Question

In lab you report a measured volume of \(128.7 \mathrm{mL}\) of water. Using significant figures as a measure of the error, what range of answers does your reported volume imply? Explain.

Short answer

The reported volume of \(128.7 \mathrm{mL}\) has 4 significant figures and an error/resolution of ±0.1 mL. The range of possible volumes implied by the measurement is between \(128.6 \mathrm{mL}\) and \(128.8 \mathrm{mL}\).

Step-by-step solution

Identifying the number of significant figures

We are given the volume measurement of 128.7 mL. In this value, all non-zero digits and the decimal point are considered significant. So, we have a total of 4 significant figures in this measurement (1, 2, 8, and 7).

Determine the error or resolution based on significant figures

The last significant figure in the measurement usually determines the resolution or error in the measurement. In our case, the last significant figure is 7, which is located in the tenths place (0.1 mL). Therefore, the error/resolution in the measurement is ±0.1 mL.

Calculate the range of possible volumes

Now that we know the error in the measurement, we can find the range of possible volumes. The lower limit of the range is the original measurement minus the error, while the upper limit is the original measurement plus the error. Lower limit: \(128.7 \mathrm{mL} - 0.1 \mathrm{mL} = 128.6 \mathrm{mL}\) Upper limit: \(128.7 \mathrm{mL} + 0.1 \mathrm{mL} = 128.8 \mathrm{mL}\) So, the range of possible volumes that the reported measurement of 128.7 mL could imply is between 128.6 mL and 128.8 mL.

Key concepts

Measurement Uncertainty

Measurement uncertainty is a concept that aims to communicate the level of confidence we have in our measurements. It reflects the limits of precision that our measuring instruments and procedures can achieve. For instance, when a volume of water is measured as 128.7 mL, this measurement is not perfectly exact. There's always a small range of error, which in the context of significant figures, is implied by the last digit reported. When we see 128.7 mL with four significant digits, we understand that the '7' is the least certain - it's in the tenths place and it signifies an uncertainty of ±0.1 mL.

This indicates that even though the measurement is reported as 128.7 mL, the actual volume could be slightly more or less. The concept of uncertainty is fundamental in chemistry and other sciences where precision is crucial. It is a form of honesty in scientific measurements, revealing that our measurements are not exact but are the best approximation given the tools and methods used.

• Understanding uncertainty helps evaluate the quality of measurements.
• Expressing measurement uncertainty guides the interpretation of results.

Scientific Notation

Scientific notation is a method of writing numbers that are too large or too small to be conveniently written in decimal form. In scientific notation, a number is expressed as the product of a number between 1 and 10 and a power of 10. For example, the distance between the Earth and the Sun, approximately 150,000,000,000 meters, can be expressed as 1.5 x 10^11 meters in scientific notation.

This is particularly useful in chemistry when dealing with very small or large quantities. In the context of significant figures, scientific notation also becomes invaluable because it clearly indicates the precision of a measurement. For instance, if a laboratory scale reads 0.0001287 grams, writing it in scientific notation as 1.287 x 10^-4 grams makes it evident that the measurement is precise to four significant figures.

• Scientific notation makes working with extreme values manageable.
• It clarifies the significant figures in a measurement.

Converting to Scientific Notation

Move the decimal point in a number until you have a coefficient between 1 and 10, count the places you moved it as the exponent on 10 (to the right is negative, to the left is positive). For instance, 0.0123 becomes 1.23 x 10^-2.

Quantitative Analysis

Quantitative analysis in chemistry involves the determination of the exact amount of a substance in a sample. It includes techniques and measurements that give numerical results, which might be the mass, volume, or concentration of the analyte. In our example with the volume of water measured as 128.7 mL, quantitative analysis is at play. This value gives a specific quantity that can support various scientific and experimental conclusions.

The role of significant figures becomes crucial here, as the precision of the results from quantitative analysis lends credibility and accuracy to the findings. Accuracy in quantitative data is significant as it affects the credibility of the analysis and subsequent decisions or experiments based on these results.

• Quantitative analysis demands precise measurements.
• The results are often expressed with significant figures to reflect the level of precision.