📚 Temperature Graphs: Heating and Cooling Curves | 温度图线：加热与冷却曲线
Temperature graphs, often called heating or cooling curves, display how the temperature of a substance changes over time while thermal energy is transferred at a steady rate. In IB Physics, these graphs are essential for visualising sensible heating, phase transitions, and the energy balance between heat supplied and internal energy changes. By analysing slopes and plateaus, we can extract specific heat capacities and latent heats, deepening our understanding of thermal physics.
温度图线（常被称为加热或冷却曲线）展示了在恒定速率传递热能时，物质温度随时间的变化情况。在 IB 物理中，这些图线对于直观呈现显热升温、相变以及供热与内能变化之间的能量平衡至关重要。通过分析斜率和平台，我们可以求出比热容和潜热，从而加深对热物理的理解。
1. What are Temperature Graphs? | 什么是温度图线？
A temperature graph is a plot of temperature on the vertical axis against time on the horizontal axis, recorded while energy is added to or removed from a sample at a nearly constant rate. The shape of the graph reveals whether the substance is experiencing a temperature rise (sensible heating) or a phase change (latent heating) at a constant temperature.
温度图线是以温度为纵轴、时间为横轴的关系图，它是在以接近恒定的速率向样品输入或从中抽取能量时记录得到的。图线的形状揭示了物质究竟在经历温度升高（显热升温）还是在恒定温度下发生相变（潜热变化）。
Such graphs are sometimes called 'thermal fingerprints' because every pure substance produces a characteristic pattern of sloping sections and flat plateaus when heated or cooled under controlled conditions. Impure substances or mixtures, by contrast, show smeared-out transitions.
这类图线有时被称为“热指纹”，因为在受控条件下加热或冷却时，每一种纯物质都会产生特有的斜线区与水平平台组合的图样。相比之下，不纯物质或混合物则会显示出弥散的转变区间。
2. The Heating Curve of a Pure Substance | 纯物质的加热曲线
Consider a solid initially well below its melting point. As heat is supplied at constant power, the temperature rises steadily, forming a straight sloped segment. Once the melting point is reached, the temperature stops increasing and a horizontal plateau appears while the solid absorbs latent heat of fusion and transforms into a liquid.
设想一块起始温度远低于熔点的固体。以恒定功率供热时，温度稳步上升，形成一段平直的斜线。一旦到达熔点，温度停止上升，此时出现水平平台，固体吸收熔化潜热并转变为液体。
After all the solid has melted, the temperature of the liquid rises again, producing a second sloped segment with a gradient that usually differs from that of the solid. When the boiling point is reached, a second plateau emerges during the absorption of latent heat of vaporisation. Finally, the gas heats up, yielding a third sloped segment.
在所有固体熔化之后，液体的温度再次上升，形成第二段斜线，其斜率通常与固体段不同。当到达沸点时，在吸收汽化潜热的过程中出现第二个平台。最后，气体继续升温，产生第三段斜线。
A complete heating curve therefore consists of five distinguishable regions: solid warming, melting plateau, liquid warming, boiling plateau and gas warming. The power is assumed constant, so the plateau duration depends directly on the latent heat and mass.
因此，一条完整的加热曲线包含五个可区分的区域：固体升温、熔化平台、液体升温、沸腾平台和气体升温。假设加热功率恒定，平台的持续时间直接取决于潜热和质量。
3. The Cooling Curve and Supercooling | 冷却曲线与过冷现象
A cooling curve is obtained by removing energy from a substance, typically by allowing it to cool in the surrounding air or in a water bath. Starting with a hot gas, the temperature falls steadily until it reaches the condensation point, where a flat plateau appears as the gas condenses into a liquid, releasing latent heat.
冷却曲线是通过从物质中抽取能量得到的，通常将物质放置在周围空气或水浴中自然冷却。从高温气体开始，温度稳定下降，直至到达冷凝点，此时出现平坦的平台，气体冷凝为液体并释放潜热。
Further cooling leads to a liquid-cooling segment, followed by a freezing plateau where the liquid solidifies. A common laboratory investigation with stearic acid (a waxy solid) often reveals supercooling: the liquid cools below its freezing point without solidifying, then suddenly crystallises, causing the temperature to jump back up to the freezing plateau.
