Lesson 4Lesson 4

Measuring & Significant DigitsMeasuring & Significant Digits

Anything in black letters = write it in your notes (‘knowts’)

Topic 1 – Units of Measurement & Metric Prefixes

Topic 2 – Scientific Notation

Topic 3 – Accuracy, Precision and Error

Topic 4 – Significant Digits

Topic 1 – Units of Measurement & Metric Prefixes

Measurements without units are useless!

“I walked 5 today.”

“The speed of light is 186,000

“I weigh 890”

“20 of water”

All measurements need units!

SI Base UnitsQuantity SI base

unitSymbol

Length meter m

Mass kilogram kg

Temperature kelvin K

Time second s

Number of Things

mole mol

Luminous intensity

candela cd

Electric current

ampere A

SI – International System of Units

We will use all of these in this class

Commonly Used Metric PrefixesPrefix Symbol Meaning Factor

mega M 1 million times larger than the base 106

kilo k 1000 times larger than the base 103

BASE Base Unit (meter, second, gram, etc)

deci d 10 times smaller than the base 10-1

centi c 100 times smaller than the base 10-2

milli m 1000 times smaller than the base 10-3

micro μ 1 million times smaller than the base 10-6

nano n 1 billion times smaller than the base 10-9

pico p 1 trillion times smaller than the base 10-12

Volume - Amount of space occupied by an object (remember?)

1 L = 1000 mL 1 mL = 1 cm3

Normal units used for volume:

Solids – m3 or cm3

Liquids & Gases – liters (L) or milliliters (ml)

The volume of a material changes with temperature, especially for gases.

Mass - Measure of inertia (remember?)

Weight - Force of gravity on a mass; measured in pounds (lbs) or Newtons.

Weight can change with location, mass does not

Energy – Ability to do work or produce heat.

Normal units used for energy:

SI – joule (J)

non-SI – calorie (cal)

How many joules are in a kilojoule?

How many calories are in a kilocalorie?

1 cal = 4.184 J

Temperature – measure of how cold or hot an object is.

Temperature – measure of the average kinetic energy of molecules.

Normal units used for temp:

SI – kelvin (K)

non-SI – celsius (°C) or Fahrenheit (°F)

yucky!

15.273 Celsiuskelvin TT

Celsius

Kelvin

100 divisions

100 divisions

100°CBoiling point

of water373.15 K

0°CFreezing point

of water273.15 K

mass

volumeDensity =

Normal units for density:

g/cm3, g/mL, g/L

Densities of Some Common Materials

Solids and Liquids Gases

MaterialDensity at

20°C (g/cm3)Material

Density at 20°C (g/L)

Gold 19.3 Chlorine 2.95

Mercury 13.6 Carbon dioxide 1.83

Lead 11.3 Argon 1.66

Aluminum 2.70 Oxygen 1.33

Table sugar 1.59 Air 1.20

Corn syrup 1.35–1.38 Nitrogen 1.17

Water (4°C) 1.000 Neon 0.84

Corn oil 0.922 Ammonia 0.718

Ice (0°C) 0.917 Methane 0.665

Ethanol 0.789 Helium 0.166

Gasoline 0.66–0.69 Hydrogen 0.084

Topic 2 – Scientific Notation

We will often work with really large or really small numbers in this class.

872,000,000 grams

0.0000056 moles

= 8.72 x 108 grams

= 5.6 x 10-6 moles

Scientific NotationStandard Notation

6.02 x 1023

coefficient exponent

The coefficient must be a single, nonzero digit, exponent must be an integer.

To multiply numbers written in scientific notation, multiply the coefficients and add the exponents.

(3 x 104) x (2 x 102) = (3 x 2) x 104+2 = 6 x 106

(2.1 x 103) x (4.0 x 10–7) = (2.1 x 4.0) x 103+(–7) = 8.4 x 10–4

Multiplication and Division

To divide numbers written in scientific notation, divide the coefficients and subtract the

exponents (top – bottom)

2

5

10 x 0.6

10 x 0.3 2510 x 0.6

0.3

310 x 5.0210 x 0.5

Coefficient needs to be between 1 and 10

Addition and Subtraction

When adding or subtracting in Sci. Not., the exponents must be the same.

(5.4 x 102) + (8.0 x 102)

= (5.4 + 8.0) x 102

= 13.4 x 102

= 1.34 x 103

Example

Solve each problem and express the answer in scientific notation.

a. (8.0 x 10–2) x (7.0 x 10–5)

b. (7.1 x 10–2) + (5 x 10–2)

a.

Multiply the coefficients and add the exponents.

(8.0 x 10–2) x (7.0 x 10–5)

= (8.0 x 7.0) x 10–2 + (–5)

= 56 x 10–7

= 5.6 x 10–6

b.

Rewrite one of the numbers so that the exponents match. Then add the coefficients

(7.1 x 10–2) + (5 x 10–2)

= (7.1 + 5) x 10–2

= 12.1 x 10–2

= 1.21 x 10–1

Topic 3 – Accuracy, Precision and Error

Accuracy - closeness of a measurement to the actual or accepted value.

