📚 Explaining Pressure | 解释压力
Pressure is one of the most fundamental concepts in physics, linking the microscopic world of colliding particles to large‑scale engineering applications. Whether you are studying the crushing depths of the ocean, the behaviour of an inflated balloon, or the force exerted by a gas on the walls of its container, a clear understanding of pressure explains why systems behave as they do. This article develops the definition of pressure, explores its behaviour in liquids and gases, examines classic methods of measurement, and finally connects pressure to the kinetic theory of matter, providing a complete A‑Level treatment for CIE learners.
压力是物理学最基本的概念之一，它将微观粒子的碰撞与宏观的工程应用联系起来。无论是研究海洋深处的巨大压强、气球膨胀后的行为，还是气体对容器壁施加的力，清晰地理解压力都能解释这些系统为何这样表现。本文从压力的定义出发，探讨压力在液体和气体中的行为，介绍经典的测量方法，最后将压力与物质的分子动理论联系起来，为学习CIE课程的同学提供一套完整的A‑Level阐述。
1. Definition and the Pressure Formula | 定义与压力公式
Pressure is defined as the force acting normally (perpendicularly) per unit area on a surface. Mathematically, this is expressed as p = F / A, where p is the pressure, F is the magnitude of the perpendicular force, and A is the area over which the force is spread. In the SI system, pressure is measured in pascals (Pa), and 1 Pa is equal to 1 N m−2. Because only the normal component of force contributes, pressure is a scalar quantity, even though force is a vector.
压力定义为垂直作用在单位面积上的力。数学表达为 p = F / A，其中 p 是压力，F 是垂直力的大小，A 是力所分布的面积。在国际单位制中，压力的单位是帕斯卡（Pa），1 Pa 等于 1 N m−2。由于只有力的垂直分量起作用，尽管力是矢量，压力却是一个标量。
Consider a sharp knife and a blunt one applying the same force: the sharp edge has a much smaller area, so the pressure is enormously greater, allowing it to cut easily. This simple relationship explains the function of nails, ice skates, and even the design of snowshoes, where a large area reduces pressure to prevent sinking into soft snow.
试想一把锋利的刀和一把钝刀施加相同的力：刀刃锋利的接触面积小得多，因此压力大得多，能轻松切割。这一简单的关系解释了钉子、冰刀的工作原理，甚至雪鞋的设计——增大面积以降低压强，防止陷入松软的雪中。
- p = F / A (perpendicular force only)
- Unit: pascal (Pa) = N m−2
- Pressure is a scalar.
2. Pressure in a Fluid at Rest | 静止流体中的压强
For a stationary liquid, the pressure at a point depends on the weight of the fluid column above that point. The pressure difference between two levels separated by a vertical height h in a fluid of uniform density ρ is given by Δp = ρ g h, where g is the acceleration of free fall. If one point is at the free surface where the pressure is atmospheric pressure patm, then the absolute pressure at depth h is p = patm + ρ g h.
对于静止的液体，某一点的压强取决于该点上方液柱的重量。在密度均匀的流体中，垂直高度相差 h 的两点间的压力差为 Δp = ρ g h，其中 g 是自由落体加速度。如果一点在自由液面处，其压力为大气压 patm，那么在深度 h 处的绝对压力就是 p = patm + ρ g h。
This relationship is derived by considering a vertical cylinder of liquid: the upward force on the bottom face must balance both the downward force on the top face and the weight of the liquid column. The formula applies to any fluid of constant density and tells us that pressure increases linearly with depth, not depending on the shape of the container.
这一关系是通过考虑一个竖直的液柱推导出来的：底面向上的力必须平衡顶面向下的力以及液柱自身的重量。该公式适用于任何密度恒定的流体，并且说明压力随深度线性增加，与容器的形状无关。
Δp = ρ g h
3. Characteristics of Fluid Pressure | 流体压力的特性
One of the most important properties of a fluid is that pressure at a point acts equally in all directions. If you press a small piston into a liquid, the liquid pushes back with the same force on the piston face as it would on any surface of the same area oriented at any angle. This isotropy of fluid pressure underpins hydraulic systems and explains why a submerged object feels compression from all sides, not just from above.
