Introducing Entropy | 引入熵

📚 Introducing Entropy | 引入熵

In A-Level Chemistry, entropy is a cornerstone of thermodynamics that allows chemists to predict whether a chemical reaction or physical change will occur spontaneously. Entropy, represented by the symbol S, quantifies the way energy and matter are distributed within a system at a given temperature. Unlike enthalpy, which focuses on heat transfer, entropy provides insight into the natural tendency of systems to evolve toward states with greater dispersal of energy.

在A-Level化学中，熵是热力学的基石，让化学家能够预测化学反应或物理变化是否自发发生。熵用符号S表示，量化了在给定温度下系统内能量和物质的分布方式。与关注热传递的焓不同，熵揭示了系统自然趋向能量更分散状态的趋势。

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1. What is Entropy? | 什么是熵？

Entropy is a thermodynamic state function, meaning its value depends only on the current state of a system, not on the path taken to reach that state. The SI unit of entropy is joules per kelvin per mole (J K⁻¹ mol⁻¹). A system with high entropy has energy spread out over many different microstates – different ways particles can be arranged and can move – while a system with low entropy has energy concentrated in fewer microstates.

熵是一个热力学状态函数，这意味着它的值只取决于系统的当前状态，与达到该状态的路径无关。熵的国际单位是焦耳每开尔文每摩尔 (J K⁻¹ mol⁻¹)。一个高熵系统，其能量散布在许多不同的微观状态（粒子排列和运动的不同方式）上；而低熵系统的能量则集中在较少的微观状态上。

is defined as the heat transferred reversibly divided by the temperature in kelvin: ΔS = q_rev / T. For a chemical reaction, we usually work with standard entropy values to calculate ΔS°.

任何过程的熵变 ΔS 定义为可逆传递的热量除以开尔文温度：ΔS = q_rev / T。对于化学反应，我们通常使用标准熵值来计算 ΔS°。

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2. Entropy as a Measure of Disorder | 熵作为无序度的量度

At an introductory level, entropy is often described as a measure of disorder or randomness. This simple picture is helpful: a solid crystal, where particles are arranged in a highly ordered lattice, has low entropy. When that solid melts to form a liquid, the particles gain freedom to move past one another, increasing the disorder and thus the entropy. When the liquid vaporises, the gas particles become completely free and the entropy rises dramatically.

在入门层面上，熵常被描述为无序度或随机度的量度。这个简单的图像很有用：固体晶体中粒子排列在高度有序的晶格中，熵值很低。当固体熔化为液体时，粒子获得相互移动的自由，无序度增加，因此熵增大。当液体蒸发时，气体粒子变得完全自由，熵急剧上升。

However, it is more accurate to say that entropy measures the dispersal of energy. In a gas, the total kinetic energy of the particles is distributed among many more translational, rotational and vibrational energy levels compared with a solid, and this wider spread of energy corresponds to a higher entropy.

然而，更准确地说，熵衡量的是能量的分散程度。在气体中，粒子总动能在平移、转动和振动能级上的分布比固体中广泛得多，这种更广泛的能量分布对应着更高的熵。

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3. Entropy and States of Matter | 熵与物质状态

Entropy values follow a clear trend with physical state: S°(gas) >> S°(liquid) > S°(solid). The standard entropy of a perfect crystal at absolute zero is zero, a statement known as the Third Law of Thermodynamics. As temperature increases, particles vibrate more vigorously, and entropy rises.

熵值随物理状态呈现明显趋势：S°(气体) 远大于 S°(液体) 又大于 S°(固体)。完美晶体在绝对零度时的标准熵为零，这被称为热力学第三定律。随着温度升高，粒子振动更加剧烈，熵值增大。

For example, water illustrates this beautifully: H₂O(s) has a very low entropy (~41 J K⁻¹ mol⁻¹ at 0 °C), H₂O(l) has a significantly higher entropy (~70 J K⁻¹ mol⁻¹), and H₂O(g) has a much larger entropy (~189 J K⁻¹ mol⁻¹). The phase changes (fusion and vaporisation) are accompanied by positive entropy changes.

例如，水很好地说明了这一点：H₂O(s) 熵值很低（0 °C 时约 41 J K⁻¹ mol⁻¹），H₂O(l) 熵值显著更高（约 70 J K⁻¹ mol⁻¹），而 H₂O(g) 的熵值却大得多（约 189 J K⁻¹ mol⁻¹）。相变（熔化和蒸发）伴随着正的熵变。

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4. Standard Molar Entropy (S°) | 标准摩尔熵 (S°)

The standard molar entropy, S°, is the entropy of one mole of a substance under standard conditions (298 K, 100 kPa). Unlike standard enthalpies of formation, S° values are absolute – they are not measured relative to an arbitrary zero. Some typical S° values are given in the table below.

