Uncertainties in A-Level Chemistry | A-Level 化学中的不确定度

📚 Uncertainties in A-Level Chemistry | A-Level 化学中的不确定度

In A-Level chemistry, every measurement carries uncertainty. Understanding how to quantify, combine, and report uncertainties is essential for accurate data analysis, practical assessment, and exam success.

在 A-Level 化学中，每一次测量都伴随着不确定度。理解如何量化、合并和报告不确定度，对于准确的数据分析、实验评估和考试成功至关重要。

1. What Is Uncertainty in Measurement? | 什么是不确定度？

Uncertainty is the range of values within which the true value of a measurement is expected to lie. It is not a mistake; it reflects the limitations of the measuring instrument and the observer.

不确定度是指测量的真实值预期所在的数值范围。它并不是错误，而是反映了测量仪器和观察者的局限性。

For example, a balance that reads to 0.001 g has a stated uncertainty of ±0.001 g, meaning the true mass could be up to 0.001 g higher or lower than the displayed value.

例如，一台可读至 0.001 g 的天平，其标注不确定度为 ±0.001 g，意味着真实质量可能比显示值高或低最多 0.001 g。

2. Absolute Uncertainty vs Percentage Uncertainty | 绝对不确定度与百分比不确定度

Absolute uncertainty has the same units as the measurement itself, such as ±0.10 cm³ or ±0.5 °C. Percentage uncertainty expresses the absolute uncertainty as a fraction of the measured value, multiplied by 100.

绝对不确定度的单位与测量值本身的单位相同，例如 ±0.10 cm³ 或 ±0.5 °C。百分比不确定度则将绝对不确定度表示为测量值的一部分，再乘以 100。

Percentage uncertainty = (Absolute uncertainty / Measured value) × 100%

If a burette reading is 25.00 cm³ with an uncertainty of ±0.05 cm³, the percentage uncertainty is (0.05 / 25.00) × 100% = 0.20%.

如果滴定管读数为 25.00 cm³，不确定度为 ±0.05 cm³，则百分比不确定度为 (0.05 / 25.00) × 100% = 0.20%。

3. Reading Uncertainty of Digital Instruments | 数字仪器的读数不确定度

For digital instruments such as balances, pH meters, and digital thermometers, the uncertainty is typically taken as half of the last decimal place displayed, unless the manufacturer specifies otherwise.

对于数字仪器，如天平、pH 计和数字温度计，除非制造商另有规定，不确定度通常取最后一位显示数字的一半。

If a digital balance displays 3.42 g, the last digit is 0.01 g, so the uncertainty is ±0.005 g. This is because the true value is rounded to the nearest 0.01 g, so it could be anywhere between 3.415 g and 3.425 g.

如果数字天平显示 3.42 g，最后一位为 0.01 g，则不确定度为 ±0.005 g。这是因为真实值四舍五入到最接近的 0.01 g，可能在 3.415 g 到 3.425 g 之间。

4. Uncertainty of Analogue Instruments | 模拟仪器的读数不确定度

For analogue instruments such as burettes, measuring cylinders, rulers, and thermometers, the uncertainty is usually half of the smallest scale division.

对于模拟仪器，如滴定管、量筒、直尺和温度计，不确定度通常为最小刻度的一半。

A burette has scale divisions of 0.10 cm³, so each reading has an uncertainty of ±0.05 cm³. Since two readings are taken (initial and final), the total uncertainty for the volume delivered is ±0.10 cm³.

滴定管的最小刻度为 0.10 cm³，因此每次读数的不确定度为 ±0.05 cm³。由于需要读取两次（初始和最终），因此滴定所传液体体积的总不确定度为 ±0.10 cm³。

5. Repeated Measurements and Mean | 重复测量与平均值

When a measurement is repeated, the uncertainty can be estimated from the range: half the range of the repeated measurements is often used as the absolute uncertainty.

当重复测量时，可以通过极差来估计不确定度：通常使用重复测量极差的一半作为绝对不确定度。

If three titres are 23.10, 23.20, and 23.15 cm³, the range is 23.20 – 23.10 = 0.10 cm³, so the uncertainty is ±0.05 cm³. The mean titre is 23.15 cm³ with uncertainty ±0.05 cm³.

如果三组滴定体积为 23.10、23.20 和 23.15 cm³，极差为 23.20 – 23.10 = 0.10 cm³，因此不确定度为 ±0.05 cm³。平均滴定体积为 23.15 cm³，不确定度为 ±0.05 cm³。

6. Combining Uncertainties When Adding or Subtracting | 加减运算中的不确定度合并

When measurements are added or subtracted, the absolute uncertainties are added together.

