📚 Chance and Spontaneous Change | 概率与自发变化
Every chemical and physical process in the universe has a natural tendency to move in one direction rather than another. An ice cube melts in a warm room but never refreezes spontaneously; a gas expands to fill its container but never contracts into a corner by itself. These observations point to a profound connection between chance, probability, and the direction of change. At the heart of this connection lies entropy – a thermodynamic quantity that measures the number of ways energy and particles can be arranged. Understanding spontaneous change through the lens of chance gives us a powerful tool to predict whether a reaction will occur and to explain why some processes are irreversible.
宇宙中的每一个化学和物理过程都有一种自然趋势，朝着某个方向进行而不是相反。冰块在温暖的房间里融化，却不会自发地重新冻结；气体膨胀充满容器，却不会自己缩到角落。这些观察揭示了一个深刻的联系：概率、机会与变化方向之间的关系。这一联系的核心就是熵——一种衡量能量和粒子排列方式数目的热力学量。通过概率的视角来理解自发变化，给了我们一个强大的工具，不仅能预测反应是否会发生，还能解释为什么有些过程是不可逆的。
1. Understanding Spontaneous Processes | 理解自发过程
A spontaneous process is one that, once initiated, continues on its own without any external input of energy. Dissolving salt in water, the rusting of iron, and the cooling of a hot drink are all spontaneous under the conditions they occur. Importantly, 'spontaneous' does not mean 'instantaneous' – rusting happens slowly – but it indicates a natural direction of change. In contrast, non-spontaneous processes, such as the electrolysis of water to produce hydrogen and oxygen, require a continuous supply of energy to force the reaction backwards.
自发过程是指一旦开始，就不需要任何外部能量输入就能自行继续的过程。盐溶于水、铁生锈、热饮冷却，这些在所处条件下都是自发的。重要的是，“自发”并不意味着“瞬间”发生——生锈是缓慢的——但它表明了变化的自然方向。相反，非自发过程，例如电解水生成氢气和氧气，需要持续供给能量才能迫使反应逆向进行。
2. The Role of Energy and Disorder | 能量与无序的作用
For many years, chemists thought that the sole driving force behind spontaneity was a decrease in energy. Exothermic reactions, which release heat, were seen as the prime candidates for spontaneous change because they lead to more stable products. Yet many endothermic processes, such as the dissolving of ammonium nitrate in water (which absorbs heat and feels cold), also occur spontaneously. This puzzle was resolved by introducing the concept of disorder – or more precisely, the distribution of energy among particles. A system tends to move towards a state of greater disorder because that state is statistically more probable.
多年来，化学家们认为自发性的唯一驱动力是能量的降低。放热反应释放热量，被视为自发变化的首选，因为它们产生更稳定的产物。然而许多吸热过程，例如硝酸铵溶于水（吸收热量，感觉冷），也是自发的。这一难题通过引入无序的概念——更准确地说是能量在粒子间的分布——得到了解决。系统倾向于朝着更无序的状态演化，因为这种状态在统计学上更有可能发生。
3. Entropy: A Measure of Chance | 熵：概率的量度
Entropy (S) is the thermodynamic property that quantifies how spread out or dispersed the energy of a system is among its possible arrangements. The greater the number of ways particles and their energy quanta can be arranged, the higher the entropy. Consider a crystal at absolute zero: all particles are fixed in a perfect lattice with minimal vibrational energy, so there is essentially only one arrangement – entropy is at its lowest. When heated, particles vibrate more vigorously and energy can be distributed in many different ways, increasing entropy. Spontaneous changes are those that lead to an overall increase in the total entropy of the system and its surroundings.
熵（S）是热力学性质，用于量化系统能量在其可能排列方式中的分散程度。粒子和它们的能量量子能够排列的方式越多，熵就越高。想象一下绝对零度下的晶体：所有粒子都固定在完美的晶格中，振动能量极小，实际上只有一种排列方式——熵处于最低值。当加热时，粒子振动更加剧烈，能量可以以许多不同方式分布，熵增加。自发变化是那些导致系统及其周围的总熵增加的变化。
4. Microstates and Macrostates | 微观状态与宏观状态
To connect entropy with probability, we need the concept of microstates. A microstate is a specific, detailed arrangement of all the particles and their energies in a system at any instant. A macrostate is defined by bulk properties like temperature, pressure, and volume, which can be realized by an enormous number of microstates. For example, when two gases mix, the mixed macrostate has vastly more microstates than the separated macrostate. Nature does not 'prefer' disorder; it simply evolves towards the macrostate with the highest number of accessible microstates because that macrostate is overwhelmingly more probable.
