📚 R1.4 Entropy and Spontaneity (HL only) | R1.4 熵与自发性（仅HL）
Entropy and spontaneity are central concepts in chemical thermodynamics, helping us predict whether a reaction or process will occur without external intervention. In this HL topic, we explore entropy as a measure of disorder, link it to microstates through Boltzmann's equation, and combine it with enthalpy to define Gibbs free energy. We will also see how temperature can switch the direction of spontaneity and how the equilibrium constant is related to the standard free energy change.
熵与自发性是化学热力学的核心概念，它们帮助我们预测一个反应或过程是否能在没有外部干预的情况下发生。在这个 HL 主题中，我们将熵作为无序度的量度来探讨，通过玻尔兹曼方程将其与微观状态联系起来，并将其与焓结合以定义吉布斯自由能。我们还会看到温度如何改变自发性的方向，以及平衡常数与标准自由能变化之间的关系。
1. The Concept of Entropy | 熵的概念
Entropy, symbol S, is a thermodynamic state function that quantifies the dispersal of energy and matter within a system. It is often described qualitatively as a measure of disorder or randomness: the greater the disorder, the higher the entropy. A gas has higher entropy than a liquid, which in turn has higher entropy than a solid at the same temperature.
熵，符号为 S，是一个热力学状态函数，它量化了系统内能量和物质的分散程度。通常被定性描述为无序度或随机度的量度：无序度越大，熵值越高。在相同温度下，气体的熵高于液体，而液体的熵又高于固体。
As a state function, the entropy change ΔS for a process depends only on the initial and final states, not on the path taken. The units of entropy are J K⁻¹ mol⁻¹. In IB Chemistry, we focus on the direction of change: entropy tends to increase in an isolated system, which is a statement of the Second Law of Thermodynamics.
作为一个状态函数，过程的熵变 ΔS 仅取决于初态和终态，与所经历的路径无关。熵的单位是 J K⁻¹ mol⁻¹。在 IB 化学中，我们关注变化的方向：在孤立系统中，熵倾向于增加，这是热力学第二定律的一种表述。
2. Entropy and Microstates – Boltzmann's Equation | 熵与微观状态——玻尔兹曼方程
At the molecular level, entropy is related to the number of ways the particles of a system can be arranged while having the same total energy. These distinguishable arrangements are called microstates, symbol W. Ludwig Boltzmann proposed the equation that links entropy to the number of microstates:
在分子水平上，熵与系统粒子在总能量相同的情况下可能存在的排列方式数目有关。这些可区分的排列称为微观状态，符号为 W。路德维希·玻尔兹曼提出了将熵与微观状态数联系起来的方程：
S = k ln W
gas at room temperature has a vast number of microstates and correspondingly high entropy.
其中 k 是玻尔兹曼常数（1.38 × 10⁻²³ J K⁻¹）。具有更多微观状态的系统熵值更高，因为能量分配的方式更多。例如，接近 0 K 的晶体只有一个微观状态（W = 1），因此 S = 0，而室温下的气体有大量的微观状态，熵值相应很高。
This microscopic interpretation explains why processes like expansion of a gas or heating increase entropy: they increase the number of accessible microstates.
这种微观解释说明了为什么像气体膨胀或加热这样的过程会使熵增加：它们增大了可及微观状态的数量。
3. The Third Law of Thermodynamics and Absolute Entropy | 热力学第三定律与绝对熵
The Third Law of Thermodynamics states that the entropy of a perfect, pure crystal at absolute zero (0 K) is exactly zero. This provides an absolute reference point, allowing us to assign absolute entropy values to substances, unlike enthalpy which is measured relative to an arbitrary zero.
热力学第三定律指出，在绝对零度（0 K）时，完美纯净晶体的熵为零。这提供了一个绝对的参考点，使我们能够给物质赋予绝对的熵值，而不像焓那样是相对于一个任意零点进行测量。
As a substance is heated, its entropy increases due to the increasing thermal motion of its particles. Phase transitions, such as melting and boiling, result in sharp increases in entropy because the particles gain greater freedom of movement. These absolute entropy values are known as standard entropies, S°, when measured under standard conditions (298 K, 1 bar).
当物质被加热时，由于粒子热运动加剧，其熵值增加。相变（如熔化和沸腾）会导致熵的急剧增加，因为粒子获得了更大的运动自由度。这些在标准条件（298 K，1 bar）下测得的绝对熵值称为标准熵 S°。
Standard entropies are always positive, unlike standard enthalpies of formation which can be negative. This is because entropy measures dispersal, and even the most ordered substance above 0 K possesses some entropy.
