📚 Entropy and Spontaneity (HL) | 熵与自发性（HL）
Entropy and spontaneity lie at the heart of chemical thermodynamics. While enthalpy changes tell us about heat transfer, they alone cannot predict whether a reaction will occur. The missing piece is entropy, a measure of disorder or the dispersal of energy. Combined with enthalpy through Gibbs free energy, entropy provides a universal criterion for spontaneity. This article explores the HL content required for understanding entropy, calculating entropy changes, applying the Second Law, and using the Gibbs free energy equation to predict feasible reactions under different conditions.
熵和自发性是化学热力学的核心。焓变告诉我们热量的传递，但仅凭焓变无法预测一个反应是否能够发生。缺失的一环就是熵——它是对系统无序度或能量分散程度的一种量度。将熵与焓通过吉布斯自由能结合在一起，就能给出自发性的普适判据。本文将梳理IB HL要求掌握的熵与自发性内容，包括理解熵的概念、计算熵变、运用热力学第二定律，以及使用吉布斯自由能方程预测不同条件下可行的反应。
1. What Is Entropy? | 什么是熵？
Entropy (symbol S) is a thermodynamic state function that measures the degree of disorder or randomness in a system. More precisely, it describes the number of possible microscopic arrangements (microstates) consistent with the system’s macroscopic state. The greater the number of microstates, the higher the entropy. In simple terms, a messy room has higher entropy than a neatly organized one because there are many more ways for the items to be scattered than to be perfectly placed.
熵（符号 S）是热力学状态函数，衡量系统中无序或混乱的程度。更准确地说，它描述了与系统宏观状态一致的微观排列方式（微观状态）数目。微观状态数目越多，熵就越高。简单来说，一个杂乱的房间比一个整洁的房间具有更高的熵，因为物品散乱分布的方式远多于整齐摆放的方式。
r matter is more mixed. The change in entropy, ΔS, is what matters for spontaneity: ΔS = Sfinal – S_initial. A positive ΔS means the system becomes more disordered.
熵的单位是 J K⁻¹ mol⁻¹，属于广度性质。当能量更加分散或物质混合更充分时，熵就会增加。对于自发性，重要的是熵的变化量 ΔS：ΔS = S_最终 – S_初始。正的 ΔS 表示系统变得更加无序。
2. Standard Entropy, S° | 标准熵 S°
Every substance has an absolute entropy value, determined at a standard state. The standard entropy S° is the entropy of one mole of a substance at 100 kPa and a specified temperature, usually 298 K. Unlike enthalpy, where we measure changes relative to arbitrary reference states, entropy has an absolute zero based on the Third Law of Thermodynamics: the entropy of a perfect crystal at 0 K is zero. This allows us to tabulate S° values for substances.
每一种物质都有一个绝对值熵，这个值是在标准状态下测定的。标准熵 S° 是指 100 kPa 下、在指定温度（通常为 298 K）时，一摩尔物质的熵。与焓不同——焓的变化是相对于任意选定的参考状态来测量的——熵拥有建立在热力学第三定律基础上的绝对零点：完美晶体在 0 K 时的熵为零。这使得我们可以将各种物质的 S° 值列成表格。
Standard molar entropy values for selected substances at 298 K are shown below. Note that gases have much higher S° values than liquids or solids because of their greater freedom of motion.
下面列出若干物质在 298 K 下的标准摩尔熵。请注意，气体的 S° 值远高于液体或固体，因为气体分子运动的自由度高得多。
| Substance | State | S° / J K⁻¹ mol⁻¹ |
|---|---|---|
| C (diamond) | s | 2.4 |
| H₂O | l | 69.9 |
| H₂O | g | 188.8 |
| CH₄ | g | 186.3 |
| CO₂ | g | 213.7 |
3. Factors Affecting Entropy | 影响熵的因素
Several factors increase the entropy of a substance, all linked to a greater dispersal of energy among more microstates.
