Chapter 1. Scientific Measurements

Discussion

• Explore: Practice using laboratory balances and glassware to understand the uses and limitations of each and be able to choose appropriate lab equipment.
• Explain: Analyze the results and explain variation in measurements by using the concepts of precision and accuracy, significant figures, and determinate and indeterminate errors.
• Apply: Be able to justify your choice of different laboratory resources, like glassware and balances, for use in future experiments.

Introduction

Two chemistry students are in a heated argument. Pete feels that “scientific experiments are done in a controlled manner and are reproducible if done correctly. Science cannot progress if measurements give different results every time.” But Claire responds that “scientists should take multiple measurements. Results are not always the same and it is wrong to simply perform one test and report the answer.”

Who do you think is right? If measured values are always the same, then why take more than one measurement? If measured values are different, does that mean that science is not reproducible? Or are “true” values never truly known?

In this experiment, you will learn more about scientific measurements by investigating the density of water at room temperature and (perhaps) resolve this dilemma. In this process you will also gain familiarity and proficiency with several important laboratory resources, including laboratory glassware and balances.

Table 1.1 Summary of properties measured in this experiment.

Reading Scale Values

The volume of a liquid is measured in graduated glassware by recording the position of the bottom of the meniscus (the curved surface of a liquid in a tube). As shown in the figure, it is important to place your eye perpendicular to the glassware to obtain a correct reading. In this example a reading of 36.5 mL is appropriate. When reading a scale value, the number you should record includes all of the digits that are known, plus one that is estimated. In this example, the bottom of the meniscus is clearly greater than 36.0 and less than 37.0; a reasonable estimate is 36.5 mL. Graduated laboratory glassware such as burets or graduated cylinders, as well as thermometers, are read in this manner.

Calculations

Dimensional analysis is a common problem-solving approach and in this experiment it can be used for density calculations. Conversion factors are used to ensure that the answers to problems are in the correct units. In this method both the numbers and the units are multiplied together, divided into each other, or canceled out. The end result will be a number and a unit.

Example: A student weighed a marble to obtain a mass of 4352.4 mg and measured a volume, by displacement, of 2.31 mL. What is the density in g/cm³ of the marble?

The question is asking for density, which is mass divided by volume, in units of g/cm³. The measurements give us the mass and volume, but not in the units we need. We can use conversion factors with dimensional analysis to solve this problem:

Question 1.1: The student then decided to cool the same marble and see if the density changed. After analysis, the density of the colder marble was determined to be 1.91 g/cm³. What are the mass and volume of the cold marble?

Experimental Measurements

Precision and accuracy are commonly mentioned in discussions of experimental measurements. Precision refers to how closely individual measurements agree with one another and accuracy refers to how closely a measured value agrees with the accepted value.

It is important to properly use laboratory resources to obtain and report the maximum number of significant figures, which are the digits that indicate the precision with which a measurement is made. For example, a mass reading on an analytical balance of 12.0409 g has six significant figures. Measuring the same object on a top-loading balance would result in only four significant figures. You should be able to explain this based on the precision of the instruments as listed in the table above. Your textbook has more information on the calculation of significant figures.

It should be recognized that the reported number of significant digits encompasses all measurements in a given procedure. Often, a single piece of glassware or instrumentation will limit the number of significant digits that can be obtained.

Returning to the above example, you will notice with Method A 1) the measured density value is consistently less than the accepted value, and 2) repeated measurements produced different density values. The idea of errors in measurement may be used to describe both of these.

Errors in measurement fall into two categories: determinate errors and indeterminate errors. A determinate error is one that has an identifiable source and can be corrected for. Examples of determinate errors include using a meter stick that is actually 1.020 meters long or reading the top of the meniscus instead of the bottom. Both of these errors can be identified and corrected. Note that determinate errors will have a directional bias. Using a “meter” stick that is 1.020 meters long will always result in an error that understates the actual length. It is likely that Method A has a determinate error that leads to a result consistently less than the accepted value.

Indeterminate errors are those that cannot be identified and corrected. For example, in our effort to obtain the maximum number of significant figures we estimate the last digit when reading a scale value. Sometimes the estimate might be too high, other times too low. In either event, it is not possible to estimate the last value in a reproducible way every time. This results in unavoidable variation in the reported value. Even when a value is recorded from a digital instrument (like with a pH meter or analytical balance), there is unavoidable uncertainty in the last digit. This means that, even if determinate errors are absent and the procedure followed flawlessly, there will be indeterminate errors that lead to variation in the measured values. Indeterminate errors will always lead to a range of measured values, and they will not display the directional bias of a determinate error. If only indeterminate error is present, taking multiple measurements and averaging the results is a good way to approximate the true value. The fact that Method A did not always provide the same measured value is consistent with the unavoidable presence of indeterminate errors.