标准熵总是正值，这与可能为负的标准生成焓不同。这是因为熵衡量的是分散度，而即使在 0 K 以上，最有序的物质也具有一定的熵。
4. Standard Entropy (S°) and Trends | 标准熵（S°）及变化趋势
Standard entropies of substances depend on several factors. In general, S° increases with increasing molar mass for similar substances because heavier atoms or molecules have more closely spaced energy levels, making more energy states accessible.
物质的标准熵取决于几个因素。通常，对于相似物质，S° 随摩尔质量增大而增大，因为较重的原子或分子能级间隔更紧密，从而可及的能量状态更多。
The physical state is crucial: S°(gas) > S°(liquid) > S°(solid). For example, S° of H₂O(g) is 188.8 J K⁻¹ mol⁻¹, while liquid water is 69.9 J K⁻¹ mol⁻¹. Complexity also matters: a more complex molecule, such as C₃H₈(g), has a higher entropy than CH₄(g) because it has more atoms and more ways to distribute energy through vibrations and rotations.
物理状态至关重要：S°(气体) > S°(液体) > S°(固体)。例如，H₂O(g) 的 S° 为 188.8 J K⁻¹ mol⁻¹，而液态水为 69.9 J K⁻¹ mol⁻¹。复杂性同样重要：较复杂的分子，如 C₃H₈(g)，比 CH₄(g) 具有更高的熵，因为它有更多的原子，可以通过振动和转动分配能量的方式更多。
Allotropes also differ: diamond has a lower S° (2.4 J K⁻¹ mol⁻¹) than graphite (5.7 J K⁻¹ mol⁻¹) due to diamond's rigid three-dimensional network.
同素异形体的熵值也有差异：金刚石的 S°（2.4 J K⁻¹ mol⁻¹）低于石墨（5.7 J K⁻¹ mol⁻¹），这是因为金刚石具有刚性的三维网络结构。
5. Calculating Entropy Changes for Reactions | 计算反应的熵变
The standard entropy change for a reaction, ΔS°ᵣₓₙ, can be calculated from the absolute entropies of products and reactants using a formula similar to that for enthalpy:
反应的标准熵变 ΔS°ᵣₓₙ 可以利用产物和反应物的绝对熵，通过类似于焓变的公式进行计算：
ΔS° = Σ S°(products) − Σ S°(reactants)
Each S° value is multiplied by the stoichiometric coefficient from the balanced equation. Because S° values are given in J K⁻¹ mol⁻¹, it is essential to watch units when combining with enthalpy values later.
每个 S° 值都要乘以配平方程中的化学计量系数。由于 S° 值的单位是 J K⁻¹ mol⁻¹，在之后与焓值结合时一定要注意单位转换。
ve because the reaction produces a liquid from gases, decreasing disorder. The calculated value is roughly −326 J K⁻¹ mol⁻¹ (per mole of reaction as written). This negative sign tells us the system becomes more ordered.
例如，对于反应：2H₂(g) + O₂(g) → 2H₂O(l)，ΔS° 为负值，因为该反应由气体生成液体，降低了无序度。计算值约为 −326 J K⁻¹ mol⁻¹（按方程式所写的每摩尔反应）。负号表明系统变得更加有序。
6. Introducing Gibbs Free Energy | 吉布斯自由能简介
The spontaneity of a chemical process depends not only on the system's entropy change but also on the surroundings. Josiah Willard Gibbs combined the system's enthalpy and entropy into a single criterion: the Gibbs free energy change, ΔG.
化学过程的自发性不仅取决于系统的熵变，还与周围环境有关。约西亚·威拉德·吉布斯将系统的焓和熵结合为一个判据：吉布斯自由能变 ΔG。
ΔG = ΔH − TΔS
If ΔG > 0, the reverse reaction is spontaneous. When ΔG = 0, the system is at equilibrium.
在恒温恒压下，若 ΔG < 0，反应是自发的（热力学上可行）。若 ΔG > 0，逆反应是自发的。当 ΔG = 0 时，系统处于平衡状态。
The TΔS term represents the energy dispersed to the surroundings at temperature T. A highly exothermic reaction (ΔH negative and large) and a reaction that produces more disordered products (ΔS positive) both favour a negative ΔG.
