CHAPTER 2: Measurements and Calculations

[Pages:19]CHAPTER 2: Measurements and Calculations

MEASUREMENT UNITS

measurement: a number with attached units To measure, one uses instruments = tools such as a ruler, balance, etc.

All instruments have one thing in common: UNCERTAINTY!

INSTRUMENTS CAN NEVER GIVE EXACT MEASUREMENTS!

2.4 - 2.5 SIGNIFICANT FIGURES (also called "Sig Figs" or "Significant Digits")

When a measurement is recorded, all the numbers known with certainty are given along with the last number, which is estimated. All the digits are significant because removing any of the digits changes the measurement's uncertainty.

Example: Using Rulers A, B, and C below, indicate the measurement to the line indicated for each ruler. Assume these are centimeter rulers, so show the each measurement has units of cm. Circle the estimated digit for each measurement.

Ruler A

0

1

2

3

4

5

Ruler B

0

1

2

3

4

5

Ruler C

4.1 4.2 4.3 4.4

A

B

C

Increment of the smallest markings on ruler

# d.p. needed

Measurement

# of sig figs

Thus, a measurement is always recorded with one more digit than the smallest markings on the instrument used, and measurements with more sig figs are usually more accurate.

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Guidelines for Determining Number of Sig Figs (if the measurement is given):

Count the number of digits in a measurement from left to right:

1. When a decimal point is present: ? For measurements 1, count all the digits (even zeros). ? 60.2 cm has 3 sig figs, 5.0 m has 2 sig figs, 150.00 g has 5 s.f.

? For measurements less than 1, start with the first nonzero digit and count all digits (even zeros) after it. ? 0.011 mL, 0.0050 g, and 0.00022 kg each have 2 sig figs

2. When there is no decimal point: ? Count all non-zero digits and zeros between non-zero digits ? 125 g has 3 sig figs and 107 mL has 3 sig figs ? Placeholder zeros may or may not be significant ? 1000 g may have 1, 2, 3 or 4 sig figs

Example: How many significant digits do the following numbers have?

a. 105 _____

b. 90.40 _____

c. 100.00 _____

d. 0.0050 _____

2.1 LARGE AND SMALL NUMBERS (or SCIENTIFIC NOTATION)

Some numbers are very large or very small difficult to express.

Avogadro's number = 602,000,000,000,000,000,000,000 an electron's mass = 0.000 000 000 000 000 000 000 000 000 91 kg

To handle such numbers, we use a system called scientific notation. Regardless of their magnitude, all numbers can be expressed in the form

N ? 10n

where N =digit term= a number between 1 and 10, so there can only be one number to the left of the decimal point: #.####

n = an exponent = a positive or a negative integer (whole #).

To express a number in scientific notation: ? Count the number of places you must move the decimal point to get N between 1 and 10.

Moving decimal point to the right (if # < 1) negative exponent. Moving decimal point to the left (if # > 1) positive exponent.

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Example: Express the following numbers in scientific notation (to 3 sig figs):

555,000 __________________ 0.000888 __________________ 602,000,000,000,000,000,000,000 ___________________________

Also, in some cases the number of sig figs in a measurement may be unclear:

For example,

Ordinary form

Scientific Notation

Express 100.0 g to 3 sig figs: ___________ ______________

Express 100.0 g to 2 sig figs: ___________ ______________

Express 100.0 g to 1 sig fig:

___________ ______________

Thus, some measurements--usually those expressing large amounts--must be expressed in scientific notation to accurately convey the number of sig figs.

ROUNDING OFF NONSIGNIFICANT DIGITS

It is safer to NEVER round or truncate, but to indicate the last significant digit by underlining it and keeping one extra digit. The textbook explains unbiased rounding, and below we use "normal" rounding (biased toward even numbers). You must be able to round answers if necessary using the normal method, but only to present final results.

How do we eliminate nonsignificant digits? ? If first nonsignificant digit < 5, just drop ALL nonsignificant digits ? If first nonsignificant digit 5, raise the last sig digit by 1 then

drop ALL nonsignificant digits

last significant digit

72.58643 g

first nonsignificant digit

For example, express 72.58643 with 3 sig figs: 72.58643 "t"o "3 s"ig"fig"s # _______________

Example: Express each of the following with the number of sig figs indicated:

a. 376.276

"t"o "3 s"ig"fig"s #

b. 500.072

"t"o "4 s"ig"fig"s #

c. 0.00654321 "t"o "3 s"ig"fig"s #

d. 1,234,567 "t"o "5 s"ig"fig"s #

_______________________ _______________________ _______________________ _______________________

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e. 2,975

"t"o "2 s"ig"fig"s #

_______________________

Be sure to express measurements in scientific notation when necessary to make it clear how

many sig figs there are in the measurement.

2.5 SIGNIFICANT FIGURES IN CALCULATIONS

ADDING/SUBTRACTING MEASUREMENTS

When adding and subtracting measurements, your final value is limited by the measurement with the largest uncertainty--i.e. the number with the fewest decimal places.

Ex 1: 106.61 + 0.25 + 0.195 = 107.055 107.055 tocorrect # of sigfigs ______________

Ex 2: 725.50 ? 103 = 622.50

622.50 tocorrect # of sigfigs ______________

MULTIPLYING/DIVIDING MEASUREMENTS

When multiplying or dividing measurements, the final value is limited by the measurement with the least number of significant figures.

