Unit B.1: Thermal Energy Transfers单元 B.1：热能传递

Particles in a substance have a distribution of speeds, so any single molecule's KE is constantly changing through collisions. Temperature is proportional to the mean translational kinetic energy of the particles, which is well-defined even when individual energies fluctuate. For an ideal monatomic gas this link is exact:

物质中的粒子具有速率分布，因此任一分子的动能因碰撞不断变化。温度正比于粒子的平均平动动能，即便个体能量起伏，这个平均值仍是良定义的。对理想单原子气体，此关系严格成立：

where $k_{B}$ is the Boltzmann constant (data booklet). The zeroth law of thermodynamics states that if body $A$ is in thermal equilibrium with $B$, and $B$ with $C$, then $A$ is in equilibrium with $C$. This is what lets a thermometer work: it equilibrates with the object, then reads its own temperature.

其中 $k_{B}$ 是玻尔兹曼常数（数据手册）。热力学第零定律指出：若物体 $A$ 与 $B$ 热平衡，$B$ 与 $C$ 热平衡，则 $A$ 与 $C$ 热平衡。这正是温度计能工作的原因：它先与物体达到平衡，再读出自身温度。

In a school determination of $c$, an electrical heater of power $P = VI$ delivers energy $Q = VIt$ to a mass $m$, and the temperature rise $\Delta T$ is measured. Equating $VIt = mc\Delta T$ gives

在学校测定 $c$ 时，功率 $P = VI$ 的电加热器在时间 $t$ 内向质量 $m$ 输入能量 $Q = VIt$，并测得温升 $\Delta T$。令 $VIt = mc\Delta T$ 得

In practice some energy leaks to the surroundings, so the measured $c$ is usually an overestimate (less of the input went into the sample than assumed). Two standard fixes: lag the sample to reduce losses, or plot temperature against time and take the gradient while the sample is near room temperature, where the loss rate is smallest. The same loss bias explains why an unlagged method-of-mixtures experiment yields a final temperature slightly below the ideal prediction.

实际中部分能量会泄漏到环境，故测得的 $c$ 通常偏大（实际进入样品的能量少于假定）。两种标准修正：给样品加隔热层以减少损失；或作温度—时间图，在样品接近室温（损失率最小）处取斜率。同样的损失偏差也解释了为何无隔热的混合法实验末温会略低于理想预测。

Evaporation is a phase change that occurs at the surface of a liquid below its boiling point: the fastest molecules escape, so the average KE of those left behind drops and the liquid cools. This is the mechanism behind sweating and the cooling effect of an alcohol swab.

蒸发是发生在液面、温度低于沸点的相变：最快的分子逸出，剩余分子的平均动能下降，液体因而降温。这正是出汗与酒精棉片致冷的机制。

Vaporisation generally needs more energy per kilogram than fusion ($L_{v} > L_{f}$) because melting only loosens a lattice into a liquid (molecules stay in contact), whereas boiling must fully separate the molecules against the intermolecular forces and do work pushing back the surrounding atmosphere. For water, $L_{v} = 2.26\times10^{6}\ \mathrm{J\,kg^{-1}}$ is roughly seven times $L_{f} = 3.34\times10^{5}\ \mathrm{J\,kg^{-1}}$.

汽化每千克所需能量通常大于熔化（$L_{v} > L_{f}$），因为熔化只是把晶格松动成液体（分子仍相互接触），而沸腾必须克服分子间作用力把分子完全分开，并对外界大气做功。对水而言，$L_{v} = 2.26\times10^{6}\ \mathrm{J\,kg^{-1}}$ 约为 $L_{f} = 3.34\times10^{5}\ \mathrm{J\,kg^{-1}}$ 的七倍。

A surface's emissivity $e$ (between $0$ and $1$) measures how well it emits radiation compared with a perfect (black-body) emitter. A matt black surface has $e$ near $1$ (good emitter and good absorber); a polished silver surface has $e$ near $0$ (poor emitter, good reflector). Crucially, a good emitter is also a good absorber at the same wavelength (Kirchhoff's idea of radiative balance), which is why solar panels and car radiators are painted matt black, while survival blankets and vacuum flasks are silvered.

