1. Measurement and Problem Solving
Chapter 2
Measurement and Problem Solving

2. Homework Exercises (optional, in Tro textbook) 1 through 27 (odd)
Problems (in Tro textbook)
29-65 (odd)
67-91 (odd)
93-99 (odd)
Cumulative Problems (in Tro textbook)
(odd)
Highlight Problems (optional, in Tro textbook)
119, 121

3. Scientific Notation: Writing Large and Small Numbers
A system in which an ordinary decimal number (m) is expressed as a product of a number between 1 and 10, multiplied by 10 raised to a power (n)
Used to write very large or very small numbers
Based on powers of 10

4. Scientific Notation: Writing Large and Small Numbers
Consists of a number (coefficient) followed by a power of 10 (x 10n)
Negative exponent: Number is less than 1
Positive exponent: Number is greater than 1
exponent
Coefficient or
decimal part
exponential term
or part

5. Scientific Notation: Writing Large and Small Numbers
In an ordinary cup of water there are:
Each molecule has a mass of:
7,910,000,000,000,000,000,000,000 molecules
gram
In scientific notation:
7.91 х 1024 molecules
2.99 х gram
In scientific work, very large or very small numbers are frequently encountered
The example of one ordinary amount of water (cup)

6. To Express a Number in Scientific Notation:
For small numbers (<1):
Locate the decimal point
Move the decimal point to the right to give a number (coefficient) between 1 and 10
Write the new number multiplied by 10 raised to the “nth power”
where “n” is the number of places you moved the decimal point. It has a negative sign
If the decimal point is moved to the right, then the exponent is a negative number (× 10-n)

7. To Express a Number in Scientific
For large numbers (>1):
Locate the decimal point
Move the decimal point to the left to give a number (coefficient) between 1 and 10
Write the new number multiplied by 10 raised to the “nth power”
where “n” is the number of places you moved the decimal point. It has a positive sign.
If the decimal point is moved to the left, the exponent is a positive number (× 10n)

8. Examples Write each of the following in scientific notation 12,500
0.0202
37,400,000

9. 12,500 Examples Decimal place is at the far right
Move the decimal place to a position between the 1 and 2
Coefficient (1.25)
The decimal place was moved 4 places to the left (large number) so exponent is positive
1.25x104

10. Examples
0.0202
Move the decimal place to a position between the 2 and 0
Coefficient (2.02)
The decimal place was moved 2 places to the right (small number) so exponent is negative
2.02x10-2

11. Examples 37,400,000 Decimal place is at the far right
Move the decimal place to a position between the 3 and 7
Coefficient (3.74)
The decimal place was moved 7 places to the left (large number) so exponent is positive
3.74x107

12. Examples
Move the decimal place to a position between the 1 and 0
Coefficient (1.04)
The decimal place was moved 5 places to the right (small number) so exponent is negative
1.04x10-5

13. Scientific Notation and Calculators
Enter the coefficient (number)
Push the key:
Then enter only the power of 10
If the exponent is negative, use the key:
DO NOT use the multiplication key:
to express a number in sci. notation EE or
EXP
(+/-)
EE key includes the X 10 value X

14. Converting Back to a Standard Number
Determine the sign of the exponent, n
If n is + the decimal point will move to the right (gives a number greater than one)
If n is – the decimal point will move to the left (gives a number less than one)
Determine the value of the exponent of 10
The “power of ten” determines the number of places to move the decimal point

15. Using Scientific Notation
To compare numbers written in scientific notation
First compare the exponents of 10
The larger the exponent, the larger the number
If the exponents are the same, then compare coefficients directly
Which number is larger?
21.8 х or х 104
2.18 х > х 104

16. Significant Figures: Writing Numbers to Reflect Precision
Two kinds of numbers
Counted (exact)
Measured
There are two kinds of numbers. Those that are counted and those that are defined. Counted numbers have an exact value, those that are measured can not have an exact value
Counted numbers do not involve a measuring device

17. Measured Numbers
Scientific numbers are reported so that all digits are certain except the last digit which is estimated
A measurement:
involves reading a measuring device
always has some amount of uncertainty
uncertainty comes from the tool used for comparison
e.g. some rulers show smaller divisions (more precise) than others

18. Measured Numbers
Always have to estimate the value between the two smallest divisions on a measuring device
Every person will estimate it slightly differently, so there is some uncertainty present as to the true value
2.8 cm
2.9 cm
2.8 to 2.9 cm

19. Significant Figures: Writing Numbers to Reflect Precision
Scientific numbers are reported so every digit is certain except the last which is estimated
To indicate the uncertainty of a single measurement scientists use a system called significant figures
Significant figures: All digits known with certainty plus one digit that is uncertain

20. Counting Significant Figures
The last digit written in a measurement is the number that is considered to be uncertain (estimated)
Unless stated otherwise, the uncertainty in the last digit is ±1
The precision of a measured quantity is determined by number of sig. figures
A zero in a measurement may or may not be significant
significant zeros
place-holding zeros (not significant)