要将熵与概率联系起来，我们需要微观状态的概念。微观状态是指系统中所有粒子及其能量在任一时刻的具体、细致的排列。宏观状态则由温度、压力、体积等宏观性质定义，这些性质可以由大量的微观状态来实现。例如，当两种气体混合时，混合后的宏观状态比分离的宏观状态拥有多得多的微观状态。自然界并非“偏爱”无序，它只是向着可到达的微观状态数最多的宏观状态演化，因为那个宏观状态具有压倒性的更高的概率。
5. The Second Law of Thermodynamics | 热力学第二定律
The second law states: the total entropy of an isolated system always increases during a spontaneous change. It never decreases. For a non-isolated system, we must consider the entropy change of both the system and its surroundings. A process can be spontaneous if the system's entropy decreases, provided the surroundings' entropy increases by an even larger amount, so that ΔStotal > 0. This is why living organisms, which maintain high internal order, can do so only by exporting entropy to their environment through respiration and waste production.
热力学第二定律指出：在自发变化过程中，孤立系统的总熵总是增加的，它永远不会减少。对于非孤立系统，我们必须同时考虑系统及其周围的熵变。即使系统的熵减少，只要周围的熵增加得更多，使得总熵变 ΔStotal > 0，过程就可能是自发的。这就是为什么生物体能够维持高度内部秩序，却只能通过呼吸和产生废物将熵输出到环境中来实现。
6. Gibbs Free Energy and Spontaneity | 吉布斯自由能与自发性
Measuring entropy changes of the surroundings directly is often inconvenient. The Gibbs free energy (G) was introduced to combine system enthalpy and entropy into a single criterion for spontaneity at constant temperature and pressure. The defining equation is:
ΔG = ΔH − TΔS
where ΔG is the change in Gibbs free energy, ΔH is the enthalpy change, T is the absolute temperature in kelvin, and ΔS is the entropy change of the system. For a process to be spontaneous, ΔG must be negative. This equation beautifully captures the tug-of-war between energy minimisation (negative ΔH) and entropy maximisation (positive ΔS, which contributes -TΔS to ΔG).
直接测量周围环境的熵变常常不方便。吉布斯自由能（G）的引入，将系统的焓和熵结合成一个在恒温恒压下的自发性判据。定义方程为：
ΔG = ΔH − TΔS
其中 ΔG 是吉布斯自由能的变化，ΔH 是焓变，T 是以开尔文为单位的绝对温度，ΔS 是系统的熵变。一个过程要自发进行，ΔG 必须为负。这个方程出色地捕捉了能量最小化（负的 ΔH）和熵最大化（正的 ΔS，它通过 −TΔS 项贡献给 ΔG）之间的拉锯战。
7. Calculating Entropy Changes | 计算熵变
Standard molar entropy values (S°) are listed for substances under standard conditions (298 K, 100 kPa). The standard entropy change for a reaction is calculated similarly to enthalpy changes:
ΔS° = Σ S°(products) − Σ S°(reactants)
The units of entropy are J K⁻¹ mol⁻¹. Note that unlike standard enthalpies of formation which are relative to elements, standard entropies are absolute values based on the third law of thermodynamics. Gases typically have much higher S° values than solids due to their highly disordered arrangement and translational freedom. Using ΔH° and ΔS° we can then compute ΔG° and predict spontaneity.
标准摩尔熵值（S°）列出了一般物质在标准条件（298 K，100 kPa）下的熵。反应的标准熵变计算与焓变类似：
ΔS° = Σ S°(产物) − Σ S°(反应物)
熵的单位是 J K⁻¹ mol⁻¹。注意，不同于标准生成焓是相对于元素的，标准熵是根据热力学第三定律得到的绝对值。气体通常比固体有高得多的 S° 值，因为它们的高度无序排列和平动自由度。利用 ΔH° 和 ΔS°，我们可以计算 ΔG° 并预测自发性。
8. Factors Affecting Entropy | 影响熵的因素
Several factors influence the entropy of a system. First, physical state: S(solid) < S(liquid) < S(gas). Second, the complexity of the molecule: larger, more complex molecules with more atoms have higher entropy because they possess more vibrational and rotational modes. Third, temperature: increasing temperature always raises entropy by populating higher energy levels. Fourth, mixing: forming a solution or mixing gases increases entropy dramatically due to the greater number of possible arrangements. Finally, volume: expanding a gas into a larger volume increases the number of positional microstates and thus entropy.
