📚 Measuring Gases | 气体的测量
Gas measurements are central to A-Level Physics, providing hands-on verification of the ideal gas laws. By controlling the mass of a dry gas and carefully recording pressure P, volume V, and thermodynamic temperature T, you can demonstrate Boyle's law, Charles's law, and the pressure law, and determine fundamental constants such as absolute zero and the molar gas constant R. Accurate use of manometers, barometers, gas syringes, and constant-volume gas thermometers is essential.
气体的测量是A-Level物理的核心内容，能够实际验证理想气体定律。通过控制干燥气体的质量并认真记录压强P、体积V和热力学温度T，你可以验证波义耳定律、查尔斯定律和压力定律，并测定绝对零度和摩尔气体常数R等基本常数。正确使用U形管压力计、气压计、气体注射器和定容气体温度计至关重要。
1. Introduction to Gas Measurements | 气体测量简介
In the laboratory, gas experiments always start with a fixed mass of dry gas. The three measurable quantities – pressure, volume, and temperature – are interrelated, and keeping one of them constant allows the relationship between the other two to be investigated. Real gases approximate ideal behaviour when the pressure is not too high and the temperature is well above the boiling point.
在实验室中，气体实验总是从固定质量的干燥气体开始。三个可测量量——压强、体积和温度——相互关联，保持其中一个恒定就能研究另外两个之间的关系。当压强不太高且温度远高于沸点时，真实气体近似表现出理想行为。
Common apparatus includes a gas syringe for volume, a U-tube manometer or pressure sensor for pressure, and a thermometer or thermocouple for temperature. All readings should be taken after thermal equilibrium is reached.
常用仪器包括测量体积的气体注射器、测量压强的U形管压力计或压强传感器，以及测量温度的温度计或热电偶。所有读数应在热平衡达成后读取。
2. Boyle’s Law: Pressure-Volume Relationship | 波义耳定律：压强与体积的关系
f gas at constant temperature, the pressure P is inversely proportional to the volume V: P ∝ law. To verify it, a gas syringe is sealed and attached to a pressure sensor; the volume is changed stepwise and the corresponding pressure recorded after allowing the gas to return to room temperature.
对于恒温条件下的固定质量气体，压强P与体积V成反比：P ∝ 1/V，即PV = 常数。这就是波义耳定律。为了验证它，将气体注射器密封并连接压强传感器；逐步改变体积，待气体恢复到室温后记录对应的压强。
A graph of P against V yields a rectangular hyperbola, whereas a graph of P against 1/V gives a straight line passing through the origin. The product PV should remain constant within experimental error.
P对V作图得到等轴双曲线，而P对1/V作图得到一条通过原点的直线。PV乘积在实验误差范围内应保持不变。
3. Charles’s Law: Volume-Temperature Relationship | 查尔斯定律：体积与温度的关系
d mass of gas at constant pressure, the volume V is directly proportional to its absolute temperature T: V ∝ ses a capillary tube containing a small plug of sulfuric acid or mercury trapping a dry air column. The tube is immersed in a water bath whose temperature is gradually raised while keeping the pressure constant (often by ensuring the open end is exposed to the atmosphere).
查尔斯定律指出，在压强不变的情况下，固定质量气体的体积V与其绝对温度T成正比：V ∝ T，即V/T = 常数。常用方法是用一根毛细管，其中一小段硫酸或水银封住一段干燥空气柱。将管子浸入水浴中逐步升高温度，同时保持压强恒定（通常让开口端与大气相通）。
The length of the air column is proportional to its volume, so a graph of length against temperature in kelvin is a straight line. Extrapolating the line to zero volume gives a temperature of approximately −273 °C, confirming the idea of absolute zero.
空气柱的长度与其体积成正比，因此长度对热力学温度作图是一条直线。将直线外推至体积为零时，得到温度约为 −273 °C，从而验证了绝对零度的概念。
4. Pressure Law: Pressure-Temperature Relationship | 压力定律：压强与温度的关系
is held constant, the pressure P is directly proportional to the absolute temperature T: P ∝ ure law. It can be demonstrated using a sealed glass bulb connected to a pressure gauge or a constant-volume gas thermometer placed in water baths at different temperatures.
