NkT
其中N为分子数，k为玻尔兹曼常数（1.38 × 10⁻²³ J K⁻¹）。
7. Kinetic Model of an Ideal Gas | 理想气体的分子运动模型
The kinetic theory explains macroscopic properties by modelling gas molecules as identical, perfectly elastic point particles in random motion. The assumptions include: large number of molecules, negligible molecular volume compared with the container, no intermolecular forces except during collisions, and random rapid motion obeying Newton’s laws. From these, we derive:
pV = ⅓ N m ⟨c²⟩
where m is the mass of one molecule and ⟨c²⟩ is the mean‑square speed. This equation links the microscopic mean‑square speed to the macroscopic pressure and volume.
分子动理论通过将气体分子视为完全弹性、无体积的质点做无规则运动来解释宏观性质。其假设包括：分子数量极大；分子本身体积与容器相比可忽略；除碰撞瞬间外无分子间力；分子做遵守牛顿定律的快速无规则运动。由此可推导出：
pV = ⅓ N m ⟨c²⟩
其中m为单个分子的质量，⟨c²⟩为方均速率。该方程将微观的方均速率与宏观压强和体积联系起来。
8. Molecular Kinetic Energy and Temperature | 分子动能与温度
Comparing pV = ⅓ N m ⟨c²⟩ with pV = NkT gives:
½ m ⟨c²⟩ = ³⁄₂ kT
The left‑hand side is the average translational kinetic energy of a single molecule. Hence the absolute temperature is a measure of the average random kinetic energy of the molecules. This profound result explains why all ideal‑gas predictions require temperature in kelvin and why there is a lower limit to temperature.
对比pV = ⅓ N m ⟨c²⟩与pV = NkT可得：
½ m ⟨c²⟩ = ³⁄₂ kT
等式左边是单个分子的平均平动动能。因此，绝对温度是分子平均无规则动能的量度。这一重要结果解释了为何所有理想气体预言都必须使用开尔文温度，也说明了温度存在下限的原因。
9. First Law of Thermodynamics | 热力学第一定律
The first law is a statement of conservation of energy for a thermodynamic system:
ΔU = Q + W
where ΔU is the increase in internal energy, Q is the heat supplied to the system, and W is the work done on the system. (Some texts use W for work done by the system, changing the sign; the CIE specification prefers the convention where work done on the system is positive.) The internal energy depends only on the state of the system, whereas Q and W depend on the path.
热力学第一定律是热力学系统的能量守恒表述：
ΔU = Q + W
其中ΔU是内能的增量，Q是传递给系统的热量，W是对系统所做的功。（有些教材用W表示系统对外做的功，那样符号会相反；CIE大纲倾向于规定对系统做功为正。）内能仅取决于系统的状态，而Q和W与过程路径相关。
10. Thermodynamic Processes and p–V Diagrams | 热力学过程与p–V图
Four special processes are often analysed using p–V graphs:
- Isothermal (ΔU = 0, T constant): pV = constant; hyperbola on p– onvention ΔU = Q + W, for an isothermal expansion, work done ON gas is negative, so Q = +Won? Let us stick to agreement: Q + W = 0, so Q = −Won. For gas expanding, Won is negative, so Q positive, heat enters.)
- Isovolumetric (W = 0, V constant): vertical line on p–V; ΔU = Q.
- Isobaric (p constant): horizontal line; W = −p ΔV (work done on gas).
- Adiabatic (Q = 0): no heat exchange; ΔU = W, steeper curve than isothermal.
在p–V图上常分析四种特殊过程：
- 等温（ΔU = 0，T恒定）： pV为定值；p–V图上为双曲线。Q = −W（所有输入热量用于气体对外做功，如果遵守ΔU = Q + W，等温膨胀时对气体做功为负，Q为正）。
- 等容（W = 0，V恒定）： p–V图上的竖直线；ΔU = Q。
- 等压（p恒定）： 水平线；对气体做功W = −p ΔV。
- 绝热（Q = 0）： 无热量交换；ΔU = W，曲线比等温线更陡。
11. Work Done by a Gas | 气体做功
When a gas expands by a small volume dV against an external pressure p, the work done on is dW = −p dV. For a finite change, the total work is the negative of the area under the p– ne is the area enclosed by the loop, and the first law links heat and work to the internal‑energy change, which is zero over a complete cycle.
当气体克服外压p膨胀微小体积dV时，对气体做的元功为dW = −p dV。对于有限变化，总功等于p–V曲线下方面积的负值。在一个循环中，净功等于循环所围的面积，而热力学第一定律将热量和功与内能变化联系起来；对于完整循环，内能变化为零。
12. Summary of Key Equations | 关键方程总结
The essential mathematical relations for A‑Level thermal physics are collected below:
| Concept | Equation | Notes |
|---|---|---|
| Heat and temperature change | Q = m c Δθ | Δθ in °C or K |
| Latent heat | Q = m L | L = Lf or Lv |
| Ideal gas law (molar) | pV = nRT | R = 8.31 J mol⁻¹ K⁻¹ |
| Ideal gas law (molecular) | pV = NkT | k = 1.38 × 10⁻²³ J K⁻¹ |
| Kinetic theory pressure | pV = ⅓ N m ⟨c²⟩ | m = mass of one molecule |
| Mean kinetic energy | ½ m ⟨c²⟩ = ³⁄₂ kT | Average translational KE |
| First law of thermodynamics | ΔU = Q + W | W = work done on system |
下表中总结了A‑Level热物理的核心数学关系：
| 概念 | 方程 | 备注 |
|---|---|---|
| 热量与温度变化 | Q = m c Δθ | Δθ 单位 °C 或 K |
| 潜热 | Q = m L | L = Lf 或 Lv |
| 理想气体定律（摩尔） | pV = nRT | R = 8.31 J mol⁻¹ K⁻¹ |
| 理想气体定律（分子） | pV = NkT | k = 1.38 × 10⁻²³ J K⁻¹ |
| 分子动理论压强 | pV = ⅓ N m ⟨c²⟩ | m = 单个分子质量 |
| 平均动能 | ½ m ⟨c²⟩ = ³⁄₂ kT | 平均平动动能 |
| 热力学第一定律 | ΔU = Q + W | W = 对系统做的功 |
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