有多个因素影响系统的熵。第一，物理状态：S(固体) < S(液体) < S(气体)。第二，分子的复杂性：更大、更复杂的分子，原子更多，拥有更多的振动和转动模式，因此熵更高。第三，温度：升高温度总是通过使更高能级被占据来提高熵。第四，混合：形成溶液或混合气体由于可能的排列方式大大增加，熵急剧增大。最后，体积：气体膨胀到更大的体积，增加了位置微观状态数，从而增加了熵。
9. Spontaneity in Chemical Reactions | 化学反应中的自发性
wever, it is also endothermic (ΔH° > 0). At low temperature the TΔS term may be smaller than ΔH, making ΔG positive and the reaction non-spontaneous. At high temperature, the −TΔS term dominates, driving ΔG to negative values and making the decomposition spontaneous. This is why heating is required to decompose limestone in a kiln. The temperature at which ΔG becomes zero marks the point where spontaneity switches; it is given by T = ΔH/ΔS (assuming ΔH and ΔS do not vary with temperature).
以碳酸钙的热分解为例：CaCO₃(s) → CaO(s) + CO₂(g)。该反应产生气体，因此 ΔS° 正值很大。但同时它也是吸热的（ΔH° > 0）。在低温下，TΔS 项可能小于 ΔH，使得 ΔG 为正，反应非自发。在高温下，−TΔS 项占主导，使 ΔG 变为负值，分解反应变得自发。这就是为什么在窑炉中分解石灰石需要加热。ΔG 变为零的温度标志着自发性切换的点；该温度由 T = ΔH/ΔS 给出（假设 ΔH 和 ΔS 不随温度变化）。
10. Chance and the Direction of Change | 概率与变化的方向
ssible microstates. Only 2 of those correspond to all four particles in one bulb (one microstate for the left bulb, one for the right). The remaining 14 microstates have the particles spread between both bulbs. Thus, the probability of finding all particles in one half is merely 2/16 = 1/8. If we had a mole of particles, the chance of all being in one half is fantastically small, effectively zero. This is why gases expand and fill a vacuum spontaneously: the most probable macrostate is the one with evenly distributed particles, which maximises entropy.
一个用四个气体粒子在双球容器中的思想实验有助于说明概率。如果粒子随机移动，总共有 16 个可能的微观状态。其中只有 2 个对应于全部四个粒子都集中在一个球里（左侧球一个微观状态，右侧球一个）。剩下的 14 个微观状态中，粒子分布在两个球之间。因此，发现所有粒子都在一半空间的概率仅仅是 2/16 = 1/8。如果我们有一摩尔粒子，全部粒子集中在一半的概率小得惊人，实际上为零。这就是为什么气体自发膨胀并充满真空：最可能的宏观状态是粒子均匀分布的状态，这使熵最大化。
11. Why Some Processes Are Slow | 为什么有些过程缓慢
Thermodynamics tells us that a process is possible (spontaneous) if ΔG is negative, but it says nothing about how quickly it will occur. The rate is governed by kinetics. For example, carbon in the form of diamond has a positive ΔG° to convert to graphite under standard conditions, yet diamonds do not spontaneously turn into graphite at a measurable rate – the reaction has an enormous activation energy barrier. Similarly, a mixture of hydrogen and oxygen gases at room temperature is thermodynamically unstable with respect to water formation, but the reaction does not happen without a spark. Entropy may favour a change, but chance alone cannot overcome kinetic barriers; molecules must still collide with sufficient energy and proper orientation.
热力学告诉我们，如果 ΔG 为负，过程在热力学上是可能（自发）的，但它完全没有说明过程会多快发生。速率由动力学控制。例如，碳以金刚石的形式存在，在标准条件下转化为石墨的 ΔG° 为正，但金刚石不会以可测量的速率自发转变为石墨——该反应有巨大的活化能势垒。同样，室温下的氢气和氧气混合物相对于生成水来说热力学上是不稳定的，但没有火花反应就不会发生。熵可能青睐某一变化，但单靠概率无法克服动力学障碍；分子仍然必须具有足够的能量和适当的取向进行碰撞。
12. Summary and Key Points | 总结与要点
his likelihood. The second law dictates that total entropy must increase for any spontaneous process. Chemists use Gibbs free energy to assess spontaneity at constant temperature and pressure, with ΔG = ΔH − TΔS. A negative ΔG signals a spontaneous reaction. While entropy can explain why a change happens, the rate is another matter entirely. Together, thermodynamics and kinetics give a complete picture of chemical reactivity, with probability and chance lying at the very core of why some changes are natural and others are not.
自发变化本质上是一场在原子尺度上演的概率游戏。作为衡量微观状态数目的量，熵量化了这种可能性。热力学第二定律规定，任何自发过程的总熵必须增加。化学家使用吉布斯自由能来评估恒温恒压下的自发性，表达式为 ΔG = ΔH − TΔS。ΔG 为负即表明反应自发。虽然熵可以解释变化为什么会发生，但反应速率完全是另一回事。热力学和动力学一起描绘了化学反应活性的完整图景，而概率与机会正是为什么有些变化是自然而然的、有些则不然的核心所在。
Published by TutorHao | Chemistry Revision Series | aleveler.com
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