当测量值相加或相减时，绝对不确定度应相加。

For example, to find the mass of a sample, you weigh a container (15.25 g ± 0.005 g) and then the container with the sample (18.40 g ± 0.005 g). The mass of the sample = 18.40 – 15.25 = 3.15 g, and the total absolute uncertainty = 0.005 + 0.005 = ±0.010 g.

例如，要计算样品质量，称量容器 (15.25 g ± 0.005 g)，再称量容器加样品 (18.40 g ± 0.005 g)。样品质量 = 18.40 – 15.25 = 3.15 g，总绝对不确定度 = 0.005 + 0.005 = ±0.010 g。

7. Combining Uncertainties When Multiplying or Dividing | 乘除运算中的不确定度合并

When measurements are multiplied or divided, the percentage uncertainties are added together.

当测量值相乘或相除时，百分比不确定度应相加。

Suppose you calculate concentration using mass = 2.50 g ± 0.01 g and volume = 100.0 cm³ ± 0.5 cm³. Percentage uncertainty in mass = (0.01 / 2.50) × 100% = 0.40%. Percentage uncertainty in volume = (0.5 / 100.0) × 100% = 0.50%. Total percentage uncertainty = 0.40 + 0.50 = 0.90%.

假设你用质量 = 2.50 g ± 0.01 g 和体积 = 100.0 cm³ ± 0.5 cm³ 计算浓度。质量的百分比不确定度 = (0.01 / 2.50) × 100% = 0.40%。体积的百分比不确定度 = (0.5 / 100.0) × 100% = 0.50%。总百分比不确定度 = 0.40 + 0.50 = 0.90%。

This total percentage uncertainty can then be converted back to an absolute uncertainty: (0.90 / 100) × concentration value.

然后可将总百分比不确定度换算回绝对不确定度：(0.90 / 100) × 浓度值。

8. Uncertainties in Titration Calculations | 滴定计算中的不确定度

In a titration, several uncertainties contribute: the balance for weighing, the volumetric flask, the pipette, and the burette. When calculating a final result such as concentration or molar mass, you must combine all relevant uncertainties.

在滴定中，多个不确定度会共同作用：称量用的天平、容量瓶、移液管和滴定管。当计算最终结果如浓度或摩尔质量时，必须合并所有相关的不确定度。

For a pipette of 25.0 cm³, the manufacturer’s tolerance is often ±0.06 cm³. A 250 cm³ volumetric flask has a tolerance of ±0.20 cm³. These are absolute uncertainties that contribute to the overall calculation.

对于 25.0 cm³ 的移液管，制造商允许的误差通常为 ±0.06 cm³。250 cm³ 容量瓶的允许误差为 ±0.20 cm³。这些绝对不确定度都会影响整体计算。

9. Recording Results and Significant Figures | 记录结果与有效数字

The measured value and its uncertainty must be quoted to the same number of decimal places. For example, write 23.15 cm³ ± 0.05 cm³, not 23.15 cm³ ± 0.1 cm³.

测量值及其不确定度必须保留相同的小数位数。例如，应写 23.15 cm³ ± 0.05 cm³，而不是 23.15 cm³ ± 0.1 cm³。

The final calculated answer should be rounded to match the precision of the least precise measurement. Do not report more significant figures than justified by the experimental data.

最终计算结果应四舍五入到与精度最低的测量值相匹配的位数。报告的有效数字不应超过实验数据所能够支持的范围。

10. Improving Accuracy by Reducing Uncertainty | 通过降低不确定度提高准确性

Uncertainty can be reduced by using more precise instruments, increasing the size of the measured quantity, performing repeats, and taking care with technique.

可以通过使用更精密的仪器、增大被测量的量、进行重复实验以及注意操作技术来降低不确定度。

Use a balance with higher resolution, e.g. 0.001 g instead of 0.01 g.

使用分辨率更高的天平，例如 0.001 g 而不是 0.01 g。
Weigh a larger mass so that the absolute uncertainty becomes a smaller percentage.

称量更大的质量，使绝对不确定度所占的百分比更小。
Use larger titre volumes where possible to reduce the percentage effect of the burette reading uncertainty.

尽可能使用更大的滴定体积，以降低滴定管读数不确定度的百分比影响。
Repeat experiments and calculate the mean to reduce random errors.

重复实验并计算平均值，以减少随机误差。

Remember: reducing the percentage uncertainty improves the reliability of your final answer, even if the absolute uncertainty of each instrument stays the same.

请记住：即使每台仪器的绝对不确定度不变，降低百分比不确定度也能提高最终答案的可靠性。

Published by TutorHao | Chemistry Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Uncertainties in A-Level Chemistry | A-Level 化学中的不确定度