Ex 1: 106.61 ? 0.25 ? 0.195 = 5.1972375 5.1972375 tocorrect # of sigfigs ____________

Ex 2: 106.61 ? 91.5 = 9754.815 9754.815 tocorrect # of sigfigs _____________

SOLVE: Ex. 1: 7.4333 g + 8.25 g + 10.781 g = _________________________

Ex. 2: 13.50 cm ? 7.95 cm ? 4.00 cm = _________________________

# s.f:

# dp:

Ex. 3: 9.75 mL - 7.35 mL = _________________________

Ex. 4:

101.755 g

= _________________________

25.75 cm ? 10.25 cm ? 8.50 cm

# s.f:

# dp:

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MULTIPLYING/DIVIDING WITH EXPONENTIAL NUMBERS:

When multiplying or dividing measurements with exponents, use the digit term (N in "N ?10n") to determine number of sig figs.

Ex. 1: (6.02?1023)(4.155?10-9) = 2.50131?1015

How do you calculate this using your scientific calculator? Step 1. Enter "6.02?1023" by pressing:

6.02 then EE or EXP (which corresponds to "?10^") then 23

Your calculator should look similar to:

6.02 x1023

Step 2. Multiply by pressing: ?

Step 3. Enter "4.155? 10-9" by pressing:

4.155 then EE or EXP (which corresponds to "?10") then (-) 9 (Be sure to push the Negate button, not the Subtract mathematical operation.)

Your calculator should look similar to:

4.155 x10-9

Step 4. Get the answer by pressing: =

Your calculator should now read

2.50131 x1015

The answer with the correct # of sig figs = ___________________

Be sure you can do exponential calculations with your calculator. Most of the calculations we do in chemistry involve very large and very small numbers with exponential terms.

Ex. 2: (3.75?1015) (8.6?104) = 3.225?1020 tocorrect#of sigfigs ___________________

Ex. 3:

1.90 !1015 2.500 !108

= 760000

tocorrect # of sigfigs ___________________

SIGNIFICANT DIGITS AND EXACT NUMBERS

Although measurements can never be exact, we can count an exact number of items. For example, we can count exactly how many students are present in a classroom, how many M&Ms are in a bowl, how many apples in a barrel.

2.6 USING UNITS IN CALCULATIONS

Unit equation: Simple statement of two equivalent values

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Conversion factor = unit factor = equivalents:

- Ratio of two equivalent quantities

Unit equation: 1 dollar = 10 dimes

Unit factor: 1dollar or 10 dimes 10 dimes 1dollar

Unit factors are exact if we can count the number of units equal to another or if both quantities are in the same system of measurement--i.e., both in the metric system (e.g. cm and meters) or in the English system (inches and feet).

For example, the following unit factors and unit equation are exact:

365.25 days 1day 12 in. 1m and 1 yard 3 feet

1 year

24 hours 1foot 100 cm

Exact equivalents have an infinite number of sig figs

never limit number of sig figs!

Note: When the relationship between two units or items is exact, the "" (meaning "equals exactly") is used instead of the basic "=" sign.

Equivalents based on measurements or relating measurements from two different systems are inexact or approximate because they contain uncertainty, such as

1.61km 1mile

65 mi hour

3.00 ?108 m s

Approximate equivalents do limit the sig figs for the final answer.

SOLVING MULTSTEP CONVERSION PROBLEMS (or DIMENSIONAL ANALYSIS PROBLEM SOLVING)

1. Write the units for the answer. 2. Determine what information to start with.

3. Arrange all unit factors (showing them as fractions with units), so all of the units cancel except those needed for the final answer.

4. Check for the correct units and the correct number of sig figs in the final answer.

Example 1: If a marathon is 26.2 miles, then a marathon is how many yards? (1 mile5280 feet, 1 yard3 feet)

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Example 2: You and a friend decide to drive to Portland, which is about 175 miles from Seattle. If you average 99 kilometers per hour with no stops, how many hours does it take to get there? (1 mile = 1.609 km)

Example 3: The speed of light is about 2.998?108 meters per second. Express this speed in miles per hour. (1 mile=1.609 km, 1000 m1 km)

2.2 Basic Units of Measurement

International System or SI Units (from French "le Syst?me International d'Unit?s") ? standard units for scientific measurement

Metric system: A decimal system of measurement with a basic unit for each type of measurement

quantity length mass volume time

basic unit (symbol) meter (m) gram (g) liter (L) second (s)

quantity length mass time

temperature

SI unit (symbol) meter (m)

kilogram (kg) second (s) Kelvin (K)

Metric Prefixes Multiples or fractions of a basic unit are expressed as a prefix

Each prefix = power of 10 The prefix increases or decreases the base unit by a power of 10.

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Prefix tera giga mega kilo

deci

Symbol T G M k

d

centi

milli micro nano pico femto

c

m ? (Greek "mu")

n p f

Metric Conversion Factors Ex. 1 Complete the following unit equations:

Multiple/Fraction

1,000,000,000,000 1012 1,000,000,000 109 1,000,000 106 1000 103 0.1 1 10-1

10

0.01 1 10-2

100

0.001 1 10-3

1000

10?6 10?9 10?12 10?15

a. 1 kg ________ g b. 1 m ________ nm c. 1 cm ________ m

d. 1 L ________ mL

g. 1 s _______ fs

e. 1 g ________ ?g

h. 1 m _______ pm

f. 1 megaton ________ tons

Note: Although scientists use ? g to abbreviate microgram, hospitals avoid using the

Greek letter ? in handwritten orders since it might be mistaken for an m for milli

-- i.e., an order for 200 ? g might be mistaken to be 200 mg which would lead to an

overdose that's 1000 times more concentrated.

Instead, hospitals use the abbreviation mcg to indicate micrograms.

Writing Unit Factors: Complete the following unit equations then write two unit factors for each equation:

a. 1 km __________ m

b. 1 g ___________ mg

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