## CHEMM

Chem 115 POGIL Worksheet – Week 1

Units, Measurement Uncertainty, and Significant Figures

Why?

All scientists the world over use metric units. Since 1960, the metric system in use has been the

Système International d’Unités, commonly called the SI units. These units facilitate

international communication by discouraging use of units peculiar to one culture or another (e.g.,

pounds, inches, degrees Fahrenheit). But regardless of the units used, we want to have some

confidence that our measured and calculated results bear a close relationship to the “true” values.

Therefore, we need to understand the limits on our measured and calculated values. One way we

convey this is by writing numerical answers with no more and no fewer than the number of digits

that are justified by the limits of our ability to measure and know the quantity.

Learning Objective

• Know the units used to describe various physical quantities

• Become familiar with the prefixes used for larger and smaller quantities

• Master the use of unit conversion (dimensional analysis) in solving problems

• Appreciate the difference between precision and accuracy

• Understand the relationship between precision and the number of significant figures in a

number

Success Criteria

• Associate units with physical quantities

• Replace prefixes by multiplying by appropriate numerical factors

• Be able to use dimensional analysis for unit conversions

• Report computed values to the correct number of significant figures.

Prerequisites

• Exponential notation

• Having read Chapter 1 in the text

Information

The SI units consist of seven base units and two supplementary units. For now, we will only use

the four base units listed below. Later we will talk about two others. We will never use the

seventh unit (candela), a unit for luminous intensity.

Quantity Unit Abbrev.

Length meter m

Mass kilogram kg

Time second s

Temperature kelvin K

Any other units can be constructed as a combination of fundamental units. For example, velocity

could be measured in meters per second (written m/s or m*s

–1), and area could be measured in

units of meters squared (m2). When a named unit is defined as a combination of base units, it is

called a derived unit. For example, the SI unit of energy is the joule (J), which is defined as a

kg*m

2

*s

–2. Note that when a unit is named for some scientist (e.g., Joule, Hertz, Kelvin) the

written name of the unit is not capitalized, but the abbreviation is capitalized.

All metric units can be related to larger or smaller units for the same quantity by use of prefixes

that imply multiplications of the stem unit by certain powers of 10. The following prefixes are

important to know.

Prefix Abbrev. 10±n Example

Mega- M 106 Megahertz (MHz)

Kilo- k 103 kilogram (kg)

Deci- d 10–1 deciliter (dL)

Centi- c 10–2 centimeter (cm)

Milli- m 10–3 milliliter (mL)

Micro- µ 10

–6 microgram (µg)

Nano- n 10–9 nanometer (nm)

Pico- p 10–12 picosecond (ps)

Femto- f 10–15 femtosecond (fs)

Key Questions & Exercises

1. Give the names and their abbreviations for the SI units of length, mass, time, and

temperature.

2. The unit of volume is the liter (L). Why is this not a base SI unit? What kind of SI unit is it?

3. A student is asked to calculate the mass of calcium oxide produced by heating a certain

amount of calcium carbonate. The student’s answer of 90.32 is numerically correct, but the

instructor marks it wrong. Why?

4. Write the number of seconds in a day (86,400 s) in exponential notation, using a coefficient

that is greater than 1 and less than 10. (This form is called scientific notation and is generally

the preferred form of exponential notation, as explained below).

5. The diameter of a helium atom is about 30 pm. Write this length in meters, using standard

scientific notation.

6. A cubic container is 2.00 cm on each edge. What is its volume in liters? What is its volume

in milliliters (mL)? Are your answers reasonable?

Information

Units can actually help in setting up and solving many problems by using a method called

dimensional analysis (also called the factor-label method). In dimensional analysis, a problem

is typically viewed as a conversion of a given value in given units into a new value in certain

desired units. Mathematically, such problems take on the general form

(given quantity & given units)(wanted units/given units) = wanted quantity & wanted units

The factor “wanted units/given units” is a conversion factor, which is always a fractional

expression of an equivalence relationship between two different units. In carrying out the

multiplication and division, the given units cancel out, leaving the wanted units.

To apply dimensional analysis, follow this general problem-solving strategy: (1) Identify and

record what is known, with its given units; (2) identify what is to be calculated with its units;

(3) identify the concepts and/or relationships that connect the given information with what

needs to be calculated; (4) set up the solution using unit relationships as one or more conversion

factors, such that all units except those desired for the answer cancel; (5) do the mathematics;

(6) check or validate your answer by asking yourself if it is a reasonable result.

Example: How many inches is 2.00 cm, given that the inch is defined as exactly 2.54 cm?

(1) We know the length in centimeters.

(2) We want the length in inches.

(3) 1 in / 2.54 cm (exactly)

(4) Possible conversion factors are 1 in/2.54 cm and 2.54 cm/1 in. We are starting with cm

and want to end up with in, so the first conversion factor will do the job.

(5) Do the mathematics.

Note that the centimeter units cancel, leaving the desired units of inches.

(6) If 2.54 cm is an inch, then 2.00 cm should be a fraction of an inch. So, 0.787 in looks

like a reasonable answer.

Key Questions & Exercises

7. In general, how can you identify whether or not you have written the correct conversion

factor for the problem?

8. One liter is 1.06 quarts (qt). Write two possible conversion factors from this relationship.

9. The posted speed limit is 60 mi/hr. You are doing 120 km/hr in your Porsche convertible that

you just bought in Germany. Are you speeding? Explain. [1.0 mi = 1.6 km]

10. In the gym, you slip on two 45-lb barbell plates to a bar that weighs 45 lb. What is the mass

of the set-up in kilograms? [1.00 kg = 2.20 lbs]

11. A tabletop is 36 in long and 24 in wide. What is the area of the tabletop in square meters?

[1 in = 2.54 cm, exactly]

Information

Measured quantities always have some experimental error. Therefore, measured quantities are

regarded as inexact. The accuracy of a measured quantity is its agreement with a standard or

true value. In reality, we generally cannot know the true value of something we wish to measure.