表面的发射率 $e$（介于 $0$ 与 $1$ 之间）衡量它相对于理想（黑体）发射体的发射能力。亚光黑表面 $e$ 接近 $1$（良发射体、良吸收体）；抛光银表面 $e$ 接近 $0$（不良发射体、良反射体）。关键是：良发射体在同一波长也是良吸收体（基尔霍夫的辐射平衡思想），故太阳能板和汽车散热器漆成亚光黑，而急救毯和保温瓶镀银。

Emissivity enters the Stefan–Boltzmann law for a real (non-ideal) body as a multiplying factor $e$: $L = e\,\sigma A T^{4}$. The idealised black body of B1.5 takes $e = 1$.

对真实（非理想）物体，发射率以乘因子 $e$ 进入斯特藩—玻尔兹曼定律：$L = e\,\sigma A T^{4}$。B1.5 的理想黑体取 $e = 1$。

A black body emits a continuous spectrum of all wavelengths, with an intensity-vs-wavelength curve that peaks at a single wavelength and falls off on either side. As temperature rises, the whole curve shifts up (more power at every wavelength) and the peak moves to shorter wavelengths (covered by Wien's law in B1.6). The Stefan–Boltzmann law is the area under this spectral curve.

黑体发射包含所有波长的连续光谱，其强度—波长曲线在某一波长处出现峰值并向两侧衰减。温度升高时整条曲线上移（每个波长功率都增大），峰值移向更短波长（由 B1.6 维恩定律给出）。斯特藩—玻尔兹曼定律即该光谱曲线下方的面积。

For a star at distance $d$, the power spreads over a sphere of area $4\pi d^{2}$, so the apparent brightness $b$ (power per unit area received at Earth) is

对距离 $d$ 处的恒星，功率分布在面积 $4\pi d^{2}$ 的球面上，故视亮度 $b$（地球处单位面积接收的功率）为

This inverse-square dilution, combined with the Stefan–Boltzmann luminosity, is the foundation of the stellar measurements taken up later in Theme E (Fusion and stars).

这种平方反比稀释，配合斯特藩—玻尔兹曼光度，构成后续主题 E（聚变与恒星）中恒星测量的基础。

Suppose a star's spectral peak gives $\lambda_{\max} = 9.7\times10^{-7}\ \mathrm{m}$, so by Wien's law $T = 2.9\times10^{-3} / 9.7\times10^{-7} \approx 3000\ \mathrm{K}$ (a cool red star). If its luminosity is independently measured as $L = 1.0\times10^{28}\ \mathrm{W}$, rearrange the Stefan–Boltzmann law for the radius:

设某恒星光谱峰值给出 $\lambda_{\max} = 9.7\times10^{-7}\ \mathrm{m}$，则由维恩定律 $T = 2.9\times10^{-3} / 9.7\times10^{-7} \approx 3000\ \mathrm{K}$（一颗冷的红色恒星）。若另测得光度 $L = 1.0\times10^{28}\ \mathrm{W}$，把斯特藩—玻尔兹曼定律对半径求解：

Substituting, $r = \sqrt{\dfrac{1.0\times10^{28}}{4\pi(5.67\times10^{-8})(3000)^{4}}} \approx 4.6\times10^{10}\ \mathrm{m}$, far larger than the Sun. A cool yet highly luminous star must be enormous, which is exactly how astronomers identify red giants. This Wien-then-Stefan–Boltzmann chain is the workhorse for stellar properties in Theme E.

代入得 $r = \sqrt{\dfrac{1.0\times10^{28}}{4\pi(5.67\times10^{-8})(3000)^{4}}} \approx 4.6\times10^{10}\ \mathrm{m}$，远大于太阳。一颗温度低却高光度的恒星必然极其庞大，这正是天文学家识别红巨星的依据。这条"先维恩、后斯特藩—玻尔兹曼"的链条是主题 E 中求恒星性质的主力工具。