继续冷却会出现液体降温段，接着是液体凝固的凝固平台。一个常见的硬脂酸（蜡状固体）实验经常暴露出过冷现象：液体冷却到凝固点以下却未凝固，然后突然结晶，使温度跃升回到凝固平台。
Supercooling occurs when the first crystal seeds are slow to form. The released latent heat raises the temperature of the sample to the true freezing point, distinguishing the experimental curve from the ideal one.
当过冷现象发生时，最初的结晶晶核形成缓慢。释放的潜热将样品温度提升至真正的凝固点，使得实验曲线与理想曲线有所区别。
4. Key Regions: Sensible Heating and Phase Changes | 关键区域：显热升温与相变
The sloped regions of a temperature graph correspond to sensible heating: the energy supplied increases the kinetic energy of particles, raising the temperature. The amount of energy required is given by Q = m c Δθ, where c is the specific heat capacity, m the mass and Δθ the temperature change.
温度图线的倾斜区域对应显热升温：所提供的能量增加了粒子的动能，从而使温度升高。所需能量由 Q = m c Δθ 给出，其中 c 是比热容，m 是质量，Δθ 是温度变化。
Flat plateaus represent phase changes at constant temperature. The added or removed energy goes into breaking or forming intermolecular bonds rather than changing kinetic energy. The energy change is Q = m L, where L is the specific latent heat (Lf for fusion, Lv for vaporisation).
水平平台代表恒温条件下的相变。输入或抽出的能量用于打破或形成分子间键，而不是改变动能。能量变化为 Q = m L，其中 L 是比潜热（Lf 表示熔化潜热，Lv 表示汽化潜热）。
Recognising these two types of region is the first step in interpreting any heating or cooling curve. The longer the plateau at a given power, the larger the latent heat.
识别这两类区域是解读任何加热或冷却曲线的第一步。在给定功率下，平台越长，潜热越大。
5. Gradient and Specific Heat Capacity | 斜率与比热容
When heating or cooling occurs at a constant net power P, the temperature change per unit time in a sensible region is governed by the energy balance: P = m c (Δθ/Δt). Rearranging gives the slope of the temperature–time graph:
当以恒定净功率 P 进行加热或冷却时，显热区域中单位时间的温度变化由能量平衡决定：P = m c (Δθ/Δt)。重新整理可得到温度–时间图线的斜率：
Δθ/Δt = P / (m c)
This relationship shows that, for a given power and mass, the slope is inversely proportional to the specific heat capacity. A substance with a low c warms up rapidly (steep slope), whereas a high-c substance shows a gentle incline.
这一关系表明，对于给定的功率和质量，斜率与比热容成反比。低比热容的物质升温迅速（斜率陡），而高比热容的物质升温平缓。
On a cooling curve, the effective cooling power is not constant but depends on the temperature difference between the substance and its surroundings. Nonetheless, the same inverse relationship broadly holds: a high specific heat capacity makes the temperature fall more slowly.
在冷却曲线中，有效散热功率并非恒定，而是依赖于物质与周围环境的温差。尽管如此，大体上同样存在反比关系：高比热容使得温度下降得更慢。
6. Plateau and Latent Heat | 平台与潜热
During a phase change, all the energy supplied (or removed) is used to change the potential energy of the particle bonding, so the temperature remains constant. For heating at constant power P, the duration of a plateau Δtplateau is related to the specific latent heat L by:
在相变期间，所有输入（或抽出）的能量都用于改变粒子间键合的势能，因此温度保持不变。对于恒定功率 P 加热，平台的持续时间 Δtplateau 与比潜热 L 的关系为：
Δtplateau = m L / P
This equation allows experimental determination of L by measuring the plateau length. If the mass and power are known, a longer melting plateau indicates a larger latent heat of fusion.
该方程使我们可以通过测量平台长度来实验测定 L。如果质量和功率已知，较长的熔化平台表明熔化潜热较大。
In a cooling curve, the same principle applies: the plateau persists while the substance releases latent heat to its surroundings. For water, Lv is roughly 7 times larger than Lf, so the boiling plateau is about 7 times longer than the melting plateau for the same mass and power.