Precision - closeness of repeated measurements to each other

The closeness of a dart to the bull’s-eye corresponds to the degree of accuracy. The closeness of several darts to one another

corresponds to the degree of precision.

Good Accuracy, Good Precision

Poor Accuracy, Good Precision

Poor Accuracy, Poor Precision

Darts on a dartboard illustrate the difference between accuracy and precision.

Accuracy and Precision

Error

Suppose you measured the melting point of a compound to be 78°C

Suppose also, that the actual melting point value (from reference books) is 76°C.

The error in your measurement would be 2°C.

alexperimentacceptederror

Error is always a positive value

How far off you are in a measurement doesn’t tell you much.

For example, lets say you have $1,000,000 in your checking account. When you balance your checkbook at the end of the month, you find that you are off by $175; error = $175

Now, lets be more realistic, you have $225 in your checking account and after balancing you are off by $175!

In both cases, there is an error of $175.

But in the first, the error is such a small portion of the total that it doesn’t matter as much as the second.

So, instead of error, percent error is more valuable.

Percent error compares the error to the size of the measurements.

100 x accepted

error error percent

Topic 4 – Significant Digits

In any measurement, the last digit is estimated

30.2°CThe 2 is estimated (uncertain) by the experimenter, another person may say 30.1 or 30.3

0.72 cm

9.3 mL

Increasing Precision

The significant figures in a measurement are the numbers that are part of the measurement.

Zeros that are NOT significant are called placeholders.

Rules for determining Significant Figures1. Every nonzero digit in a reported measurement is assumed to be significant.

2. Zeros appearing between nonzero digits are significant.

3. Leftmost zeros appearing in front of nonzero digits are not significant. They act as placeholders. By writing the measurements in scientific notation, you can eliminate such placeholding zeros.

4. Zeros at the end of a number and to the right of a decimal point are always significant.

5. Zeros at the rightmost end of a measurement that lie to the left of an understood decimal point are not significant if they serve as placeholders to show the magnitude of the number.

5 (continued). If such zeros were known measured values, then they would be significant. Writing the value in scientific notation makes it clear that these zeros are significant.

6. There are two situations in which numbers have an unlimited number of significant figures. The first involves counting. A number that is counted is exact.

6 (continued). The second situation involves exactly defined quantities such as those found within a system of measurement.

HOLY SMOKES!!

“Line Through” Method for Counting Sig Figs

1. If there is a decimal, start from the left and draw a line through any zeros, the numbers remaining are significant.

2. If there is no decimal, start the line from the right.

SOURCE: Skylar Morben, 2014 MRHS Graduate

A Shorter Method

How many significant digits?

100 1.00 0.23

0.0034 1.01 1005.4

0.10 100.0 54.0

How many significant digits are in the following measurements?

a) 150.31 grams b) 10.03 mL

c) 0.045 cm d) 4.00 lbs

e) 0.01040 m f) 100.10 cm

g) 100 grams h) 1.00 x 102 grams

i) 11 cars j) 2 molecules

An answer can’t be more accurate than the measurements it was

calculated from

Rules for Add/Subtracting Sig FigsThe answer to an +/- calculation should be rounded to the same number of decimal places as the measurement with the least number of decimal places.

3.2 cm

2.05 cm

+ 1.1 cm 3.15 cm

32.10 g

+ 5.0012 g 37.1012 g

37.10 g

Rules for Mult/Division Sig Figs The answer to a x/÷ calculation should be rounded to the same number of sig figs as the measurement with the least number of sig figs.

2.0 cm

x 1.89 cm 2 3.78 cm

3.8 cm2

8.19

8.1

g

mL 1.01111111... g mL

1.0 g mL

Always round your final answer off to the correct number of significant digits.

Draw a box around the significant digits in the following measurements.

a)2.2000 b) 0.0350 c) 0.0006d)0.0089 e) 24,000 f) 4.360 x 104

g)0.0708 h) 1200 i) 0.6070k) 21.0400 l) 0.007 m) 5.80 x 10-3

Round off each of the following numbers to two significant figures.

a) 86.048 b) 29.974 c) 6.1275

d)0.008230 e) 800.7 f) 0.07864

g) 0.06995 h) 7.096 i) 8000.10

Express each of the following numbers in standard scientific notation with the correct number of significant digits.

a)0.00000070b)25.3c)825,000d)826.7e)43,500f)65.0g)0.000320h)0.0432

Perform the following arithmetic. Round the answers to the proper number of sig. figs. Box in your final answer and don’t forget units!

a)2.41 cm x 3.2 cm b) 4.025 m x 18.2 m

c) 81.4 g 104.2 cm3

Perform the following arithmetic. Round the answers to the proper number of sig. figs. Box in your final answer and don’t forget units!

d) 822 mi 0.028 hr

e) 10.89 g / 1 mL