Finally, please note that a scientist’s treatment of error differs from how others view error. For a scientist, “error” is not forgetting to close the door or writing down the wrong number; if these mistakes occur, the experiment should be started over! Instead, errors are an inevitable part of measurement. To have the very best precision and accuracy it is important to identify and remove determinate errors and reduce the magnitude of indeterminate errors.

Experimental Design (Procedure)

In this experiment, your task is to prepare plots of mass/volume for distilled water at room temperature using four different pieces of laboratory glassware. Each mass/volume plot should have at least four different measured values that are between 5 mL and 25 mL (see the following figure). The mass will be measured with a laboratory balance.

Glassware to use:

Materials Provided

Equipment

• 2 100-mL beakers (~50 g each)
• 1 250-mL beaker (~110 g)
• 1 400-mL beaker (~136 g)
• 1 600-mL beaker (~172 g)
• 2 125-mL Erlenmeyer flasks (~82 g each)
• 3 250-mL Erlenmeyer flasks (~120 g)
• 1 25-mL buret and buret clamp
• 1 10-mL graduated cylinder
• 1 100-mL graduated cylinder
• 1 5-mL volumetric pipet
• 1 10-mL volumetric pipet
• rubber pipet bulb
• 1 thermometer (–20°C–100°C)

Common Equipment

• top-loading balances
• analytical balances
• Parafilm (covers glassware)

Chemicals

• none

Cautions

Goggles must be worn in the laboratory.

Procedure

You are responsible for recording your procedure in your notebook. You should provide a sufficient level of detail such that another scientists could repeat your experiment.

Waste Disposal

All solutions may be rinsed down the drain.

Data (Use the correct number of significant figures)

Water temperature: ______________

Water, accepted density at room temperature (calculated): ____________

At 4°C water’s density is 1.000 g/cm³. For the range of temperatures associated with “room temperature” (from about 20°C to 30°C), the density of pure water is:

Density (g/cm³) = [–0.00030 (g/°C · cm³) X Temperature (°C)] + 1.0042 (g/cm³)

¹Error = experimental value – true value.

The relative error = error/true value.

The % error = relative error X 100%.

Report Details

1. Reproduce your completed data table.
2. Prepare and submit four graphs of mass/volume. Each graph should have a title identifying the glassware employed, the axes should be properly labeled (including units), and a best-fit line fit (y = mx + b format) and the r² value reported on the graph. Use a program like Excel for preparing the graphs.

Answer the following questions.

1. A discussion of precision and accuracy is included in this experiment. Use these terms to rank the precision and accuracy of each type of glassware tested. Support your analysis with your data.
2. What does the slope of each mass/volume plot represent? What information does the plot provide that a single measurement does not? What would the plot look like if a determinate error were present?
3. A mass/volume plot includes two measured properties (mass and volume) and one calculated property (density). How does the plot illustrate the intensive or extensive nature of each of these properties?

Answer Clinic

Scientific Measurements

Question 1.1

A student analyzed a marble to obtain a mass of 4352.4 mg and measured a volume, by displacement, of 4.31 mL. The resulting density was 1.88 g/cm³. The student then decided to cool the same marble and see if the density changed. After analysis, the density of the colder marble was determined to be 1.91 g/cm³. What are the mass and volume of the cold marble?

Answer

The density = 1.91 g/cm³. This is a ratio of mass/volume. In the original question the mass = 4352.4 mg and the volume = 2.31 mL. With a change in temperature will the mass change or the volume? It must be the volume because the amout of matter in the marble has not changed, only the amount of space it takes up.

Mass = 4352.4 mg

Question 1.2

A chemist analyzed a liquid sample multiple times by two different methods. The sample has a true density of 0.9855 g/mL.

Which method has better precision?

Answer

The range of values for method A is much narrower. Method A has better precision. Which method has better accuracy?

Answer

The true value = 0.9855.

The average for method A = 0.9840. The average for method B = 0.9814

Method A has better accuracy. Note: It is difficult to determine the accuracy if the precision is poor as the results will be widely scattered. Simply calculating an average may be misleading under these circumstances.