Question

A student used a \(250 \mathrm{~mL}\) graduated cylinder having volume markings every \(2 \mathrm{~mL}\) to carefully measure \(100 \mathrm{~mL}\) of water for an experiment. A fellow student said that by reporting the volume as "100 mL" in her lab notebook, she was only entitled to one significant figure. The first student disagreed. Why did her fellow student say the reported volume had only one significant figure?

Short answer

The student's report of '100 mL' is actually to three significant figures because the smallest division is 2 mL, allowing an estimation to the nearest 1 mL.

Step-by-step solution

Understanding Significant Figures

Significant figures in a measurement include all the digits that are known precisely, plus one last digit that is somewhat uncertain or estimated. The precision of the measurement is often limited by the smallest division on the measurement scale, which in this case is 2 mL.

Analyzing the Smallest Division

The graduated cylinder used in the experiment has markings every 2 mL. This means that any measurement made with this graduated cylinder can be precise up to 1 mL above the nearest marking (due to the estimation between the lines). Therefore, a measurement of 100 mL is understood to be between 99 mL and 101 mL.

Determining the Number of Significant Figures

Since the volume can be estimated to the nearest 1 mL, the measurement of 100 mL is known to three significant figures. The '1' is known precisely, as is the first '0', while the second '0' is the uncertain digit but still significant as it denotes the precision of the measurement.

Key concepts

Significant Figures in Measurement

When you measure something, you're always limited by the tool you're using. In our daily lives, we might not think much about whether we've drunk 250.00 mL of water or just about 250 mL – but in the world of science, this distinction is crucial. This is because scientific measurements aim for both accuracy and precision, which brings us to the concept of significant figures.

Significant figures, often referred to as 'sig figs' for short, tell us how trustworthy our measurements are. They include all the digits that are surely known, plus one extra digit that's kind of a best guess. That last one is there because measuring instruments aren't perfect. For instance, if our ruler only has marks for each centimeter, we may have to guess how much more than 3 centimeters something is. This 'guess' is an educated one and is important to record because it conveys how precise our measurement is.

In the realm of chemistry and other strict sciences, significant figures come into play with every measurement. They are crucial for conveying to others exactly how precise your data really is and are equally important when it comes to calculations. Using the correct number of significant figures in any calculation keeps the results as consistent and accurate as possible, and tells others exactly how much they can trust your data.

Precision in Chemistry

Precision in chemistry is all about how consistently you can get the same measurement. Think of it like throwing darts. If you keep hitting the same spot on the dartboard – whether it's the bullseye or not – your throws are precise. Similarly, in chemistry, if you can measure out 20 mL of water multiple times and always get 20 mL, you're being precise.

But precision is only as good as the tools you use. If you've got a beaker marked in 10 mL increments, your 20 mL measurement could be a smidge more or less each time you try it. So while the tools may limit precision, chemists strive to use tools that give them as much precision as possible – and this is where significant figures come in. They tell the story of the tool's precision. The more significant figures you can reliably report, the more precise your tool, and, therefore, your measurement is thought to be.

Back to the dart analogy, precision is not about hitting the bullseye, which would be accuracy. It's about clustering your throws closely together, and you want your measurements in chemistry to cluster closely around the true value. This eliminates wide swings in data and gives everyone confidence in the consistency and reliability of your work.

Graduated Cylinder Readings

A graduated cylinder is the quintessential chemistry tool for measuring out liquids. Just like a ruler with ticks marked at every inch or centimeter, graduated cylinders have tick marks at certain intervals for milliliters. To read them correctly, there's a bit of know-how involved.

First, you need to position your eyes at the level of the liquid's meniscus—that's the curved surface of the liquid. Usually, you’ll line up the lowest part of the meniscus with the closest tick mark to determine the volume. However, there's often a bit of liquid between that last tick mark and the meniscus; this is where estimation becomes important and where significant figures come into play.

The smallest increment on the graduated cylinder will limit how precisely you can read the volume. In the exercise, the cylinder is marked every 2 mL, so the readings are estimated to the nearest 1 mL. When you write down 100 mL, that '100' carries an implication that it's specifically between 99 and 101 mL, and because of this, all three digits are significant. This understanding translates into increased trust in the data: it means that someone taking that measurement is aware of and accounting for the limitations of the tool, ensuring the reported figures are as precise as the tool allows.