标准摩尔熵 S° 是一摩尔物质在标准条件下（298 K，100 kPa）的熵。与标准生成焓不同，S° 值是绝对值——它们不是相对于某个任意零点的测量值。下表列出了一些典型的 S° 值。

Substance	Formula	State	S° / J K⁻¹ mol⁻¹
Diamond	C	s	2.4
Graphite	C	s	5.7
Water	H₂O	l	70.0
Steam	H₂O	g	188.8
Sodium chloride	NaCl	s	72.1
Calcium carbonate	CaCO₃	s	92.9
Calcium oxide	CaO	s	39.8
Carbon dioxide	CO₂	g	213.8

Notice that more complex structures and gases have higher S° values. Diamond's very rigid, tightly bound lattice gives it an exceptionally low entropy, while carbon dioxide gas, with its many translational, rotational and vibrational modes, has a very high entropy.

请注意，更复杂的结构和气体具有较高的 S° 值。金刚石非常刚性、结合紧密的晶格使其熵值异常低，而二氧化碳气体由于其众多的平移、转动和振动模式而具有很高的熵。

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5. Calculating Standard Entropy Change (ΔS°) | 计算标准熵变 (ΔS°)

For a chemical reaction, the standard entropy change, ΔS°, is calculated from the standard molar entropies of the products and reactants:

对于化学反应，标准熵变 ΔS° 由产物和反应物的标准摩尔熵计算得出：

ΔS° = Σ S°(products) − Σ S°(reactants)

The sum of the S° values of the products is taken, each multiplied by its stoichiometric coefficient, and the sum for the reactants is subtracted.

取产物 S° 值的总和（每项乘以其化学计量系数），再减去反应物 S° 值的总和。

osition of calcium carbonate: CaCO₃(s) → CaO(s) + CO₂(g). Using the values from the table:

考虑碳酸钙的分解反应：CaCO₃(s) → CaO(s) + CO₂(g)。利用表中的数值：

ΔS° = S°(CaO) + S°(CO₂) − S°(CaCO₃) = 39.8 + 213.8 − 92.9 = +160.7 J K⁻¹ mol⁻¹

ΔS° = S°(CaO) + S°(CO₂) − S°(CaCO₃) = 39.8 + 213.8 − 92.9 = +160.7 J K⁻¹ mol⁻¹

The positive ΔS° is expected because one mole of solid reactant produces one mole of solid and one mole of gas – a substantial increase in the dispersal of matter and energy.

ΔS° 为正值是预期之中的，因为一摩尔固体反应物生成了一摩尔固体和一摩尔气体——物质和能量的分散显著增加。

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6. The Second Law of Thermodynamics | 热力学第二定律

The Second Law of Thermodynamics states that the total entropy of an isolated system always increases for a spontaneous process. In other words, for any change to occur without outside intervention, the entropy of the universe (system + surroundings) must increase.

热力学第二定律指出，孤立系统的总熵在自发过程中总是增加的。换言之，任何无需外界干预就能发生的变化，其宇宙（系统 + 环境）的熵必须增加。

When we apply this to chemical reactions, we must consider not only the entropy change of the chemical system (ΔS_system), but also the entropy change of the surroundings (ΔS_surroundings). The total entropy change is given by:

当我们将这一定律应用于化学反应时，必须不仅考虑化学系统本身的熵变 (ΔS_system)，还要考虑环境的熵变 (ΔS_surroundings)。总熵变由下式给出：

ΔS_total = ΔS_system + ΔS_surroundings

A reaction is thermodynamically feasible (spontaneous) under given conditions only if ΔS_total > 0.

只有在给定条件下 ΔS_total > 0 时，反应才是热力学上可行的（自发的）。

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7. Entropy Change of the Surroundings (ΔS_surroundings) | 环境的熵变

The entropy change of the surroundings is related to the enthalpy change of the reaction. When a reaction releases heat (exothermic, ΔH negative), the surroundings gain energy, which increases their entropy. Conversely, an endothermic reaction (ΔH positive) removes heat from the surroundings, decreasing their entropy. The quantitative relationship is:

环境的熵变与反应的焓变相关。当反应放热（放热，ΔH 为负）时，环境获得能量，从而增加其熵。相反，吸热反应（ΔH 为正）从环境中吸走热量，降低其熵。定量关系为：

ΔS_surroundings = −ΔH / T

where ΔH is the enthalpy change of the system (in J mol⁻¹) and T is the temperature in kelvin. The negative sign ensures that an exothermic reaction (ΔH < 0) gives a positive ΔS_surroundings.