TΔS 项表示在温度 T 下分散到周围环境中的能量。强放热反应（ΔH 负值且数值较大）以及生成更混乱产物的反应（ΔS 正值）都有利于 ΔG 为负值。
7. Standard Free Energy of Formation and ΔG° | 标准生成自由能与ΔG°
Similar to standard enthalpy of formation, the standard free energy of formation, ΔG°f, is the free energy change when one mole of a compound is formed from its elements in their standard states. By definition, ΔG°f of any element in its standard state is zero.
与标准生成焓类似，标准生成自由能 ΔG°f 是在标准状态下由元素生成一摩尔化合物时的自由能变化。根据定义，任何处于标准状态的元素的 ΔG°f 为零。
The standard free energy change for a reaction can be calculated using:
反应的标准自由能变可通过以下公式计算：
ΔG° = Σ ΔG°f(products) − Σ ΔG°f(reactants)
re versatile because it can be applied at any temperature, whereas the ΔG°f method is usually tabulated only for 298 K.
该方法与在 298 K 下使用 ΔG° = ΔH° − TΔS° 得到的结果相同，前提是 ΔH° 和 ΔS° 是标准值。但是，方程 ΔG = ΔH − TΔS 更通用，因为它可以在任何温度下应用，而 ΔG°f 方法通常仅提供 298 K 下的数据。
8. Temperature Dependence of Spontaneity | 温度对自发性的影响
The sign of ΔG, and therefore spontaneity, depends on the interplay between ΔH and ΔS, weighted by temperature. The following table summarises the four possibilities:
ΔG 的符号以及由此决定的自发性，取决于 ΔH 和 ΔS 的相互作用，并通过温度加权。下表总结了四种可能的情况：
| ΔH sign | ΔS sign | Spontaneity |
|---|---|---|
| Negative (−) | Positive (+) | Always spontaneous (ΔG < 0 at all T) |
| Negative (−) | Negative (−) | Spontaneous at low T (│ΔH│ > │TΔS│) |
| Positive (+) | Positive (+) | Spontaneous at high T (│TΔS│ > │ΔH│) |
| Positive (+) | Negative (−) | Never spontaneous (ΔG > 0 at all T) |
中文释义：ΔH为负、ΔS为正时，所有温度下均自发；ΔH为负、ΔS为负时，低温自发；ΔH为正、ΔS为正时，高温自发；ΔH为正、ΔS为负时，所有温度下均不自发。
s spontaneous above 0 °C. The temperature at which ΔG = 0 is the melting point: T = ΔH/ΔS.
例如，冰的熔化：H₂O(s) → H₂O(l)，ΔH > 0（吸热）且 ΔS > 0（液体更无序）。在 0 °C 以下不自发，但在 0 °C 以上变为自发。ΔG = 0 时的温度即为熔点：T = ΔH/ΔS。
9. The Relationship between ΔG° and Equilibrium Constant | ΔG°与平衡常数的关系
standard free energy change ΔG° is related to the equilibrium constant K by the equation:
当反应混合物不处于平衡状态时，决定变化方向的是 ΔG 而非 ΔG°。在平衡时 ΔG = 0。标准自由能变 ΔG° 与平衡常数 K 通过下式相关联：
ΔG° = −RT ln K
R is the gas constant (8.31 J K⁻¹ mol⁻¹) and T is the absolute temperature in Kelvin. If ΔG° is negative, ln K is positive and K > 1, meaning products are favoured at equilibrium. If ΔG° is positive, K < 1 and reactants are favoured.
R 是气体常数（8.31 J K⁻¹ mol⁻¹），T 是绝对温度（以开尔文计）。若 ΔG° 为负，则 ln K 为正且 K > 1，说明平衡时产物更占优势。若 ΔG° 为正，则 K < 1，反应物更占优势。
1. This equation also shows that a small change in ΔG° leads to a large change in K due to the logarithmic relationship.
常见错误是用 ΔG 来计算 K。请记住，只有 ΔG° 可用于该方程。当 ΔG° 为零时，K = 1。该方程还表明，由于对数关系，ΔG° 的微小变化会导致 K 的显著变化。
10. Entropy Changes in Phase Transitions | 相变过程中的熵变
brium and ΔG = 0. This allows us to calculate the entropy change for the phase transition:
在恒定温度下的相变过程中，例如熔融或汽化，系统处于平衡状态，ΔG = 0。这使得我们可以计算相变的熵变：
ΔS = ΔH / T
J K⁻¹ mol⁻¹. The positive sign reflects the large increase in disorder when a liquid becomes a gas.