有若干因素会使物质的熵增加，它们都与能量在更多微观状态间的分散相关。
Change of state: S(solid) < S(liquid) ≪ S(gas). When a solid melts, particles gain translational freedom, increasing entropy. Vaporisation produces an even larger jump because gas particles occupy a much larger volume and move almost independently.
状态变化：S(固体) < S(液体) ≪ S(气体)。固体熔化为液体时，粒子获得了平动自由度，熵增大。汽化所产生的熵增幅度更大，因为气体粒子占据的体积大得多，并且几乎独立地运动。
Temperature: Raising the temperature of a substance increases its entropy because the particles have greater kinetic energy and access to a larger number of energy levels and microstates.
温度：升高物质的温度会使其熵增大，因为粒子的动能增大，能够达到更多的能级和微观状态。
Complexity of molecules: For substances in the same state, larger, more complex molecules (with more atoms, bonds, and vibrational modes) have higher molar entropy than simpler ones. For example, S°(C₃H₈) > S°(C₂H₆).
分子复杂性：对于同物态的物质，更大、更复杂的分子（原子数更多、化学键更多、振动模式更多）的摩尔熵高于较简单的分子。例如 S°(C₃H₈) > S°(C₂H₆)。
Mixing: When two pure substances are mixed, entropy increases because there are more ways to arrange the particles. This is why gases mix spontaneously and why dissolution of a solid in a liquid often has a positive ΔS.
混合：两种纯物质混合时，熵会增加，因为排列粒子的方式变多了。这就是气体会自发混合、以及固体溶于液体时常伴有正 ΔS 的原因。
4. Calculating Entropy Changes of a Reaction | 计算反应的熵变
, ΔS°_reaction, can be calculated from standard molar entropies of reactants and products.
反应的标准熵变 ΔS°_reaction 可以通过反应物和产物的标准摩尔熵来计算。
ΔS° = Σ S°(products) – Σ S°(reactants)
Each S° term is multiplied by the stoichiometric coefficient from the balanced equation.
每个 S° 项均乘以配平方程式中对应的化学计量系数。
Example: Calculate ΔS° for the combustion of methane at 298 K.
例题：计算 298 K 下甲烷燃烧的 ΔS°。
CH₄(g) + 2 O₂(g) → CO₂(g) + 2 H₂O(l)
S° values: CH₄ = 186.3, O₂ = 205.0, CO₂ = 213.7, H₂O(l) = 69.9 J K⁻¹ mol⁻¹.
ΔS° = [213.7 + 2 × 69.9] – [186.3 + 2 × 205.0] = (353.5) – (596.3) = –242.8 J K⁻¹ mol⁻¹.
The negative sign is expected because the reaction reduces the number of gas molecules from three moles to one, lowering disorder.
负号是预料之中的，因为该反应使气体分子数从三摩尔减少至一摩尔，降低了无序度。
5. The Second Law of Thermodynamics | 热力学第二定律
The Second Law states that the total entropy of the universe (system plus surroundings) increases for any spontaneous process:
热力学第二定律指出，任何自发过程都会使整个宇宙（系统加环境）的总熵增加：
ΔS_universe = ΔSsystem + ΔS_surroundings > 0
system is at equilibrium. Notice that a reaction can be spontaneous even if the system itself decreases in entropy (ΔSsystem < 0), provided the surroundings increase in entropy by a larger amount.
若 ΔS_universe 为负，则该过程在该条件下是非自发的。当 ΔS_universe = 0 时，系统处于平衡。请注意，即使系统自身的熵减小（ΔSsystem < 0），只要环境的熵增更大，反应仍然可以是自发的。
change of the surroundings depends on the enthalpy change of the system: ΔS_surroundings = –ΔHsystem surroundings, increasing their entropy and making a positive contribution to ΔS_universe.
