📚 Thermal Physics: Deriving Key Equations | 热力学：关键公式推导
Thermal physics lies at the heart of the A-Level Physics syllabus. Beyond memorising pV = nRT or ΔU = Q + W, being able to derive the equations from the kinetic model or from the first law provides a much deeper understanding. This article works through the core derivations of thermal physics, carefully building from empirical gas laws to the adiabatic equation, with each step explained in both English and Chinese. The notation and approach are fully aligned with the OxfordAQA International A-Level Physics specification.
热力学是A-Level物理的核心。除了记住 pV = nRT 或 ΔU = Q + W，能够从气体动理论或热力学第一定律出发推导这些方程，可以让你获得更深刻的理解。本文系统梳理热力学的核心推导，从经验气体定律逐步推进到绝热方程，每一步都用中英双语解释。采用的符号和处理方式完全符合OxfordAQA国际A-Level物理大纲的要求。
1. The Ideal Gas Law from Empirical Observations | 从经验观察建立理想气体定律
Three historically established gas laws can be combined to obtain the equation of state for an ideal gas. Boyle’s law states that at constant temperature, the pressure p of a fixed mass of gas is inversely proportional to its volume V: p ∝ 1/V. Charles’s law states that at constant pressure, the volume is directly proportional to the absolute temperature T: V ∝ T. Gay‑Lussac’s law (the pressure law) states that at constant volume, the pressure is directly proportional to the absolute temperature: p ∝ T. Combining these three proportionalities gives pV ∝ T for a fixed mass of gas.
三条历史上建立的气体定律可以合并得到理想气体的状态方程。玻意耳定律指出，在温度恒定时，一定质量气体的压强 p 与其体积 V 成反比：p ∝ 1/V。查理定律指出，在压强恒定时，体积与热力学温度 T 成正比：V ∝ T。盖‑吕萨克定律（压强定律）指出，在体积恒定时，压强与热力学温度成正比：p ∝ T。将这三个比例关系综合起来，对于一定质量的气体有 pV ∝ T。
Introducing the amount of substance n (in moles) and the universal gas constant R = 8.31 J K⁻¹ mol⁻¹ gives the familiar ideal gas equation:
引入物质的量 n（以摩尔计）和普适气体常数 R = 8.31 J K⁻¹ mol⁻¹，就得到熟悉的理想气体方程：
pV = nRT
If the total number of gas molecules N is used, the equation can be written pV = NkT, where k = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant. The two forms are linked by nR = Nk. This equation holds for ideal gases; real gases deviate from it at high pressure or low temperature.
如果使用气体分子总数 N，方程可写为 pV = NkT，其中 k = 1.38 × 10⁻²³ J K⁻¹ 为玻尔兹曼常数。两种形式通过 nR = Nk 联系。该方程适用于理想气体；真实气体在高压或低温下会偏离它。
2. Kinetic Theory Assumptions | 气体动理论的基本假设
The microscopic model that explains pressure and temperature relies on a set of simplifying assumptions about an ideal gas:
解释压强和温度的微观模型依赖于对理想气体的一组简化假设：
-
The gas consists of a very large number of identical, tiny particles (molecules) that are in constant random motion.
气体由极大量相同的微小粒子（分子）组成，它们不停地做无规则运动。
-
The volume of the molecules themselves is negligible compared with the volume of the container.
分子本身的体积与容器体积相比可以忽略不计。
-
There are no intermolecular forces except during collisions; between collisions the molecules move in straight lines at constant speed.
除碰撞瞬间外，分子间没有相互作用力；在两次碰撞之间，分子以恒定速率做直线运动。
-
Collisions between molecules, and between molecules and the container walls, are perfectly elastic. Kinetic energy is conserved.
分子之间以及分子与器壁之间的碰撞是完全弹性的，动能守恒。
-
The molecules obey Newton’s laws of motion, and the large number of particles allows statistical averages to be used.
分子服从牛顿运动定律，粒子数量极大，因此可以使用统计平均。
3. Derivation of Pressure Exerted by an Ideal Gas | 理想气体压强的推导
Imagine an ideal gas enclosed in a cube of side length L, so the volume is V = L³. Consider one molecule of mass m travelling with velocity components vx, vy, vz parallel to the edges. When this molecule collides elastically with a wall perpendicular to the x‑axis, its x‑component of velocity reverses from +vx to –vx. The change in momentum is Δpx = –2mvx, so the magnitude of momentum change imparted to the wall is 2mvx.