21. Counting Significant Figures
Nonzero integers are always significant
Zeros (may or may not be significant)
It is determined by its position in a sequence of digits in a measurement
Leading zeros never count as significant figures
Captive (interior) zeros are always significant
Trailing zeros are significant if the number has a decimal point

22. Exact Numbers They are either
Exact numbers occur in definitions or in counting
Numbers known with no uncertainty
Unlimited number of significant figures (never limit the no. of sig. figures in a calculation)
They are either
Counting numbers
7 pennies, 6 pills, 4 chairs
Defined numbers
12 in = 1 ft
1 gal = 4 quarts
1 minute = 60 seconds
Exact value is known
Do not affect the number of sig figs in an answer

23. Significant Figures in Calculations
Calculations cannot improve the precision of experimental measurements
The number of significant figures in any mathematical calculation is limited by the least precise measurement used in the calculation
Two operational rules to ensure no increase in measurement precision
addition and subtraction
multiplication and division

24. Significant Figures in Calculations: Multiplication and Division
Product or quotient has the same number of significant figures as the factor with the fewest significant figures
Count the number of significant figures in each number. The least precise factor (number) has the fewest significant figures
Rounding
Round the result so it has the same number of significant figures as the number with the fewest significant figures

25. Rounding
To round the result to the correct number of significant figures
If the last (leftmost) digit to be removed:
is less than 5, the preceding digit stays the same (rounding down)
is equal to or greater than 5, the preceding digit is rounded up
In multiple step calculations, carry the extra digits to the final result and then round off

26. Multiplication/Division Example:
4 SF
5 SF
3 SF
2.1
2 SF
2 SF
The number with the fewest significant figures is 1.1 so the answer has 2 significant figures

27. Multiplication/Division Example:
3 SF
4 SF
5 SF
3 SF
× × 273 =
2.29
The number with the fewest significant figures is 273 so the answer has 3 significant figures

28. Significant Figures in Calculations: Addition and Subtraction
Sum or difference is limited by the quantity with the smallest number of decimal places
Find quantity with the fewest decimal places
Round answer to the same decimal place

29. Addition/Subtraction Example:
1 d.p.
3 d.p.
2 d.p.
236.2
1 d.p.
The number with the fewest decimal places is so the answer should have 1 decimal place

30. Measurement The most useful tool of the chemist
Most of the basic concepts of chemistry were obtained through data compiled by taking measurements
How much…?
How long…?
How many...?
These questions cannot be answered without taking measurements
The concepts of chemistry were discovered as data was collected and subjected to the scientific method

31. Measurement
The estimation of the magnitude of an object relative to a unit of measurement
Involves a measuring device
e.g. meter stick, scale, thermometer
The device is calibrated to compare the object to some standard (inch/centimeter, pound/kilogram)
Quantitative observation with two parts: A number and a unit
Number tells the total of the quantity measured
Unit tells the scale (dimensions)
Quantitative observation: A numerical measurement of a quantity or value

32. Measurement A unit is a standard (accepted) quantity
Describes what is being added up
Units are essential to a measurement
For example, you need “six of sugar”
teaspoons?
ounces?
cups?
pounds?

33. The Basic Units of Measurement
Units tells the magnitude of the standard
Two most commonly used systems of units of measurement
U.S. (English) system: Used in everyday commerce (USA and Britain*)
Metric system: Used in everyday commerce and science (The rest of the world)
SI Units (1960): A modern, revised form of the metric system set up to create uniformity of units used worldwide (world’s most widely used)
Common measurements in commerce (gas stations, supermarkets)

34. The Standard Units: The Metric/SI System
A decimal system of measurement based on the meter and the gram
It has a single base unit per physical quantity
All other units are multiples of 10 of the base unit
The power (multiple) of 10 is indicated by a prefix

35. The Standard Units: The Metric System
In the metric system there is one base unit for each type of measurement
length
volume
mass
The base units multiplied by the appropriate power of 10 form smaller or larger units
The prefixes are always the same, regardless of the base unit
milligrams and milliliters both mean 1/1000 of the base unit

36. The Standard Units: Length
Meter
Base unit of length in metric and SI system
About 3 ½ inches longer than a yard
1 m = yd
Not convenient for description of small things
Can be subdivided or multiplied by use of metric prefixes
The linear extent in space from one end to another

37. The Standard Units: Length
Other units of length are derived from the meter
Commonly use centimeters (cm)
1 m = 100 cm
1 inch = 2.54 cm (exactly)
Not convenient for description of small things
Can be subdivided or multiplied by use of metric prefixes
The linear extent in space from one end to another

38. The Standard Units: Volume
Volume = side × side × side
Measure of the amount of three-dimensional space occupied by a object
Derived from length
SI unit = cubic meter (m3)
Metric unit = liter (L) or 10 cm3
Commonly measure smaller volumes in cubic centimeters (cm3)
Volume = side × side × side
No base unit, derived from base unit of length
Cubic meter too large for most lab work so use smaller units such as liter or milliliter