其中 ΔH 是相变的焓变（如 ΔHfus 或 ΔHvap），T 是以开尔文为单位的相变温度。例如，对于水在 373 K 下，ΔHvap = 40.7 kJ mol⁻¹，因此 ΔSvap ≈ +109 J K⁻¹ mol⁻¹。正值表明液体变成气体时无序度大幅增加。
Trouton's rule states that the entropy of vaporisation for many liquids is approximately 85 J K⁻¹ mol⁻¹. Deviations from this value occur for liquids with significant hydrogen bonding, like water, which have a higher ΔSvap because the liquid is more ordered.
特劳顿规则指出，许多液体的汽化熵大约为 85 J K⁻¹ mol⁻¹。对于有明显氢键的液体，如水的 ΔSvap 会偏高，因为其液态更加有序。
11. Entropy of Mixing and Solution Formation | 混合熵与溶液形成
When two ideal gases or two miscible liquids are mixed, the entropy of the system increases even if there is no energy change. This entropy of mixing arises because the particles have more available volume and more possible arrangements, increasing the number of microstates.
当两种理想气体或两种可混溶液体混合时，即使没有能量变化，系统的熵也会增加。这种混合熵的产生是因为粒子有更大的可用体积和更多可能的排列方式，从而增加了微观状态的数量。
ᵢ ln xᵢ, where xᵢ is the mole fraction. Although you do not need to perform this calculation at SL, HL students should recognise that mixing always contributes a positive entropy term.
对于理想混合物，每摩尔的混合熵由 ΔSₘᵢₓ = −R Σ xᵢ ln xᵢ 给出，其中 xᵢ 是摩尔分数。虽然 SL 不要求计算，但 HL 学生应当认识到混合总是贡献一个正的熵项。
In the dissolution of ionic solids, two entropy effects compete: the entropy increases as ions leave the ordered crystal lattice, but decreases as water molecules become ordered around the ions (hydration). The overall ΔS for dissolving can be positive or negative, which affects solubility. For example, dissolving NH₄NO₃ has ΔS > 0, while dissolving CaSO₄ has ΔS < 0 due to extensive ordering of water.
在离子固体的溶解中，有两种熵效应相互竞争：离子离开有序晶格使熵增加，但水分子在离子周围形成有序结构（水合）使熵减少。溶解的总 ΔS 可正可负，从而影响溶解度。例如，溶解 NH₄NO₃ 的 ΔS > 0，而溶解 CaSO₄ 的 ΔS < 0，这是因为水分子发生了高度有序排列。
12. Summary and Exam Tips | 总结与考试技巧
neity at constant T and P. A reaction is spontaneous if ΔG < 0, and at equilibrium ΔG = 0.
熵（S）衡量能量和物质的分散程度，通过 S = k ln W 量化。第三定律提供了一个绝对零点的参考。标准熵可用于计算反应的 ΔS°。吉布斯自由能（ΔG = ΔH − TΔS）是恒温恒压下自发性的最终判据。若 ΔG < 0，反应自发；平衡时 ΔG = 0。
ays check units: ΔH often given in kJ while ΔS in J, so convert one to the other before multiplying by T. When predicting the effect of temperature on feasibility, analyse the signs of ΔH and ΔS. A reaction that is non-spontaneous at low temperature may become spontaneous at high temperature if ΔS is positive.
需要记住的关键公式：ΔG° = −RT ln K 以及对于相变 ΔS = ΔH/T。务必检查单位：ΔH 常以 kJ 给出，而 ΔS 以 J 给出，因此在乘以 T 之前要统一单位。当预测温度对可行性的影响时，应分析 ΔH 和 ΔS 的符号。若 ΔS 为正，在低温下非自发的反应可能在高温下变为自发。
Finally, practice using data booklet values for S° and ΔG°f, and be comfortable estimating the sign of ΔS from the balanced equation (e.g., fewer gas molecules or formation of a solid generally means negative ΔS).
最后，练习使用数据手册中的 S° 和 ΔG°f 值，并能熟练地从配平方程式中估计 ΔS 的符号（例如，气体分子数减少或生成固体通常意味着 ΔS 为负）。
Published by TutorHao | Chemistry Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图