Measured Values

These are basic skills that many students know or have already been exposed to. Make sure you are up to speed with these. If you need a refresher or just want to make sure you are up to speed with these, these materials are designed to do that. We will talk a little about significant figures, because there always seems to be confusion surrounding them, you are mostly on your own to learn them. However, if you need help or have questions, do not hesitate to ask! We need everyone on the same page as we move forward.

Fundamental Knowledge and Skill—Measurements: Accuracy and Precision

What You Need to Know

Every measurement has some degree of uncertainty associated with it. Thus, every time we make a measurement with any measuring device, there is always some deviation from its true value. Engineers tend to think of this uncertainty as a margin of error. A yardstick has a larger margin of error than calipers. A yardstick may give you measurements that are within .5” of the true value, while calipers may only deviate from the true value by .01”.

Precision is consistency. When you have a precise measuring device, it will give you consistent results. Often, you will see precision indicated by the number of digits that a measuring device will give you. A scale that shows mass as 1.0002 g is more precise than a scale that shows mass as 1.0 g. By and large, the more digits present in the result, the more precise the device.

Accuracy is the degree to which a measured value fits the true value. An accurate measuring device will give you results that are extremely close to the accepted standard value.

How to Learn It

We will use significant figures to illustrate the degree of uncertainty associated with a measurement. If a metric yardstick has a margin of error of 0.5 cm, then a measured value of 24 cm has a true value somewhere between 23.5 cm and 24.5 cm. Instead of following the example of the engineers and writing 24 cm (\(+/- \).5 cm) as the measured length of the object, we will simply write 24 cm. The one's place here is the last significant figure, and is therefore the limit of our certainty associated with our measurement. Writing 24 cm tells us the same thing as 24 cm (\(+/- \).5 cm), if you understand significant figures.

A classic example of precision is Robin Hood shooting his bow at the target and splitting the arrow because he hit the same location each time. Even if his initial arrow had not hit the bullseye, if he had hit the same location each time, he would be incredibly precise. On the other hand, a classic example of accuracy is William Tell shooting an arrow through the apple on the top of his son’s head. If he were off by just a few inches, he would either kill his son or be executed for missing the apple. His accurate shooting resulted in saving both his and his son’s lives.

Let’s say that an object’s true mass is 7 g. If we use a precise analytical balance that has not been calibrated to give accurate results, on several weighings, we may get 6.444 g, 6.440 g, 6.448 g, 6.445 g. These results are precise, within several thousandths of a gram, but they are not accurate, as they do not reflect the true mass of the object. However, if we were to use a calibrated kitchen scale to weigh this object several times, we might get results like 6.9, 7.1, 6.8, 7.2, 7.0. This kitchen scale is not very precise, as the results are not very close to each other. However, if we were to average our results, we would get the true mass. This kitchen scale is accurate, but not precise.

Why It Matters

You’ll likely be involved in some kind of research either here at BYU or later in your life. Research, especially chemical research, needs to be accurate and precise, and it is your responsibility to understand how accurate your results actually are; otherwise, they are meaningless. In your life, it is especially important to recognize the importance of accuracy and precision and the limitations of measurements.

Fundamental Knowledge and Skill—SI Units of Measurement

_{Image from the National Institute of Standards and Technology (NIST)}

How to Recognize When You Need to Do It

Anytime you are communicating in the sciences, the metric system is preferred to the English or imperial system of weights and measures. The vast majority of the chemistry problems you will encounter in this class will be presented in metric units, and answers will be expected to be in metric units.

How to Do It

The metric system is awesome because it uses the same prefixes to describe deviations from the basic units of length, mass, and volume. That way, you only have to memorize one set of prefixes. The main ones you’ll need to know are kilo- (10\(^3\)), centi- (10\(^2\)), milli- (10\(^{-3}\)), micro- (10\(^{-6}\)), and nano- (10\(^{-9}\)). Look at the chart for reference.

Length

The basic unit of length in the metric system is the meter. A meter is about 10% longer than a yard or three feet, or about the length of an average man’s stride.

Mass

The basic unit of mass in the metric system is technically the kilogram, but the prefixes are based on the gram. One kilogram (10\(^3\) grams) is about \(2.2\) pounds, or about the weight of a pineapple.

Volume

The basic unit of volume in the metric system is the liter. One liter is about a quart. You may also see volume expressed as cubic centimeters (cc). One cubic centimeter of water is equivalent to one milliliter of water. There are 1000 cubic centimeters or milliliters in one liter.

Conversions in the metric system are easy. An object that is 50 meters long is \(50 \text{ meters} \times \frac{1 \text{km}}{1000\text{m}} = .05 \text{ kilometers}\) 

A 50 meter long object is \(50 \text{ meters} \times \frac{100 \text{cm}}{ 1\text{m}} = 5000 \text{ centimeters}\).

Why

For purposes of clarity, the metric system has been almost universally adopted by the scientific community. You need to be able to use and understand it.

Fundamental Knowledge and Skill—Significant Figures and Calculations Using Significant Figures

How to Recognize When You Need to Do It

When calculating a numerical answer using measured numerical values, almost every question we ask will require an answer in significant figures, except when you have an exact number. In that case, there are no significant figures.

How to Do It

The zeros present and implied in a numerical value are key. Look to the zeros in the values and those that specify the location of the decimal point. Zeros have a special place in significant figures. If there are zeros to the left of the decimal point, then they are ignored in significant figures. If there are zeros to the right, then they represent the limits of measurement and are thus counted in significant figures. 

For example, .0002 has one significant figure, 2.000 has 4 significant figures, and .0002000 has 4 significant figures. The easiest way to determine what the significant figures are is to convert the number to scientific notation. 