流体最重要的特性之一是：同一点的压强在各个方向上都相等。如果你将一个小活塞压入液体，液体会以相同的力推回活塞表面，就像它对相同面积、但朝向任何角度的表面所做的那样。流体压强的这种各向同性是液压系统的基础，也解释了为什么浸没的物体会感受到来自各个方向的挤压，而不仅仅是来自上方。
Furthermore, in connected bodies of the same liquid at rest, all points at the same horizontal level are at the same pressure, provided the liquid is homogeneous. This fact makes the U‑tube manometer possible. Pressure also does not depend on the total mass of liquid, but only on the vertical depth below the free surface.
此外，在连通且静止的同种液体中，只要液体均匀，同一水平面上的所有点压强都相等。这一事实使U形管压力计成为可能。压强也不取决于液体的总质量，而只取决于自由液面以下的竖直深度。
4. The U‑tube Manometer | U形管压力计
A U‑tube manometer consists of a transparent tube bent into a U‑shape, partially filled with a liquid, usually water or mercury. One end is connected to the gas supply whose pressure is to be measured, while the other end is left open to the atmosphere. The difference in the liquid levels h gives the gauge pressure of the gas relative to the atmosphere. The relationship is pgas = patm + ρ g h when the gas pressure is greater than atmospheric, or pgas = patm − ρ g h if it is lower.
U形管压力计由一个弯成U形的透明管组成，管内部分注入一种液体，通常是水或水银。一端连接到待测气体，另一端敞开与大气相通。液面高度差 h 给出了气体相对于大气压的表压。当气体压力大于大气压时，关系式为 pgas = patm + ρ g h；若气体压力较小，则为 pgas = patm − ρ g h。
In the laboratory, the manometer is used to monitor small pressure differences, for example in a gas‑law apparatus. The choice of liquid affects sensitivity: a low‑density liquid gives a larger column height for the same pressure difference, improving readability. Care must be taken to keep the tube vertical and to read the meniscus correctly.
在实验室中，压力计用于监测微小的压力差，比如气体定律实验装置中。液体的选择会影响灵敏度：对于相同的压力差，低密度液体能产生更高的液柱，从而提高读数精度。使用时必须确保管子竖直，并正确读取弯液面。
pgas − patm = ρ g h
5. Atmospheric Pressure and the Mercury Barometer | 大气压与水银气压计
Atmospheric pressure is the force per unit area exerted by the weight of the entire column of air above a surface. At sea level, standard atmospheric pressure is approximately 1.013 × 105 Pa, which is equivalent to 760 mmHg (millimetres of mercury) in a barometer. The classic Torricelli mercury barometer consists of a long glass tube, sealed at one end, filled with mercury, and inverted into a dish of mercury. The space above the mercury column is a vacuum, so the height of the column is a direct measure of atmospheric pressure.
大气压是地表上方整个空气柱的重量在单位面积上产生的力。在海平面处，标准大气压约为 1.013 × 105 Pa，相当于气压计中的 760 mmHg（毫米汞柱）。经典的托里拆利水银气压计由一根一端封闭的长玻璃管构成，管内注满水银后倒置在水银槽中。水银柱上方的空间是真空，因此水银柱的高度直接衡量了大气压。
The relationship patm = ρ g h holds for the column, where ρ is the density of mercury. This instrument demonstrates that atmospheric pressure supports a column of mercury about 0.76 m high, and a column of water about 10.3 m high, because water is much less dense. Changes in weather cause fluctuations in this height, making the barometer a simple but powerful tool for weather forecasting.
水银柱满足关系式 patm = ρ g h，其中 ρ 为水银的密度。这个仪器表明，大气压能够支撑约0.76米高的水银柱，以及约10.3米高的水柱，因为水的密度小得多。天气变化会引起液柱高度的波动，使得气压计成为一个简单而强大的气象预报工具。
6. Pascal’s Principle and Hydraulic Machines | 帕斯卡原理与液压机械
Pascal’s principle states that any change in pressure applied to an enclosed incompressible fluid is transmitted undiminished to every part of the fluid and to the walls of its container. This principle is the basis of hydraulic presses, brakes, and lifts. If two pistons of different cross‑sectional areas A1 and A2 are connected by a fluid, a small force F1 applied to the smaller piston creates a pressure p = F1 / A1 that is transmitted throughout. The larger piston then experiences an upward force F2 = p × A2 = (A2 / A1) F1. Thus a small input force can be multiplied into a much larger output force, at the cost of moving the larger piston a smaller distance.