环境的熵变取决于系统的焓变：ΔS_surroundings = –ΔHsystem / T。放热反应（ΔH < 0）加热环境，使环境的熵增加，从而对 ΔS_universe 做出正贡献。
6. Gibbs Free Energy | 吉布斯自由能
Combining the entropy changes of system and surroundings leads to a convenient single function, the Gibbs free energy G, defined as:
将系统与环境的熵变整合起来，就得到了一个便利的单值函数——吉布斯自由能 G，定义为：
G = H – TS
At constant temperature and pressure, the change in Gibbs free energy is:
在恒温恒压条件下，吉布斯自由能的变化量为：
ΔG = ΔH – TΔS
This equation links enthalpy, entropy, and temperature to predict spontaneity without having to calculate the surroundings separately. The sign of ΔG directly indicates whether a reaction is spontaneous:
- ΔG < 0 : spontaneous (feasible) in the forward direction.
- ΔG = 0 : equilibrium.
- ΔG > 0 : non-spontaneous; the reverse reaction is spontaneous.
该方程将焓、熵和温度联系起来预测自发性，而无须单独计算环境的熵变。ΔG 的符号直接表明反应是否自发：
- ΔG < 0 ：正向反应自发（可行）。
- ΔG = 0 ：平衡。
- ΔG > 0 ：非自发；逆向反应自发。
For standard conditions, we use ΔG° obtained from ΔH° and ΔS° at 298 K.
对于标准条件，我们使用由 298 K 下的 ΔH° 和 ΔS° 求得的 ΔG°。
7. Calculating ΔG° | 计算标准吉布斯自由能变
There are two principal methods to calculate the standard Gibbs free energy change for a reaction.
计算反应的标准吉布斯自由能变有两种主要方法。
Method 1: Using standard free energies of formation ΔGf°:
方法一：使用标准生成自由能 ΔGf°
ΔG° = Σ ΔGf°(products) – Σ ΔGf°(reactants)
ΔGf° of any element in its standard state is zero.
任何标准状态下的单质的 ΔGf° 均为零。
Method 2: Using ΔH° and ΔS° with the Gibbs equation:
方法二：使用 ΔH° 和 ΔS°，通过吉布斯方程计算：
ΔG° = ΔH° – TΔS°
Example: For the reaction 2SO₂(g) + O₂(g) → 2SO₃(g) at 298 K, ΔH° = –196.6 kJ mol⁻¹ and ΔS° = –189.7 J K⁻¹ mol⁻¹. Calculate ΔG°.
例题：对于反应 2SO₂(g) + O₂(g) → 2SO₃(g)，298 K 时 ΔH° = –196.6 kJ mol⁻¹，ΔS° = –189.7 J K⁻¹ mol⁻¹。计算 ΔG°。
Convert ΔS° to kJ: –0.1897 kJ K⁻¹ mol⁻¹.
ΔG° = –196.6 – (298 × –0.1897) = –196.6 + 56.5 = –140.1 kJ mol⁻¹. ΔG° < 0, so the reaction is spontaneous under standard conditions.
将 ΔS° 转化为 kJ：–0.1897 kJ K⁻¹ mol⁻¹。ΔG° = –196.6 – (298 × –0.1897) = –196.6 + 56.5 = –140.1 kJ mol⁻¹。ΔG° < 0，故该反应在标准条件下是自发的。
8. Effect of Temperature on Spontaneity | 温度对自发性的影响
Because ΔG = ΔH – e depending on the signs of ΔH and ΔS. The table below summarises the four possible cases.
