The First Law of Thermodynamics and Its Applications | 热力学第一定律及其应用

📚 The First Law of Thermodynamics and Its Applications | 热力学第一定律及其应用

The first law of thermodynamics is essentially a statement of energy conservation, extended to systems where heat and work can change the internal energy. It forms the foundation of thermal physics and is widely used in engines, refrigerators, and everyday phase changes.

热力学第一定律本质上是能量守恒定律在热学中的表述，它说明系统内能的变化等于外界对系统传递的热量与外界对系统做功之和。它是热学的基础，广泛应用于发动机、冰箱以及日常相变过程。

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1. The Statement of the First Law | 热力学第一定律的表述

The first law states: the change in internal energy of a system, ΔU, equals the heat added to the system, Q, plus the work done on the system, W.

热力学第一定律指出：系统内能的变化 ΔU 等于外界向系统传递的热量 Q 与外界对系统所做的功 W 之和。

ΔU = Q + W

In this sign convention, Q > 0 means heat is added to the system, and W > 0 means work is done on the system by the surroundings. In many physics textbooks, especially those using the chemistry convention, work done by the system is treated as positive; however, for A-level physics, we consistently use the form above where work done on the system is positive.

这里采用如下符号约定：Q ＞ 0 表示热量传入系统，W ＞ 0 表示外界对系统做功。在部分教材（尤其是化学）中会把系统对外做功取为正，但在 A-level 物理中我们统一采用上式，即外界对系统做功为正。

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2. Internal Energy and Its Physical Meaning | 内能及其物理意义

Internal energy is the total random kinetic and potential energy of the particles within a system. For an ideal gas, the potential energy between molecules is zero, so the internal energy depends only on temperature.

内能是系统内所有粒子无规则运动的动能与势能之和。对于理想气体，分子间势能为零，因此内能只与温度有关。

For a monatomic ideal gas, the internal energy is given by:

对于单原子理想气体，内能表达式为：

U = (3/2)nRT

where n is the number of moles, R is the molar gas constant, and T is the absolute temperature. Thus, if the temperature of an ideal gas does not change, its internal energy remains unchanged.

其中 n 为物质的量，R 为摩尔气体常量，T 为热力学温度。因此，若理想气体的温度不变，其内能就不变。

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3. Work Done in Volume Changes | 体积变化中的功

When a gas expands or is compressed, the work done depends on pressure and the change in volume. For an infinitesimal change, the work done on the gas is:

当气体膨胀或被压缩时，功的大小取决于压强和体积变化。对于微小变化，外界对气体做的功为：

dW = –p dV

If the pressure is constant (isobaric process), then the total work done on the gas is:

若压强恒定（等压过程），则外界对气体做的总功为：

W = –p ΔV

Notice that when the gas expands, ΔV > 0, so W < 0, meaning the gas does work on the surroundings. Conversely, when the gas is compressed, W > 0.

注意：气体膨胀时 ΔV ＞ 0，W ＜ 0，表示气体对外界做功；反之，气体被压缩时 W ＞ 0。

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4. Heat Capacity and Specific Heat Capacity | 热容与比热容

Heat Q absorbed or released by a substance is related to its mass m, specific heat capacity c, and temperature change ΔT:

物质吸收或放出的热量 Q 与其质量 m、比热容 c 和温度变化 ΔT 有关：

Q = mcΔT

For gases, there are two important heat capacities: at constant volume, C_V, and at constant pressure, C_P. For an ideal monatomic gas, C_V = (3/2)R and C_P = (5/2)R per mole. Since at constant pressure the gas does external work, C_P is greater than C_V.

对于气体，存在两个重要的热容：定容热容 C_V 和定压热容 C_P。对于单原子理想气体，摩尔定容热容 C_V = (3/2)R，摩尔定压热容 C_P = (5/2)R。由于定压过程中气体对外做功，所以 C_P 大于 C_V。

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5. Isochoric Process | 等容过程

An isochoric process occurs at constant volume, so the gas does no work: W = 0. From the first law, ΔU = Q. All heat added increases the internal energy, which raises the temperature.

等容过程是体积不变的过程，因此气体不做功：W = 0。由热力学第一定律得 ΔU = Q。所有吸收的热量都用来增加内能，从而使温度升高。

The ratio of heat to temperature change at constant volume gives the molar heat capacity C_V:

定容条件下热量与温度变化之比给出摩尔定容热容 C_V：

Q = nC_VΔT = ΔU

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6. Isobaric Process | 等压过程

An isobaric process occurs at constant pressure. The work done on the gas is W = –pΔV, and the heat added is Q = nC_PΔT. The first law becomes:

等压过程是压强保持不变的过程。外界对气体做功 W = –pΔV，吸收热量 Q = nC_PΔT。热力学第一定律可写成：

ΔU = nC_PΔT – pΔV

For an ideal gas, using ΔU = nC_VΔT and the ideal gas law pΔV = nRΔT, we obtain the important relation C_P – C_V = R.

对理想气体，利用 ΔU = nC_VΔT 和理想气体状态方程 pΔV = nRΔT，可得到重要关系 C_P – C_V = R。

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7. Isothermal Process | 等温过程

In an isothermal process, the temperature remains constant, so for an ideal gas ΔU = 0. The first law then gives Q + W = 0, meaning all heat added is converted into work done by the gas (and vice versa).