当固定质量气体的体积保持不变时，压强P与绝对温度T成正比：P ∝ T，即P/T = 常数。这便是压力定律。可使用一个连有气压计的密封玻璃球，或将定容气体温度计置于不同温度的水浴中来演示。
A graph of P against T is a straight line that, when extrapolated backwards, cuts the temperature axis at around −273 °C, giving another route to absolute zero.
P对T作图是一条直线，向后外推时与温度轴的交点约为 −273 °C，提供了获得绝对零度的另一途径。
5. The Ideal Gas Equation and Molar Gas Constant | 理想气体方程与摩尔气体常数
Combining Boyle’s law, Charles’s law, and Avogadro’s principle yields the ideal gas equation:
pV = nRT
r gas constant. This equation is used to calculate any one of the four variables when the other three are known.
结合波义耳定律、查尔斯定律和阿伏伽德罗原理，得到理想气体方程：pV = nRT，其中p为压强（Pa），V为体积（m³），n为摩尔数，T为温度（K），R = 8.31 J mol⁻¹ K⁻¹ 是摩尔气体常数。该方程可用于在已知其他三个量的情况下计算第四个量。
6. Measuring Gas Pressure: Manometers and Barometers | 气体压强测量：U形管压力计与水银气压计
ure difference Δp = ρgh, where ρ is the density of the liquid, g is the acceleration of free fall, and h is the height difference between the two liquid surfaces. The absolute pressure of the gas is then patm ± ρgh, depending on whether the gas pressure is above or below atmospheric pressure.
U形管压力计通常充有水或水银。若一臂连接气源，另一臂通大气，则压差Δp = ρgh，其中ρ是液体密度，g是自由落体加速度，h是两液面高度差。气体绝对压强则为patm ± ρgh，取决于气体压强高于还是低于大气压。
A mercury barometer gives atmospheric pressure directly: a glass tube filled with mercury is inverted in a mercury reservoir, and the height of the mercury column, typically around 760 mm, measures atmospheric pressure. Corrections for temperature must be applied for high accuracy.
水银气压计直接给出大气压强：将充满水银的玻璃管倒置在水银槽中，水银柱高度通常约为760 mm，用于测量大气压强。高精度测量时需进行温度修正。
7. Measuring Gas Volume: Syringes and Burettes | 气体体积测量：注射器与量气管
Gas syringes with low-friction plungers are widely used to measure gas volumes at atmospheric pressure. The syringe must be dry and leak-free. For gases collected over water, an inverted burette or measuring cylinder filled with water is used, but the collected gas contains water vapour. The partial pressure of the dry gas is patm − pwater, where pwater is the saturated vapour pressure of water at that temperature.
低摩擦活塞的气体注射器被广泛用于测量大气压下的气体体积。注射器必须干燥且无泄漏。对于排水集气法，使用充满水的倒置量气管或量筒，但收集到的气体含有水蒸气。干燥气体的分压为 patm − pwater，其中pwater是该温度下水的饱和蒸气压。
Always allow the gas to reach the temperature of the surroundings before reading the volume, and avoid parallax errors by reading the bottom of the meniscus at eye level.
读取体积前务必让气体达到环境温度，并在视线水平处读取液面底部以避免视差。
8. The Constant-Volume Gas Thermometer | 定容气体温度计
The constant-volume gas thermometer exploits the pressure law to measure temperature with high precision. A bulb containing a fixed mass of dry gas (often helium) is connected to a mercury manometer. The volume of the gas is kept constant by adjusting the manometer reservoir so that the mercury level in one arm always touches a reference mark. The pressure is then read from the difference in heights.
定容气体温度计利用压力定律来高精度测量温度。一个装有固定质量干燥气体（常用氦气）的球泡连接至水银压力计。通过调节压力计储液池，使一侧水银面始终接触参考标记，从而保持气体体积恒定，然后从液面高度差读取压强。
By calibrating the thermometer at the triple point of water (273.16 K) and using T = Ttp × (p/ptp), unknown temperatures can be determined. This method extends to very low and very high temperatures and is used as a primary standard.
