📚 Measuring Gases: Pressure, Volume, Temperature and Amount of Substance | 气体的测量：压强、体积、温度与物质的量
In A-level physics, a gas is described by four measurable macroscopic quantities: pressure p, volume V, temperature T and amount of substance n. Measuring these variables accurately is central to testing the gas laws and to applying the ideal gas equation in practical situations.
在 A-level 物理中，气体由四个可测量的宏观量来描述：压强 p、体积 V、温度 T 和物质的量 n。准确测量这些变量，是验证气体定律以及在实际情境中应用理想气体方程的核心。
rement turns a simple equation such as pV = nRT into a testable experimental relationship.
本文将梳理测量各个状态参量所用的仪器、方法与注意事项，并说明严谨的测量如何把 pV = nRT 这样简洁的方程变成可以实验检验的关系式。
1. Why Measuring Gases Matters | 为什么测量气体很重要
Gases are used as model systems because their macroscopic behaviour is relatively simple and can be linked directly to molecular motion through kinetic theory. A gas sample contains an enormous number of particles moving randomly, yet its bulk behaviour is captured by just four variables, which is why the ideal gas model is so powerful.
气体被用作模型系统，因为其宏观行为相对简单，并可通过分子动理论直接与分子运动联系起来。一个气体样品中包含数量极多的随机运动的粒子，但其整体行为却只用四个变量就能描述，这正是理想气体模型如此有力的原因。
tities such as molar mass and gas density, calibrating instruments, and solving real engineering problems involving engines, weather balloons, scuba tanks or gas storage. Measurements also reveal the limits of the ideal model: at high pressure or near the liquefaction temperature, real gases deviate from pV = nRT, and only careful data can show where the approximation fails.
因此，准确测量压强、体积、温度和物质的量，对于验证气体定律、测定摩尔质量与气体密度、校准仪器，以及解决涉及发动机、气象气球、潜水气瓶或气体储运的实际工程问题都至关重要。测量还能揭示理想模型的适用边界：在高压或接近液化温度时，真实气体会偏离 pV = nRT，只有严谨的数据才能显示该近似在何处失效。
2. State Variables and SI Units | 状态参量及其单位
Before taking any measurement, you must express all state variables in SI units. Pressure is in pascals (Pa), volume in cubic metres (m³), temperature in kelvins (K), and amount of substance in moles (mol). A very common source of error in examinations and in the laboratory is forgetting to convert degrees Celsius to kelvin, or cubic centimetres to cubic metres.
在进行任何测量之前，必须把所有状态参量换算成国际单位制。压强单位为帕斯卡（Pa），体积单位为立方米（m³），温度单位为开尔文（K），物质的量单位为摩尔（mol）。在考试和实验中，最常见的错误来源就是忘记把摄氏度转换为开尔文，或把立方厘米转换为立方米。
| Quantity / 量 | SI unit / 国际单位 | Common conversion / 常用换算 |
| pressure p / 压强 | Pa | 1 atm = 1.01 × 10⁵ Pa; 1 kPa = 10³ Pa; 1 atm = 760 mmHg |
| volume V / 体积 | m³ | 1 m³ = 10⁶ cm³ = 10³ dm³; 1 dm³ = 1 litre = 10⁻³ m³ |
| temperature T / 温度 | K | T (K) = θ (°C) + 273.15 |
| amount of substance n / 物质的量 | mol | 1 mol = 6.02 × 10²³ particles; n = m ÷ M |
A useful check is that all temperatures used in gas calculations must be thermodynamic temperatures. Doubling a Celsius temperature does not double a pressure or a volume, because the Celsius scale has an arbitrary zero.
一个实用的检验方法是：气体计算中使用的温度必须是热力学温度。把摄氏温度翻倍并不会使压强或体积翻倍，因为摄氏温标的零点是人為约定的。
3. Measuring Pressure | 压强的测量
meter measures atmospheric pressure using a mercury column: the column height h supports the atmosphere, so p = hρg, where ρ is the density of mercury (about 13 600 kg m⁻³) and g is the gravitational field strength (9.81 N kg⁻¹). Standard atmospheric pressure supports about 0.760 m of mercury.
压强是单位面积上的力，p = F ÷ A；对气体而言，它来自分子撞击容器壁时动量变化率。气压计用汞柱测量大气压：汞柱高度 h 支撑着大气，因此 p = hρg，其中 ρ 为汞的密度（约 13 600 kg m⁻³），g 为重力场强度（9.81 N kg⁻¹）。标准大气压约支撑 0.760 m 高的汞柱。
iquid levels h gives the gauge pressure p = hρg, and the absolute pressure of the gas equals atmospheric pressure plus or minus this difference, depending on which side is lower. Mercury is chosen for large pressures because its high density keeps the tube short; water or oil is used for small pressure differences because the level difference is then easier to read.