等温过程中温度保持不变，因此理想气体的内能变化 ΔU = 0。热力学第一定律变为 Q + W = 0，即吸收的热量全部转化为气体对外做的功（或反之）。

The work done on an ideal gas during an isothermal expansion from volume V₁ to V₂ is:

理想气体从体积 V₁ 等温膨胀到 V₂ 时，外界对气体做的功为：

W = –nRT ln(V₂/V₁)

Since ln(V₂/V₁) > 0 for expansion, W is negative, confirming that the gas does work on the surroundings.

因为膨胀时 V₂/V₁ ＞ 1，ln 为正值，所以 W 为负，表明气体对外界做功。

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8. Adiabatic Process | 绝热过程

An adiabatic process occurs with no heat exchange with the surroundings, so Q = 0. The first law becomes ΔU = W. If the gas expands adiabatically, it does work on the surroundings (W < 0), so its internal energy decreases and its temperature drops.

绝热过程是系统与外界没有热量交换的过程，Q = 0。热力学第一定律变为 ΔU = W。若气体绝热膨胀，则气体对外做功（W ＜ 0），内能减少，温度降低。

For an ideal gas in a reversible adiabatic process, pressure and volume obey:

理想气体在可逆绝热过程中满足：

pV^γ = constant, where γ = C_P/C_V

This relation is essential for solving problems involving rapid compressions or expansions, such as in diesel engines.

这个关系式对于求解快速压缩或膨胀问题（如柴油机中的过程）至关重要。

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9. Cyclic Processes and Thermal Efficiency | 循环过程与热机效率

In a cyclic process, the system returns to its initial state, so ΔU = 0. The first law implies that the net heat added equals the net work done on the system:

循环过程中系统回到初始状态，因此 ΔU = 0。热力学第一定律表明净吸热等于外界对系统做的净功：

Q_net + W_net = 0

For a heat engine, the efficiency η is the useful work output divided by the heat absorbed from the hot reservoir:

对于热机，效率 η 等于对外输出的有用功除以从高温热源吸收的热量：

η = W_output / Q_hot = 1 – Q_cold / Q_hot

In A-level problems, you may need to read a p–V graph, compute the work as the area enclosed by the cycle, and apply the first law to each stage.

在 A-level 考题中，常需要读取 p–V 图，计算循环曲线围成的面积作为净功，并对每个阶段应用热力学第一定律。

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10. Applying the First Law to Phase Changes | 热力学第一定律在相变中的应用

During melting or boiling, temperature does not change, so for a pure substance the internal energy still changes because the potential energy of the particles changes. The heat absorbed is given by Q = mL, where L is the latent heat.

在熔化或沸腾过程中，温度不变，但纯物质的内能仍然改变，因为粒子势能发生了变化。吸收的热量由 Q = mL 给出，其中 L 为潜热。

Since the volume usually changes during a phase change, a small amount of work is done against the atmosphere. The first law then links the heat input to both the internal energy change and the pΔV work.

由于相变时体积通常发生变化，系统会对抗大气压做少量功。热力学第一定律将吸收的热量与内能变化及 pΔV 功联系起来。

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11. Solved Example: Isothermal Compression | 典型例题：等温压缩

One mole of an ideal monatomic gas at 300 K is compressed isothermally from 20 dm³ to 10 dm³. Calculate the work done on the gas and the heat released.

一摩尔单原子理想气体在 300 K 下从 20 dm³ 等温压缩到 10 dm³，求外界对气体做的功和气体放出的热量。

Using W = –nRT ln(V₂/V₁), with n = 1, R = 8.31 J mol⁻¹ K⁻¹, T = 300 K, V₁ = 20 dm³, V₂ = 10 dm³:

利用 W = –nRT ln(V₂/V₁)，其中 n = 1，R = 8.31 J mol⁻¹ K⁻¹，T = 300 K，V₁ = 20 dm³，V₂ = 10 dm³：

W = –1 × 8.31 × 300 × ln(10/20) = –2493 × (–0.693) ≈ +1728 J

Since the process is isothermal, ΔU = 0, so Q = –W = –1728 J. The negative sign means 1728 J of heat is released to the surroundings.

因为过程等温，ΔU = 0，所以 Q = –W = –1728 J。负号表示气体向外界放出 1728 J 的热量。

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12. Common Mistakes and Exam Tips | 常见错误与应考要点

First, always define the sign convention clearly before applying ΔU = Q + W. If you accidentally use W as work done by the gas, the signs of all terms will flip.

第一，在应用 ΔU = Q + W 之前，必须明确符号约定。如果把 W 误当作气体对外做功，所有项的符号都会相反。

Second, remember that for an ideal gas, ΔU is determined only by ΔT. Even if Q and W are both non-zero, their sum must equal nC_VΔT.

第二，牢记对于理想气体，ΔU 只由 ΔT 决定。即使 Q 和 W 都不为零，它们的和也必定等于 nC_VΔT。

Third, practise reading p–V diagrams: the area under a curve represents work done by/on the gas, and the area of a closed loop equals the net work in a cycle.

第三，多练习阅读 p–V 图：曲线下的面积代表功，闭合曲线围成的面积等于循环过程的净功。

Finally, in adiabatic problems, never use Q = mcΔT directly, because Q = 0. Instead, combine the ideal gas law with the adiabatic relation pV^γ = constant.

最后，在绝热问题中，不能直接使用 Q = mcΔT，因为 Q = 0。应结合理想气体状态方程与绝热关系 pV^γ = 常量来求解。

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