设想一种理想气体盛放在边长为 L 的立方体盒子中，体积 V = L³。考虑一个质量为 m 的分子，其速度沿边方向的分量为 vx、vy、vz。当这个分子与垂直于 x 轴的器壁发生弹性碰撞时，其 x 方向速度分量由 +vx 反向变为 –vx。动量变化为 Δpx = –2mvx，因此传递给器壁的动量大小为 2mvx。
The molecule travels back and forth across the cube. The time between successive collisions with the same wall is the round‑trip travel time Δt = 2L / vx. The average force exerted by this one molecule on that wall is therefore:
该分子在盒子中来回运动。与同一器壁相邻两次碰撞的时间间隔是往返时间 Δt = 2L / vx。因此单个分子对该器壁的平均作用力为：
f = (2mvx) / (2L / vx) = m vx² / L
Summing over all N molecules, the total force F on the wall is F = (m / L) Σ vx², where the sum runs over all molecules. Pressure is defined as force per unit area: p = F / L² = (m / L³) Σ vx² = (m / V) Σ vx².
对所有 N 个分子求和，该器壁上的总力为 F = (m / L) Σ vx²，求和遍及所有分子。压强定义为作用在单位面积上的力：p = F / L² = (m / L³) Σ vx² = (m / V) Σ vx²。
Introduce the mean square of the x‑component of velocity: <vx²> = (1/N) Σ vx². Then p = (Nm / V) <vx²>. Because the motion is random and isotropic, <vx²> = <vy²> = <vz²>. The actual speed c of a molecule satisfies c² = vx² + vy² + vz², so the mean square speed is <c²> = <vx²> + <vy²> + <vz²> = 3 <vx²>. Hence <vx²> = ⅓ <c²>.
引入 x 方向速度分量的均方值：<vx²> = (1/N) Σ vx²。于是 p = (Nm / V) <vx²>。由于运动无规且各向同性，<vx²> = <vy²> = <vz²>。分子的实际速率 c 满足 c² = vx² + vy² + vz²，故均方速率为 <c²> = <vx²> + <vy²> + <vz²> = 3 <vx²>。因此 <vx²> = ⅓ <c²>。
Substituting gives the fundamental kinetic‑theory equation for pressure:
代入后得到气体动理论的基本压强方程：
p = ⅓ (Nm / V) <c²>
Or equivalently, p = ⅓ ρ <c²>, where ρ = Nm / V is the gas density. In terms of the volume, pV = ⅓ Nm <c²>.
或者等价地写为 p = ⅓ ρ <c²>，其中 ρ = Nm / V 为气体密度。用体积表示就是 pV = ⅓ Nm <c²>。
4. Linking Mean Kinetic Energy to Temperature | 平均动能与温度的联系
Equating the two expressions for pV — from the kinetic theory and from the ideal gas law — provides the crucial microscopic interpretation of temperature:
将动理论的 pV = ⅓ Nm <c²> 与理想气体定律的 pV = NkT 联立，就得到温度的微观解释：
⅓ Nm <c²> = NkT → ⅓ m <c²> = kT
Multiplying by 3/2 gives the average translational kinetic energy of a single molecule:
两边乘以 3/2 即得单个分子的平均平动动能：
½ m <c²> = ³/₂ kT
This shows that the absolute temperature T is a measure of the average random kinetic energy of the gas particles. For a monatomic ideal gas, this is the only form of energy; for polyatomic gases, rotational and vibrational energies also contribute according to the equipartition theorem.
这表明热力学温度 T 是气体粒子平均无规动能的量度。对单原子理想气体，这是唯一的能量形式；对多原子气体，转动能和振动能也会根据能量均分定理贡献能量。
The root‑mean‑square (rms) speed of the molecules follows directly: crms = √(<c²>) = √(3kT / m). Using molar mass M (kg mol⁻¹), where m = M / NA and R = NAk, we obtain the useful form crms = √(3RT / M).
分子的方均根速率 (rms) 可直接得出：crms = √(<c²>) = √(3kT / m)。利用摩尔质量 M（kg mol⁻¹），m = M / NA 且 R = NAk，可得常用的形式 crms = √(3RT / M)。
5. Internal Energy of an Ideal Gas | 理想气体的内能
For a mon
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导