\(.0002\) can be written as \(2 \times 10^{-4}\), \(2.000\) can be written as \(2.000 \times 10^1\), and \(.0002000\) can be written as \(2.000 \times 10^{-4}\).

Every measurement has some uncertainty associated with it. The last number or smallest measured unit amount of a reported value is considered the last significant figure and represents the certainty associated with the measurement. For example, if you use a bathroom scale, it may report a value of 72 kilograms. The value, 72 kilograms, is a measured value that has two significant figures. However, the scale has a limit to its accuracy and precision. If the reported weight is 72 kilograms, your true weight is somewhere between 71.5 kg and 72.5 kg. Say, you used a more precise (and accurate!) scale, and your weight read 71.79 kg. This reported value has 4 significant figures and could be written as 71.79 kg.

When adding or subtracting values with significant figures, the final result is limited by the least precise significant digit or last significant figure. For example, suppose you use an analytical balance to discover that a small watermelon weighs 1.5067 kg, and you use a bathroom scale to determine that your weight is 72 kg. The unrounded combined weight of the watermelon and your weight is 73.5067 kg. However, you are limited by the precision of the least precise measurement, in this case, the 72 kg. Consequently, the rounded weight that you must report is limited to the one's place or 74 kg, because of the bathroom scale’s limited precision, which only determines your weight accurately to within 0.5 kg.

When multiplying or dividing values with significant figures, the final result is also limited by the value with the least number of significant figures. For example, suppose you have determined that the density of iron is 7.874 grams/cm\(^3\). Say you weigh an irregularly shaped sample of iron that weighs 82.1 grams and need to know the volume of the sample, you could calculate using the density information or \(82.1 \text{grams} \times \frac{1 \text{cm}^{3}}{ 7.874 \text{grams}} = 10.42672085344171 \text{cm}^3\) (the unrounded value, read directly from your calculator). 

How should we round this value? The weight 82.1 grams had 3 significant figures (or sig figs for short), while 7.874 grams/cm\(^3\) has 4 significant figures. Thus, we round the reported value to the least number of sig figs, in this case, three or 10.4 cm\(^3\). Any further digits would be superfluous and untruthful, as we are limited by our measurements in this question.

It is a good practice while performing multi-step calculations to use the unrounded values for the secondary steps and not to round until the end of all calculations. Only round for significant figures at the end, as you report your final answer.

Why

Significant figures communicate only the meaningful digits in a numerical answer. Significant figures in the final answer help us understand how precisely we know that answer. Extra digits after the final significant figure are a deception, indicating that we know the answer to a greater degree of precision than we truly do. Beware of this.

Fundamental Knowledge and Skill—Conversion Factors and Stoichiometry

How to Recognize When You Need to Do It

Chemistry problems often will provide you with the reactant in one unit and ask for the quantity of product expressed in a different unit. You need to be comfortable converting substances from one unit to another.

Common stoichiometry problems include converting mass to volume using density, converting mass to moles of atoms, and converting moles of atoms to individual atoms.

How to Do It

If you wish to tell a European your height, they won’t understand you very clearly if you use inches. However, if you know that there are 12 inches in one foot and 2.54 cm in 1 inch, you can convert your height to centimeters. A 5’6” person is \(5\, \text{foot} \times \frac{12\, \text{inches}}{1\, \text{foot}} + 6\, \text{inches} = 66\, \text{inches}\). To convert to centimeters, multiply \(66\, \text{inches} \times \frac{2.54\, \text{cm}}{1\, \text{inch}}\) to get your height expressed in units of cm or \(167.64\, \text{cm}\). Then, if you would like to convert cm to meters, use the fact that there are 100 cm in 1 m in this way, \(167.64\, \text{cm} \times \frac{1\, \text{meter}}{100\, \text{cm}} = 1.67\, \text{meters}\).

Conversion factors are the means by which you convert one unit to another. They do not in any way change the amount measured; they just provide a way to express that amount using a different set of units.

For example, lead has a density of \(11.34\, \text{g}/1.0\, \text{cm}^3\). Say we had a piece of lead that had a mass of \(34.0\, \text{g}\) and we wanted to know the volume of that piece of lead. From the density we know that \(11.34\, \text{g}\) of lead = \(1.0\, \text{cm}^3\) of lead. We need units of \(\text{cm}^3\) and have to lose units of g. If we take this relationship and divide both sides by \(11.34\, \text{g}\) of lead, we get \(1 = \frac{1.0\, \text{cm}^3}{11.34\, \text{g}}\) of lead. When we multiply this relationship by the mass of our piece of lead or \(34.0\, \text{g} \times \frac{1.0\, \text{cm}^3}{11.34\, \text{g}}\) of lead, we get a value of \(3.0\, \text{cm}^3\) of lead. Notice that the value \(\frac{1.0\, \text{cm}^3}{11.34\, \text{g}}\) of lead is just 1. So, when we multiply by \(\frac{1.0\, \text{cm}^3}{11.34\, \text{g}}\), we are multiplying by 1.

Always make sure that you cancel out the units as you go. Many people like to write the starting unit on the left side, draw a long horizontal line, then write the ending unit on the right side. Then, use conversion factors to cancel out units until only the desired unit is left.

For example, let’s express .023 pounds of H\(_2\)O in units of molecules of water. The conversion factor of pounds to kilograms is 1 pound to 2.2046 kg.

Conversion factors can be inverted because they have a value of 1.

Why

Nearly every chemistry problem has a stoichiometry aspect to it. Get this right early, and you’ll be better off. This concept will also help you in other courses, including biology and physics.

General Notes

This is a key skill. Take it slowly, and you’ll be just fine. Always look at the question to determine the units you are starting with and the final units you should end up with, and always double-check your work.