通过在水的三相点（273.16 K）标定温度计，并利用 T = Ttp × (p/ptp)，可测定未知温度。该方法适用于极低和极高温度，被用作一级标准。
9. Determining Absolute Zero by Extrapolation | 通过外推法测定绝对零度
Absolute zero is the temperature at which an ideal gas would have zero volume (Charles’s law extrapolation) or zero pressure (pressure law extrapolation). In practice, real gases liquefy before reaching such low temperatures, so the value is obtained by extending a straight-line graph of V against T or P against T from data taken above room temperature.
绝对零度是指理想气体体积为零（查尔斯定律外推）或压强为零（压力定律外推）的温度。实际上，真实气体在达到如此低温之前已液化，因此该值是通过将从室温以上测得的数据绘制的V-T图或P-T图的直线段外推得到的。
The intersection with the temperature axis is consistently −273.15 °C (0 K). The linearity of the extrapolation confirms the usefulness of the ideal gas model.
与温度轴的交点始终为 −273.15 °C（0 K）。外推的线性关系证实了理想气体模型的有用性。
10. Measuring the Gas Constant R Experimentally | 实验测定气体常数R
A reliable school experiment to measure R involves reacting a known mass of magnesium ribbon with excess dilute hydrochloric acid:
Mg(s) + 2HCl(aq) → MgCl₂(aq) + H₂(g)
The hydrogen gas is collected in a gas syringe or over water in an inverted burette. From the mass of Mg, the number of moles of H₂, n, is calculated. The volume V of gas collected, the room temperature T, and the atmospheric pressure patm are recorded. If collected over water, the pressure of dry hydrogen is p = patm − pwater. Substituting into pV = nRT gives an experimental value for R.
一个可靠的学校实验是使已知质量的镁条与过量稀盐酸反应：Mg + 2HCl → MgCl₂ + H₂。生成的氢气用气体注射器收集或用排水法收集于倒置的量气管中。根据镁的质量计算出氢气的物质的量 n。记录收集到的气体体积V、室温T和大气压 patm。若使用排水集气，干燥氢气的压强为 p = patm − pwater。代入 pV = nRT 可得到R的实验值。
Typical results yield R close to 8.31 J mol⁻¹ K⁻¹. Discrepancies arise from gas leakage, incomplete reaction, or temperature difference between the gas and the thermometer.
典型实验结果R接近 8.31 J mol⁻¹ K⁻¹。误差主要来自漏气、反应不完全或气体与温度计之间的温差。
11. Experimental Errors and Precautions | 实验误差与注意事项
Systematic errors in gas measurements include uncalibrated sensors, friction in gas syringes, and failure to allow for water vapour pressure. Random errors are reduced by repeating readings and allowing sufficient time for thermal equilibrium. The gas must be dry to prevent condensation affecting volume or pressure readings.
气体测量中的系统误差包括传感器未校准、注射器活塞存在摩擦以及未考虑水蒸气压。多次读数并给予足够时间建立热平衡可减小随机误差。气体必须干燥，以防冷凝影响体积或压强读数。
Leak-tight connections are crucial; silicone grease can be used on joints. When using mercury manometers, safety precautions are essential. All volumes should be recorded at the same laboratory temperature and pressure, and the meniscus in liquid columns should be read at the tangent to avoid parallax.
气密连接至关重要；接口处可使用硅脂。使用水银压力计时须注意安全。所有体积读数应在相同的实验室温度和压强下进行，液柱液面应切线读取以避免视差。
12. Summary of Key Equations and Concepts | 关键方程与概念总结
Boyle’s law (T constant): P₁V₁ = P₂V₂ or PV = constant.
波义耳定律（T恒定）：P₁V₁ = P₂V₂ 或 PV = 常数。
Charles’s law (P constant): V₁/T₁ = V₂/T₂ or V/T = constant.
查尔斯定律（P恒定）：V₁/T₁ = V₂/T₂ 或 V/T = 常数。
Pressure law (V constant): P₁/T₁ = P₂/T₂ or P/T = constant.
压力定律（V恒定）：P₁/T₁ = P₂/T₂ 或 P/T = 常数。
Ideal gas equation: pV = nRT, with R = 8.31 J mol⁻¹ K⁻¹. Standard molar volume at STP
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