由于 ΔG = ΔH – TΔS，根据 ΔH 和 ΔS 的符号，反应的自发性随温度可能发生变化。下表总结了四种可能的情况。
| ΔH | ΔS | ΔG = ΔH – TΔS | Spontaneity |
|---|---|---|---|
| – (exothermic) | + (more disorder) | Always negative | Spontaneous at all T |
| + (endothermic) | – (less disorder) | Always positive | Non‑spontaneous at all T |
| – (exothermic) | – (less disorder) | Negative at low T, positive at high T | Spontaneous only at low temperatures |
| + (endothermic) | + (more disorder) | Positive at low T, negative at high T | Spontaneous only at high temperatures |
of the last case is the melting of ice: H₂O(s) → H₂O(l) with ΔH > 0 and ΔS > 0. It is non‑ eous above 0 °C. The temperature at which ΔG = 0 is the equilibrium point (melting point).
最后一种情况的经典例子是冰的熔化：H₂O(s) → H₂O(l)，ΔH > 0、ΔS > 0。在 0 °C 以下是非自发的，但在 0 °C 以上变成自发。ΔG = 0 时的温度就是平衡点（熔点）。
r temperature can be calculated by setting ΔG = 0, giving T = ΔH / ΔS (with ΔS in kJ K⁻¹).
转换温度可以通过让 ΔG = 0 来求得：T = ΔH / ΔS（注意 ΔS 单位为 kJ K⁻¹）。
9. Gibbs Free Energy and Equilibrium | 吉布斯自由能与平衡
ΔG° tells us about the extent of a reaction under standard conditions, but reactions in real systems often proceed to equilibrium. The relationship between standard Gibbs free energy change and the equilibrium constant K is:
ΔG° 告诉我们反应在标准条件下的限度，但在实际体系中反应常常进行至达到平衡。标准吉布斯自由能变与平衡常数 K 之间的关系为：
ΔG° = –RT ln K
where R = 8.31 J K⁻¹ mol⁻¹ and T is in Kelvin. If ΔG° is large and negative, K ≫ 1, meaning the reaction goes almost to completion. If ΔG° is large and positive, K ≪ 1, meaning the reaction barely proceeds forward. When ΔG° = 0, K = 1.
其中 R = 8.31 J K⁻¹ mol⁻¹，T 为开尔文温度。若 ΔG° 为大的负值，则 K ≫ 1，意味着反应几乎进行到底；若 ΔG° 为大的正值，则 K ≪ 1，意味着正向反应几乎不发生。当 ΔG° = 0 时，K = 1。
For a reaction not at standard conditions, the non‑standard ΔG is given by the reaction quotient Q:
对于非标准条件下的反应，非标准 ΔG 通过反应商 Q 来表达：
ΔG = ΔG° + RT ln Q
At equilibrium, Q = K and ΔG = 0, which verifies the above relation.
在平衡时 Q = K 且 ΔG = 0，这验证了上述关系式。
10. Key Points Summary | 重点总结
- Entropy (S) measures disorder; its change ΔS = Sfinal – S_initial.
- Standard molar entropies S° are absolute values based on the Third Law.
- ΔS°_reaction = Σ S°(products) – Σ S°(reactants).
- The Second Law: spontaneous processes increase total entropy of the universe.
- Gibbs free energy combines enthalpy and entropy: ΔG = ΔH – TΔS.
- ΔG < 0 indicates a spontaneous reaction; ΔG > 0 non‑spontaneous.
- Temperature can switch spontaneity for reactions where ΔH and ΔS have the same sign.
- ΔG° = –RT ln K links thermodynamics to equilibrium position.
- 熵（S）量度无序度；ΔS = S_最终 – S_初始。
- 标准摩尔熵 S° 是基于第三定律的绝对值。
- ΔS°_反应 = Σ S°(产物) – Σ S°(反应物)。
- 热力学第二定律：自发过程增加宇宙总熵。
- 吉布斯自由能综合了焓与熵：ΔG = ΔH – TΔS。
- ΔG < 0 表示自发反应；ΔG > 0 非自发。
- 对于 ΔH 和 ΔS 同号的反应，温度可以逆转其自发性。
- ΔG° = –RT ln K 将热力学与平衡位置联系在一起。
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