压强计是一根 U 形管，内装液体，一端与大气相通，另一端连接到待测气体。两侧液面高度差 h 给出表压 p = hρg；气体的绝对压强等于大气压加上或减去该差值，具体取决于哪一侧液面更低。测量较大压强时选用汞，因为其密度大、管长可较短；测量微小压强差时则用水或油，因为此时液面差更易读数。
For modern laboratory work, Bourdon gauges, aneroid gauges and digital pressure sensors are common. A digital sensor uses a flexible diaphragm whose capacitance or strain changes with pressure, giving a voltage that a data logger converts into pascals. Such sensors have a short response time, avoid parallax error and can record pressure continuously, which is invaluable when a gas is compressed or heated quickly.
在现代实验中，常用布尔登压力表、无液气压计和数字压强传感器。数字传感器利用随压强而形变的弹性膜片，其电容或应变发生变化，输出与压强成正比的电压，再由数据采集器换算为帕斯卡。这类传感器响应快、无读数视差，并能连续记录压强，在气体被迅速压缩或加热时尤为宝贵。
4. Measuring Volume | 体积的测量
The volume of a gas is never fixed by the gas itself - it is fixed by the container. Common measuring instruments include the gas syringe (typically 100 cm³ with a scale reading to about 0.5 cm³ or finer), the graduated burette used upside down over water, the eudiometer tube, and calibrated glass vessels of known volume for constant-volume experiments.
气体的体积并不是由气体本身决定的，而是由容器决定的。常用测量器具有：气体注射器（通常为 100 cm³，分度值约 0.5 cm³ 或更小）、倒置于水面上的带刻度滴定管、量气管，以及在等容实验中使用的已知体积的校准玻璃容器。
When a gas is trapped above a liquid, the measured volume is the volume of the gas plus any vapour present, so the pressure must be corrected for the saturated vapour pressure of the liquid, especially with water. If the gas is soluble in the liquid, a gas syringe or a mercury-filled tube should be used instead. For Boyle's law experiments, the gas is often trapped in a capillary tube by a short oil or mercury thread, and the length of the trapped column is taken as a measure of volume, since the cross-sectional area of the tube is uniform.
当气体被液面封住时，所测体积是气体与所含蒸气的总体积，因此压强必须对液体的饱和蒸气压进行修正，用水时尤其重要。若气体易溶于该液体，则应改用气体注射器或充汞的管。在玻意耳定律实验中，气体常被封在一根毛细管内，用一小段油柱或汞柱封住；由于管的内截面积均匀，被封气柱的长度即可作为体积的量度。
Because volume changes with temperature and pressure, the volume reading is only meaningful when the temperature and pressure are stated at the same moment. In practice this means allowing the system to reach thermal equilibrium before reading the volume, and reading the bottom of the meniscus at eye level to avoid parallax error.
由于体积会随温度和压强变化，只有在同一时刻记录温度和压强，体积读数才有意义。实际操作中，这意味着要等系统达到热平衡后再读取体积，并且要在眼睛与液面同一水平处读取弯月面最低点，以避免视差。
5. Measuring Temperature | 温度的测量
Temperature is measured with liquid-in-glass thermometers, thermocouples, resistance thermometers or thermistors, and increasingly with digital probes linked to a data logger. A thermistor has a resistance that falls rapidly as temperature rises, so it is sensitive to small changes and is well suited to monitoring a water bath during a gas-law experiment.
温度可用玻璃液体温度计、热电偶、电阻温度计或热敏电阻测量，如今也常使用连接数据采集器的数字探头。热敏电阻的阻值随温度升高而迅速下降，因此对小温度变化很灵敏，非常适合在气体定律实验中监测水浴温度。
Any thermometer must be calibrated at two fixed points: the ice point (0 °C, pure ice in equilibrium with water at standard pressure) and the steam point (100 °C, steam above boiling water at standard pressure). If the two readings differ from 0 °C and 100 °C, the scale must be corrected before the results are used.
任何温度计都必须用两个固定点进行校准：冰点（0 °C，纯净冰与纯水在标准压强下共存）和汽点（100 °C，标准压强下沸腾水面上方的蒸气）。若两个读数偏离 0 °C 与 100 °C，则必须先修正刻度，再使用实验结果。
The thermometer must be in thermal equilibrium with the gas, so the bulb should be placed close to the gas but not touching the container wall, and the water bath should be stirred to remove temperature gradients. Remember to convert to kelvin before substituting into the gas equation, and take account of the fact that the thermometer itself has a finite heat capacity and a finite response time.