帕斯卡原理指出，施加在密闭、不可压缩流体上的任何压力变化，都会无损地传递到流体的各个部分和容器壁上。这一原理是液压机、液压制动器和升降机的基础。如果用流体连接两个截面积不同的活塞 A1 和 A2，那么施加在小活塞上的小力 F1 会产生一个压力 p = F1 / A1，该压力会被传递到各处。大活塞则受到向上的力 F2 = p × A2 = (A2 / A1) F1。这样，较小的输入力就可以被放大为很大的输出力，代价是大活塞移动的距离较小。
Hydraulic systems find widespread use in car brake circuits and heavy earth‑moving machinery. The energy is conserved because the work done by the small piston (F1 d1) equals the work done by the large piston (F2 d2), assuming no friction. The volume of fluid displaced is the same, so d2 = (A1/A2) d1.
液压系统广泛应用于汽车制动回路和重型土方机械中。由于小活塞做的功（F1 d1）等于大活塞做的功（F2 d2），能量是守恒的（忽略摩擦）。排出的流体体积相同，因此 d2 = (A1/A2) d1。
7. Boyle’s Law: Pressure–Volume Relationship for a Gas | 波义耳定律：气体的压力–体积关系
For a fixed mass of an ideal gas kept at constant temperature, the pressure exerted by the gas is inversely proportional to its volume. This is Boyle’s law, often written as pV = constant or p1V1 = p2V2. Its graph on a p–V diagram is a rectangular hyperbola, and a plot of p against 1/V yields a straight line through the origin.
对于一定质量、温度恒定的理想气体，气体的压强与其体积成反比。这就是波义耳定律，通常写作 pV = 常量 或 p1V1 = p2V2。在 p–V 图上，图像是一条等轴双曲线；若以 p 对 1/V 作图，则得到一条过原点的直线。
Boyle’s law can be demonstrated experimentally using a gas syringe connected to a pressure gauge, or by trapping air in a sealed U‑tube and compressing it. The microscopic explanation is that halving the volume doubles the number of molecules per unit volume and therefore doubles the rate of collisions with the walls, provided temperature and therefore molecular speed remain unchanged, leading to twice the pressure.
波义耳定律可以通过连接压力计的气体注射器，或者通过在封闭U形管中压缩空气来实验演示。微观解释是：若体积减半，单位体积内的分子数就加倍，因此与容器壁的碰撞频率加倍——前提是温度不变，从而分子速率不变——导致压力变为原来的两倍。
pV = constant (T constant)
8. The Ideal Gas Equation | 理想气体状态方程
Combining Boyle’s law, Charles’s law, and the pressure law gives the ideal gas equation for n moles of gas: pV = nRT, where R is the molar gas constant (8.31 J mol−1 K−1). An alternative form uses the number of molecules N and the Boltzmann constant k: pV = NkT. This equation links the macroscopic quantities pressure, volume, and temperature, and is valid only for an ideal gas that obeys all the gas laws perfectly.
综合波义耳定律、查理定律和压力定律，得到n摩尔理想气体的状态方程：pV = nRT，其中 R 是摩尔气体常数（8.31 J mol−1 K−1）。另一种形式利用分子数 N 和玻尔兹曼常数 k：pV = NkT。这个方程将宏观量压力、体积和温度联系起来，并仅对完全遵守所有气体定律的理想气体严格成立。
The equation is a powerful tool for calculating any one variable when the other three are known. It also highlights that for a fixed mass of gas, (pV)/T = constant. In practice, real gases approximate ideal behaviour at low pressures and high temperatures, when intermolecular forces and the volume of molecules themselves become negligible.
该方程是当已知其中三个量时计算第四个量的有力工具。它还表明了对于一定质量的气体，(pV)/T = 常量。实际上，在低压和高温下，分子间力和分子自身体积可以忽略不计，此时真实气体趋于理想气体的行为。
| Symbol | Meaning | Unit |
| p | Pressure | Pa |
| V | Volume | m3 |
| n | Number of moles | mol |
| R | Molar gas constant | J mol−1 K−1 |
| T | Thermodynamic temperature | K |
9. Kinetic Theory Explanation of Gas Pressure | 气体压力的分子动理论解释
The pressure exerted by a gas can be understood by considering the motion of its molecules. In the simple kinetic model, a gas consists of a large number of tiny, identical particles in continuous, random motion. They collide elastically with each other and with the walls of the container. When a molecule strikes a wall, it undergoes a change in momentum, which results in a force on the wall. The average force per unit area from countless collisions per second is the macroscopic pressure.