39. The Standard Units: Volume
Since it is a three-dimensional measure, its units have been cubed
SI base unit = cubic meter (m3)
This unit is too large for practical use in chemistry
Take a volume 1000 times smaller than the cubic meter, 1dm3
No base unit, derived from base unit of length
Cubic meter too large for most lab work so use smaller units such as liter or milliliter

40. The Standard Units: Volume
Metric base unit = 1dm3 = liter (L)
1L = qt
Commonly measure smaller volumes in cubic centimeters (cm3)
Take a volume 1000 times smaller than the cubic decimeter, 1cm3

41. The Standard Units: Volume
Metric base unit = 1dm3 = liter (L)
1L = qt
Commonly measure smaller volumes in cubic centimeters (cm3)
Take a volume 1000 times smaller than the cubic decimeter, 1cm3

42. The Standard Units: Volume
The most commonly used unit of volume in the laboratory: milliliter (mL)
1 mL = 1 cm3
1 L= 1 dm3 = 1000 mL
1 m3 = 1000 dm3 = 1,000,000 cm3
Use a graduated cylinder or a pipette to measure liquids in the lab

43. The Standard Units: Mass
Measure of the total quantity of matter present in an object
SI unit (base) = kilogram (kg)
Metric unit (base) = gram (g)
Commonly measure mass in grams (g) or milligrams (mg)
1 kg = 1000 g
1 g = 1000 mg
1 kg = pounds
1 lb = g
See table 2.1 page 21

44. Prefixes Multipliers One base unit for each type of measurement
Length (meter), volume (liter), and mass (gram*)
The base units are then multiplied by the appropriate power of 10 to form larger or smaller units
The names of larger and smaller units are made by attaching a prefix to the base unit name. The meaning of the prefix remains constant.
base unit = meter, liter, or gram

45. Prefixes Multipliers (memorize)
× base unit
Mega (M) 1,000,
Kilo (k) 1,
Base
Deci (d)
Centi (c)
Milli (m)
Micro (µ)
Nano (n)
meter
liter
gram

46. n < µ < m < c < base < k < M
Prefix Multipliers
For a particular measurement:
Choose the prefix which is similar in size to the quantity being measured
Keep in mind which unit is larger
e.g. A kilogram is larger than a gram, so there must be a certain number of grams in one kilogram
Choose the prefix most convenient for a particular measurement
n < µ < m < c < base < k < M

47. Converting from One Unit to Another: Equalities
A fixed relationship between two quantities
Shows the relationship between two units that measure the same quantity
The relationships are exact, not measured
1 min = 60 s
12 inches = 1 ft
1 dozen = 12 items (units)
1L = 1000 mL
16 oz = 1 lb
4 quarts = 1 gallon
The relationships are exact by definition, whether it measuring length, volume, or mass

48. Converting from One Unit to Another: Conversion Factors
Many problems in chemistry involve a conversion of units
Conversion factor: An equality expressed as a fraction
Used as a multiplier to convert a quantity in one unit to its equivalent in another unit
May be exact or measured
Both parts of the conversion factor should have the same number of significant figures

49. Solving Multistep Conversion Problems: Dimensional Analysis Example (Conversion Factors Stated within a Problem)
The average person in the U.S. consumes one-half pound of sugar per day. How many pounds of sugar would be consumed in one year?
State the initial quantity given (+unit): One year
State the final quantity to find (+unit): Pounds
Write a sequence of units (map) which begins with the initial unit and ends with the desired unit:
year day pounds

50. Solving Multistep Conversion Problems: Dimensional Analysis Example
For each unit change,
State the equalities:
Every equality will have two conversion factors
365 days = 1 year
0.5 lb sugar =1day
year day pounds

51. Solving Multistep Conversion Problems: Dimensional Analysis Example
State the conversion factors:
Set Up the problem:

52. Guide to Problem Solving when Working Dimensional Analysis Problems
Identify the known or given quantity and the units of the new quantity to be determined
Write out a sequence of units which starts with your initial units and ends with the desired units (“solution map”)
Write out the necessary equalities and conversion factors
Perform the mathematical operations that connect the units
Check that the units cancel properly to obtain the desired unit
Does the answer make sense?

53. Density
The ratio of the mass of an object to the volume occupied by that object
Tells how tightly the matter within an object is packed together
Units for solids and liquids =
1 cm3 = 1 mL so can also use
Unit for gases = g/L
Density: solids > liquids >>> gases
g/cm3
g/mL

54. Density Can use density as a conversion factor between mass and volume
Density of some common substances given in Table 2.4, page 33
You will be given any densities on tests EXCEPT water
Density of water is 1.0 g/cm3 at room temperature
1.0 mL of water weighs how much?
How many mL of water weigh 15 g?

55. Density To determine the density of an object
Use a scale to determine the mass
Determine the volume of the object
Calculate it if possible (cube shaped)
Can also calculate volume by determining what volume of water is displaced by an object
Volume of Water Displaced = Volume of Object

56. Density Problem
Iron has a density of 7.87 g/cm3. If 52.4 g of iron is added to 75.0 mL of water in a graduated cylinder, to what volume reading will the water level in the cylinder rise?

57. Solve for volume of iron
Density Problem
Solve for volume of iron

58. End