温度计必须与气体达到热平衡，因此感温泡应靠近气体但不要接触容器壁，水浴还需要搅拌以消除温度梯度。代入气体方程之前，切记换算成开尔文，并考虑温度计本身具有有限热容和有限响应时间这一事实。
6. Determining the Amount of Substance | 物质的量的测定
routes require care, since the molar mass M must be known accurately and the mass m must refer to the gas alone, not to the container.
气体物质的量可以用两种方法确定：一是通过称量直接求得，n = m ÷ M；二是由状态参量间接求出，n = pV ÷ (RT)。两种途径都需谨慎，因为摩尔质量 M 必须准确已知，而且质量 m 必须只是气体的质量，不包含容器。
, the flask is placed in a water bath at a known temperature so that the liquid evaporates completely, and the pressure is taken as atmospheric pressure once the excess vapour has escaped. Then M = mRT ÷ (pV) gives the molar mass to within a few per cent.
一个经典的实验是测定挥发性液体的摩尔质量。将已知质量的液体注入体积已测定的烧瓶，把烧瓶放入已知温度的水浴中使其完全汽化，待多余蒸气逸出后，把压强取为当时的大气压。于是 M = mRT ÷ (pV) 就能把摩尔质量测到百分之几的精度。
pV = nRT is limited by the accuracy of p, V and T. This is why a large volume and a modest temperature difference give better precision than a small syringe volume.
摩尔气体常数通常取 R = 8.31 J mol⁻¹ K⁻¹。由于 R 本身来自测量，由 pV = nRT 得出的 n 的准确度受制于 p、V、T 的准确度。这正是为什么用较大体积、适当温度差比用小体积注射器能获得更好精度的原因。
7. The Ideal Gas Equation in Measurement Form | 理想气体方程的测量形式
y as good as the measurements you feed into them, so every quantity must be in SI units and every temperature in kelvin.
对理想气体而言，各状态参量由 pV = nRT 联系；对一定质量的气体，其合并形式为 p₁V₁ ÷ T₁ = p₂V₂ ÷ T₂。这两个式子的可靠性完全取决于代入的测量值，因此每个量都必须用国际单位制，每个温度都必须是开尔文。
pV = nRT = (m ÷ M) RT = NkT
y to the molar mass. Rearranging for molar mass gives M = mRT ÷ (pV), which is the working equation of the volatile-liquid experiment. In both cases the gas must be treated as ideal, which is a reasonable assumption at pressures near atmospheric and temperatures well above the boiling point.
在实验工作中有两个变形特别有用。将两边除以体积即得到密度形式 pM = ρRT，它把可测的密度与摩尔质量联系起来。整理成摩尔质量的形式则得到 M = mRT ÷ (pV)，这正是挥发性液体实验的工作方程。这两种情形都要求把气体当作理想气体处理，而在压强接近大气压、温度远高于沸点时，这是一个合理的假设。
8. Testing Boyle's Law Experimentally | 玻意耳定律的实验验证
Boyle' temperature, the pressure is inversely proportional to the volume, so pV = constant and p ∝ g weights, reading the length of the trapped column and the applied pressure at each step.
玻意耳定律指出：一定质量的气体在温度不变时，压强与体积成反比，故 pV = 常数，p ∝ 1 ÷ V。标准实验把空气封在毛细管内，用油柱或汞柱封口，通过螺旋或加砝码缓慢压缩，逐步读取被封气柱长度和所加压强。
The most convincing analysis is a graph of p against 1 ÷ V, which should be a straight line through the origin. Alternatively, plotting log p against log V should give a straight line of gradient -1, and any deviation from gradient -1 indicates a systematic fault. Plotting pV against p should give a horizontal line whose value is the constant.
最有说服力的分析是作 p 对 1 ÷ V 的图像，它应为一条过原点的直线。另一种方法是作 log p 对 log V 的图像，应为斜率等于 -1 的直线；若斜率偏离 -1，则说明存在系统误差。作 pV 对 p 的图像则应得到一条水平直线，其纵坐标值即为该常数。
The main precautions are: keep the temperature constant by handling the tube as little as possible and waiting for equilibrium; compress slowly so the gas remains at the surrounding temperature; check for leaks at the seal; correct the total pressure for atmospheric pressure when weights are added above a piston; and avoid reading the column length while the oil thread is still moving.