We gain confidence that our measured value is close to the truth by repeating the measurement

many times. If our repeated measurements yield a set of data that differ very little from each

other, we have some confidence that the average of these measured values is close to the true

value. The repeatability of the measurements is called its precision. In general, we assume that

greater precision in a set of numbers makes it more likely that the average value will be accurate.

However, it is possible for a very precise set of values to be inaccurate. For example, a scientist

could make the same error in each of a set of measurements, which could happen if a key

measuring device were mis-calibrated. Conversely, it is possible that a set of widely scattered

values (poor precision) could have an average value that is very close to the true value, therefore

fortuitously resulting in high accuracy.

We express the precision of a number by writing all the repeatable digits and the first uncertain

digit from a measurement or calculation. The retained digits are called the significant figures

(sig. figs.) of the number. The following rules should be used to determine the number of

significant figures of a number and to establish the correct number of significant figures in the

answer to a calculation.

(1) For decimal numbers with absolute value >1, all digits are significant.

2.620 4 sig. figs. (The final 0 is significant.)

50.003 5 sig. figs.

(2) If there is no decimal point, zeroes that set magnitude only are not significant.

103,000 3 sig. figs. (The final three zeroes are not significant.)

But, 103,000• 6 sig. figs. (All digits to the left of the decimal are significant.)

(3) For decimal numbers with absolute value <1, start counting significant figures at the first

non-zero digit to the right of the decimal point.

0.0012 2 sig. figs.

0.00070 2 sig. figs.

But, 2.0070 5 sig. figs.

Decimal numbers with absolute value <1 are frequently expressed with standard scientific

exponential notation, particularly when the number is smaller than 0.1. When doing this, all

digits in the coefficient are significant; the power of 10 does not relate to significant digits.

1.2 x 10-3 2 sig. figs.

7.0 x 10-4 2 sig. figs

(4) In multiplication and division, the answer may have no more significant figures than the

number in the chain with the fewest significant figures.

(Here, 6.5 is the limit on the number of significant digits for the answer.)

(5) When adding or subtracting, the answer has the same number of decimal places as the

number with the fewest decimal places. The number of significant figures for the result,

then, is determined by the usual rules after establishing the appropriate number of decimal

places.

3.0081

+7.41

10.4181 = 10.42 2 decimal places and 4 sig. figs.

The rules for addition and subtraction may radically alter the number of significant figures for

the answer in a chain of mathematical calculations, as the following shows.

(Here, the subtraction 1.963 !1.960 results in a number with three decimal places but only

one significant figure; this limits the result for the division.)

(6) Exact numbers, which are inherently integers or are set by definition, are not limited in their

significant digits. Some exact numbers:

(a) All integer fractions: e.g., ½,⅓,⅞

(b) Counted numbers; e.g., “15 people”

(c) Conversions within a unit system:

12 inches / 1 foot (exactly!)

Large integer numbers that are clearly estimates are not exact; e.g., “15,000 people attended

the rally.”

Relationships between units in different unit systems are usually not exact:

2.2 lb. = 1.0 kg 2 sig. figs.

2.2046223 lb. = 1.0000000 kg 8 sig. figs.

But, the following inter-system conversion factors are now set by definition and are exact:

2.54 cm / 1 inch (exactly)

1 calorie / 4.184 Joules (exactly)

A way of getting around the ambiguity in significant figures for numbers like 103,000 is to

use standard scientific exponential notation, consisting of a coefficient whose magnitude is

greater than 1 and less than 10 multiplied by the appropriate power of ten. All digits in the

coefficient are significant.

1.03 x 105 3 sig. figs. 1.030 x 105 4 sig. figs.

1.0300 x 105 5 sig. figs. 1.03000 x 105 6 sig. figs.

Key Questions & Exercises

12. A one-gram standard reference weight is repeatedly weighed on an analytical balance. The

readings from the balance are as follows: 1.003 g, 0.9998 g, 1.005 g, 0.9995 g. Comment on

the precision and accuracy of these data.

13. The same one-gram reference weight is weighed on another analytical balance. The readings

from this balance are as follows: 0.9845 g, 1.0114 g, 0.9879 g, 1.0208 g. Comment on the

precision and accuracy of these data.

14. The same one-gram reference weight is weighed on a third analytical balance. The readings

from this balance are as follows: 1.237 g, 1.243 g, 1.238 g, 1.245 g. Comment on the

precision and accuracy of these data.

15. How is precision represented in reporting a measured value?

16. How many significant figures are there in each of the following numbers?

0.0037 20.03 300 300. 3.000 x 102

17. Use your calculator to carry out the following calculations and report the answers to the

correct number of significant figures:

x = (2)(39.0983) + (2)(51.996) + (7)(15.9994) (The first number in each

multiplication is an integer.)

18. A supermarket in London is selling cod for 12.98 £/kg. If the rate of exchange is $1.6220 =

1.0000 £, what is the price in dollars per pound? 1.000 kg = 2.205 lb

19. A hollow metal sphere has an outer diameter (o.d.) of 4.366 cm and an inner diameter (i.d.)

of 4.338 cm. What is the volume of the metal in the sphere? Express your answer to the

proper number of significant figures. [V = (4/3)πr3]