式中 ΔH 是系统的焓变（单位为 J mol⁻¹），T 是开尔文温度。负号确保放热反应 (ΔH < 0) 给出正的 ΔS_surroundings。

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8. Total Entropy Change and Spontaneity | 总熵变与自发性

pontaneity. For the decomposition of CaCO₃ at 298 K, ΔH = +178 kJ mol⁻¹ (endothermic), so:

通过结合系统和环境的贡献，我们可以预测自发性。对于 298 K 下 CaCO₃ 的分解，ΔH = +178 kJ mol⁻¹（吸热），因此：

ΔS_surroundings = −(+178000 J mol⁻¹) / 298 K = −597 J K⁻¹ mol⁻¹

ΔS_surroundings = −(+178000 J mol⁻¹) / 298 K = −597 J K⁻¹ mol⁻¹

Even though ΔS_system = +161 J K⁻¹ mol⁻¹, the total entropy change is ΔS_total ⁻¹ mol⁻¹. The negative total indicates the reaction is not spontaneous at room temperature – as we observe: CaCO₃ does not decompose at 298 K.

尽管 ΔS_system = +161 J K⁻¹ mol⁻¹，但总熵变为 ΔS_total = +161 + (−597) = −436 J K⁻¹ mol⁻¹。负的总熵变表明该反应在室温下不是自发的——正如我们观察到的：CaCO₃ 在 298 K 下不会分解。

At a much higher temperature, the −ΔH/T term becomes less negative, and the total entropy change can become positive, making the reaction feasible.

在更高的温度下，−ΔH/T 项变得不那么负，总熵变就有可能变为正值，从而使反应变得可行。

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9. From Total Entropy to Gibbs Free Energy | 从总熵到吉布斯自由能

Working with total entropy changes can be cumbersome. J. Willard Gibbs introduced a more convenient function, the Gibbs free energy, G, defined so that the spontaneity condition can be assessed using only system properties. Since ΔS_total = ΔS_system − ΔH/T, multiplying by T gives:

处理总熵变可能很繁琐。J. Willard Gibbs 引入了一个更方便的函数——吉布斯自由能 G，使得只需使用系统性质即可评估自发性条件。由于 ΔS_total = ΔS_system − ΔH/T，乘以 T 得到：

TΔS_total = TΔS_system − ΔH

or equivalently,

或等价地，

−TΔS_total = ΔH − TΔS_system

The left-hand side, −TΔS_total, is defined as the change in Gibbs free energy, ΔG. Hence the celebrated Gibbs equation:

左边 −TΔS_total 被定义为吉布斯自由能变 ΔG。因此得到著名的吉布斯方程：

ΔG = ΔH − TΔS

(From this point onward, ΔS without a subscript refers to the entropy change of the chemical system.) A reaction is spontaneous when ΔG < 0.

（从这里开始，不带下标的 ΔS 指化学系统的熵变。）当 ΔG < 0 时，反应自发进行。

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10. The Gibbs Equation (ΔG = ΔH – TΔS) | 吉布斯方程 (ΔG = ΔH – TΔS)

The Gibbs equation elegantly combines enthalpy, entropy and temperature. For a reaction to be feasible at constant temperature and pressure, ΔG must be negative. The equation reveals that the sign of ΔG depends on the relative magnitudes of ΔH and TΔS:

吉布斯方程优雅地结合了焓、熵和温度。要使反应在恒温恒压下可行，ΔG 必须为负。该方程表明，ΔG 的符号取决于 ΔH 和 TΔS 的相对大小：

• If ΔH < 0 and ΔS > 0, then ΔG is always negative – the reaction is always spontaneous.
• 如果 ΔH < 0 且 ΔS > 0，则 ΔG 始终为负——反应总是自发的。
• If ΔH > 0 and ΔS < 0, then ΔG is always positive – the reaction is never spontaneous.
• 如果 ΔH > 0 且 ΔS < 0，则 ΔG 始终为正——反应永远不能自发。
• If ΔH > 0 and ΔS > 0, the reaction becomes feasible at high temperatures (when TΔS outweighs ΔH).
• 如果 ΔH > 0 且 ΔS > 0，反应在高温下变得可行（当 TΔS 超过 ΔH 时）。
• If ΔH < 0 and ΔS < 0, the reaction is feasible only at low temperatures.
• 如果 ΔH < 0 且 ΔS < 0，反应仅在低温下可行。

This explains why some endothermic processes, like the thermal decomposition of limestone, only occur when the temperature is high enough.

这就解释了为什么某些吸热过程，例如石灰石的热分解，只有在温度足够高时才会发生。

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11. Worked Example: Calculating ΔS° and ΔG | 计算实例：计算ΔS°和ΔG

Let us examine the reaction for the synthesis of ammonia: N₂(g) + 3H₂(g) ⇌ 1 J K⁻¹ mol⁻¹, S°(NH₃) = 193 J K⁻¹ mol⁻¹. The standard enthalpy change ΔH° = −92 kJ mol⁻¹.

我们来看氨的合成反应：N₂(g) + 3H₂(g) ⇌ 2NH₃(g)。标准熵数据为：S°(N₂) = 192 J K⁻¹ mol⁻¹，S°(H₂) = 131 J K⁻¹ mol⁻¹，S°(NH₃) = 193 J K⁻¹ mol⁻¹。标准焓变 ΔH° = −92 kJ

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