主要注意事项包括：尽量少用手接触管子以保持温度恒定，并等待达到平衡；缓慢压缩，使气体始终处于环境温度；检查封口是否漏气；当在活塞上方加砝码时，要用大气压对总压强进行修正；油柱仍在移动时不要读取气柱长度。
9. Testing Charles's Law and the Pressure Law | 查理定律与压强定律的验证
Charles' e the volume of a fixed mass of gas is directly proportional to its thermodynamic temperature, V ∝ as is trapped in a capillary tube with a movable oil thread at atmospheric pressure and the tube is placed in a water bath whose temperature is varied from about 0 °C to 100 °C.
查理定律指出：一定质量的气体在压强不变时，其体积与热力学温度成正比，即 V ∝ T，故 V ÷ T = 常数。实验中通常把气体封在毛细管内，用可移动的油柱使其维持在大气压下，再把管子放入温度在 0 °C 至 100 °C 之间变化的水浴中。
A graph of V against θ (°C) is a straight line that, when extrapolated backwards, cuts the temperature axis at about -273 °C, which is the experimental route to the idea of absolute zero. A graph of V against T (K) is a straight line through the origin, which is the preferred way to present the data because it does not depend on extrapolation.
以 V 对 θ（°C）作图得到一条直线，向低温方向外推，它与温度轴的交点约为 -273 °C，这是通向绝对零度概念的实验途径。以 V 对 T（K）作图则是过原点的直线，这是更可取的数据呈现方式，因为它不依赖外推。
The pressure law (often called Gay-Lussac's law) states that at constant volume, p ∝ a rigid metal or glass vessel of fixed volume connected to a pressure gauge and immersed in a bath. A graph of p against T (K) should pass through the origin, and the temperature must be allowed to equilibrate at each setting; the vessel must not expand noticeably as it warms.
压强定律（常称盖-吕萨克定律）指出：气体体积不变时 p ∝ T，故 p ÷ T = 常数。该定律用体积固定的刚性金属或玻璃容器与压强计相连，并浸入恒温浴中进行验证。以 p 对 T（K）作图应过原点；每个温度点都需等待平衡，且容器受热时不能有明显膨胀。
10. Sources of Error and How to Improve Accuracy | 误差来源与精度提升
Random errors in gas experiments come from parallax when reading a meniscus, fluctuation of bath temperature, small leaks, and the finite resolution of thermometers and gauges. These can be reduced by taking repeated readings, using digital sensors, reading at eye level, and plotting graphs to average out scatter.
气体实验中的随机误差来自读取弯月面时的视差、水浴温度波动、微小漏气，以及温度计和压力表的分辨率限制。可以通过重复读数、使用数字传感器、在视线水平处读数，以及作图取平均来减小这些误差。
Systematic errors are more serious because they shift every reading in the same direction. Examples include a zero error on a pressure gauge, a trapped air bubble in a manometer, an uncalibrated thermometer reading 1 °C too high, forgetting to subtract the saturated vapour pressure of water, and failing to allow for the volume of the connecting tube in a constant-volume apparatus. Systematic errors usually show up as a non-zero intercept or a wrong gradient, which is why linear graphs are so valuable.
系统误差更严重，因为它使每个读数都朝同一方向偏移。例如：压力表存在零点误差、压强计中残留气泡、温度计未校准而偏高 1 °C、忘记扣除水的饱和蒸气压，以及在等容装置中未计入连接管的体积。系统误差通常表现为图像截距不为零或斜率错误，这正是线性图像如此有价值的原因。
Precision is improved by making the measured change as large as possible relative to the resolution of the instrument: use a larger volume, a wider temperature range, or a longer trapped air column. Always record the number of significant figures justified by the instrument, quote the uncertainty, and, where possible, take the mean of several runs with fresh gas samples to reduce contamination effects.
要提升精度，应使所测变化相对于仪器分度值尽可能大：采用更大体积、更宽温度范围或更长的被封气柱。始终按仪器精度记录有效数字，标明不确定度，并在可能时用新取的气体样品做多次测量取平均，以减小污染带来的影响。
11. Worked Example and Exam Technique | 例题与考试技巧
nt, p₁V₁ = p₂V₂, so p₂ = p₁V₁ ÷ V₂ = (1.0 × 10⁵ × 24) ÷ 8.0 = 3.0 × 10⁵ Pa. Note that the volume units cancel, so no conversion to m³ is needed for an isothermal calculation, but the temperature must still be converted if it appears in the equation.
例题：一定质量的气体在 20 °C、压强 1.0 × 10⁵ Pa 时体积为 24 cm³。在温度不变的情况下被压缩到 8.0 cm³。因 T 不变，故 p₁V₁ = p₂V₂，所以 p₂ = p₁V₁ ÷ V₂ = (1.0 × 10⁵ × 24) ÷ 8.0 = 3.0 × 10⁵ Pa。注意体积单位相消，因此等温计算中
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