气体的压力可以通过其分子的运动来理解。在简单的分子动理论模型中，气体由大量微小的等同粒子组成，它们持续地做无规则运动，彼此以及和容器壁发生弹性碰撞。当一个分子撞击器壁时，它的动量发生改变，这就对器壁产生了一个力。每秒无数次碰撞在单位面积上造成的平均力就是宏观的压力。
An important derivation yields the fundamental kinetic equation pV = ⅓ N m <c2>, where N is the number of molecules, m is the mass of one molecule, and <c2> is the mean square speed. Comparing this with pV = NkT gives the relationship between kinetic energy and temperature: ½ m <c2> = (3/2) kT. Thus, the absolute temperature is a measure of the average random kinetic energy of the gas molecules.
一个重要的推导得到了基础分子动理论方程 pV = ⅓ N m <c2>，其中 N 是分子数，m 是一个分子的质量，<c2> 是方均速率。将这个式子与 pV = NkT 比较，就得到动能与温度的关系：½ m <c2> = (3/2) kT。因此，绝对温度是气体分子平均无规则平动动能的量度。
This model elegantly explains Boyle’s law: reducing the volume increases the frequency of collisions, raising the pressure. It also explains why heating a gas at constant volume increases its pressure – the molecules move faster, strike harder and more often, and exert a greater force on the walls.
这个模型优雅地解释了波义耳定律：减小体积会增加碰撞频率，从而提高压力。它也解释了何以在定容下加热气体会使压力升高——分子运动变快，撞击更猛烈、更频繁，从而对器壁施加更大的力。
10. The Pressure Law (Gay‑Lussac’s Law) | 压力定律（盖–吕萨克定律）
For a fixed mass of gas at constant volume, the pressure is directly proportional to its absolute temperature. This is the pressure law, written as p / T = constant or p1 / T1 = p2 / T2. A graph of p against T is a straight line that, when extrapolated backwards, crosses the temperature axis at absolute zero (−273.15 °C).
对于体积恒定的一定质量气体，压强与其热力学温度成正比。这就是压力定律，写作 p / T = 常量 或 p1 / T1 = p2 / T2。p 对 T 的图线是一条直线，向后延长会与温度轴交于绝对零度（−273.15 °C）处。
Experimentally, the law is studied using a flask of air immersed in water baths of different temperatures, connected to a pressure gauge. The temperature must be measured in kelvin for the proportionality to hold. The microscopic reason is that increasing temperature increases the molecules’ kinetic energy and thus their momentum change per collision, as well as the collision frequency if the volume remains fixed. Both effects combine to raise the pressure linearly with T.
实验上，可将一个盛有空气的烧瓶浸在不同温度的水浴中，并连接到压力计来研究该定律。温度必须使用开尔文温标，比例关系才成立。微观原因是，升高温度会增加分子的动能，从而增加每次碰撞的动量变化；若体积固定，碰撞频率也会增加。这两个效应相结合，使压力随温度线性增大。
p ∝ T (V constant) or p/T = constant
11. Pressure in Real Fluids and Applications | 真实流体中的压力及应用
Although the principles above are derived for idealised liquids and gases, they apply remarkably well to everyday situations. Dams are built thicker at the bottom because water pressure increases with depth. Scuba divers must equalise pressure to avoid injury, and submarines are designed with strong hulls to withstand crushing deep‑sea pressures. Aeroplanes rely on cabin pressurisation systems to maintain a safe, breathable atmosphere at high altitudes where external air pressure is very low.
虽然上述原理是从理想化的液体和气体推导而来的，但它们在日常情况下非常适用。水坝的底部建得更厚，因为水压随深度增加。潜水员必须平衡内外压力以免受伤，潜艇设计了坚固的壳体以承受深海的巨大压力。飞机依靠座舱加压系统，在外部气压很低的高空维持安全可呼吸的环境。
In meteorology, pressure maps (isobars) indicate regions of high and low atmospheric pressure, driving winds and weather systems. In medicine, blood pressure is routinely measured using a sphygmomanometer – essentially a pressurised cuff that follows the same hydrostatic principles. Even the sensation of your ears ‘popping’ in a lift or aircraft is a direct experience of pressure equalisation between the middle ear and the outside atmosphere.
在气象学中，气压图（等压线）标示出高压和低压区域，驱动风与天气系统。在医学上，血压常规使用血压计测量——它本质上是一个加压袖带，遵循相同的流体静力学原理。甚至你在电梯或飞机里感到耳朵“砰”的一声，那也是中耳与外界气压平衡的直接体验。
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