在冷却曲线中，同样的原理也适用：当物质向周围环境释放潜热时，平台会持续。以水为例，Lv 大约是 Lf 的 7 倍，因此在相同质量和功率下，沸腾平台大约比熔化平台长 7 倍。
7. Interpreting Slopes: Different Phases | 解读不同物态的斜率
The slope of the sensible-heating segments reflects the specific heat capacity of the substance in that particular phase. In many materials, csolid < cliquid < cgas, so the solid warming slope is steeper, the liquid slope is gentler, and the gas slope is shallowest.
显热升温段的斜率反映了该物质在该物态下的比热容。在许多材料中，csolid < cliquid < cgas，因此固体升温斜率最陡，液体斜率更缓，气体斜率最平缓。
Water is an exception: cwater ≈ 4200 J kg-1 K-1 is significantly larger than cice ≈ 2100 J kg-1 K-1, so the liquid slope is actually shallower than the solid slope. This anomaly is important to recognise when analysing heating curves of water–ice systems.
水是一个例外：cwater ≈ 4200 J kg-1 K-1 显著大于 cice ≈ 2100 J kg-1 K-1，因此液体斜率实际上比固体斜率更平缓。在分析冰–水系统的加热曲线时，认识到这一异常情况很重要。
| Phase / 相 | Specific heat capacity (J kg⁻¹ K⁻¹) | Heating slope (P=100 W, m=0.5 kg) |
|---|---|---|
| Ice (solid) | 2100 | 0.095 K s⁻¹ (steeper) |
| Water (liquid) | 4200 | 0.048 K s⁻¹ (shallower) |
| Steam (gas) | 2000 | 0.100 K s⁻¹ (steeper, but depends on conditions) |
The table above compares slopes for water phases under the same heating conditions. Both the specific heat capacity and the effective heating rate (taking into account losses) determine the final slope seen on the graph.
上表比较了相同加热条件下水各相的斜率。比热容和有效加热速率（考虑热损失）共同决定了图上所见的最终斜率。
8. Mixtures and Alloys: No Sharp Plateaus | 混合物与合金：无尖锐平台
Pure substances display perfectly horizontal plateaus because a phase change occurs at a single, well-defined temperature. Mixtures, alloys and amorphous materials such as wax or glass, however, soften or melt over a range of temperatures. Their heating curves show curved transitions rather than flat steps.
纯物质显示出完全水平的平台，因为相变发生在一个单一、明确的温度下。然而，混合物、合金和无定形材料（如石蜡或玻璃）会在一个温度范围内软化或熔化。它们的加热曲线显示出弯曲的过渡区，而不是平坦的台阶。
During a cooling experiment, a paraffin wax mixture will exhibit a gradual slope change instead of a crisp freezing plateau. This behaviour can be used to distinguish pure from impure substances and explains why temperature graphs are helpful in quality control.
在冷却实验中，石蜡混合物会呈现出渐变的斜率变化，而不是清晰的凝固平台。这一行为可用于区分纯物质与不纯物质，并解释了为何温度图线在质量控制中有用。
In an IB context, students should be able to identify by eye whether a substance is likely pure or a mixture by examining the flatness of its phase-change segments.
在 IB 课程中，学生应能通过肉眼观察相变段的平坦程度来判断物质是纯物质还是混合物。
9. The Cooling Curve of Stearic Acid | 硬脂酸的冷却曲线
One of the classic IB required practicals involves recording the cooling curve of stearic acid (octadecanoic acid), which has a freezing point around 69 °C. A test tube of molten stearic acid is placed in a beaker of warm water, and the temperature is recorded every 30 seconds while it cools.
经典的 IB 必做实验之一就是记录硬脂酸（十八烷酸）的冷却曲线，其凝固点约为 69 °C。将装有熔融硬脂酸的试管置于盛有温水的烧杯中，冷却过程中每 30 秒记录一次温度。
The resulting graph typically shows a steep liquid-cooling section,
Published by TutorHao | IB Physics Revision Series | aleveler.com
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