CHAPTER 2 NOTESMeasurements and Solving Problems

Problem Set:

Scientific Method1. Problem

state it clearly – usually as a question2. Gather information

do some research on your problem3. Hypothesis

a suggested solution4. Procedure

experiment and examine the situation to check the hypothesis5. Data

Note everything your senses can gather. Record the data and keep careful records.6. Analysis

Put the data in order- charts/tables. Figure out the meaning of the data7. Conclusion

Explain the data. State whether or not it supports the hypothesis.

Scientific Method• Theory

• A hypothesis that has been rigorously tested, and not found faulty, usually also having been found somewhat useful.

• Law• A readily demonstrable fact, that cannot be disproven.

2.1 Units of Measurement Measurement – comparison of an object to a standard. The problem is, what do you use as a standard?

Standard should be an object or natural phenomenon of constant value, easy to preserve and reproduce, and practical in size.

2.1 Units of Measurement• The SI System• SI = Standard International• Important base units to know:

Quantity SI Base Unit English Equivalent

Mass Kilogram (Kg) 1 m = 39.36 in

Length Meter (m) 1 kg = 2.2 lbs

Volume Liter or cm3

Time Second (s)

Temperature Kelvin (K) K = °C + 273.15

Amount of a substance Mole (m)

2.1 Units of Measurement• Important prefixes(multiples of base units) to know:

Prefix Abbreviation Meaning Example

tera  T 1012 1 terameter = 1,000,000,000,000

giga G  109 1 gigameter = 1,000,000,000

mega-  M 106 1 megameter = 1,000,000

kilo- k  103 1 kilogram = 1000

hecto-  H 102 1 hectometer = 100

deka D  101 1 dekameter = 10

BASE Meter, liter, gram, second

1

deci- d  10-1 1 deciliter = 10 .1

centi-  c 10-2 1 centimeter = 100 .01

milli- m  10-3 1 millimeter= 1,000 .001

micro-  u 10-6 1 micrometer = 1,000,000 .000001

nano- n  10-9 1 nanometer = 1,000,000,000 .000000001

pico- p  10-12 1 picometer = 1,000,000,000,000 .000000000001

2.3 Using Scientific Measurement• Significant Figures (Digits) - “Sig Figs”

Definition: digits in a measurement that are known + 1 estimated digit 1.15 ml implies 1.15 + 0.01 ml

The more significant digits, the more reproducible the measurement is.

• These are the numbers that “count!”

Ex1: π = 22/7 = 3.1415927 what do math teachers let you use?

Ex2: You collect a paycheck for a 40 hour week – what’s the difference between getting paid pi vs. 3.14 ?

Rules for finding the # of sig figs1. All non-zeros are significant ex. 7 [1] 77 [2] 4568 [4] 2. Zeros between non-zeros are significant

ex. 707 [3] 7053 [4] 7.053 [4]3. Zeroes to the left of the first nonzero digit serve only to fix the

position of the decimal point and are not significantex: 0.0056 [2] 0.0789 [3] 0.0000001 [1]

4. In a number with digits to the right of a decimal point, zeroes to the right of the last nonzero digit are significant

ex: 43 [2] 43.00 [4] 43.0 [3] 0.00200 [3]0.040050 [5]

5. In a number that has no decimal point, and that ends in zeroes (ex. 3600), the zeroes at the end may or may not be significant (it is ambiguous). To avoid ambiguity, express in scientific notation and show in the coefficient the number of significant digits. ex. 3600 = 3.6 x 103 [2]

Scientific Notation• A way to express very small or very large numbers• Example:

• 12345 = 1.2345 x 104

• 0.00456 = 4.56 x 10-3

Coefficient – must be between 1 and 9

Base

Exponent – the # of times the decimal was moved

(+) to the left

(-) to the right

Scientific Notation• 56934 =

• 0.0000037 =

• 2.347 x 10-3 =

• 8.98736 x 105 =

Reverse it!

(+) right

(-) left

Counting significant digits1. Convert to scientific notation2. Disappearing zeroes just hold the decimal point, they aren’t significant

• Ex1:700 [ ] - means “about 700 people at a football game”700. [ ] - means “exactly 700 ......”700.0 [ ] - means “teacher weighs exactly 700.0 lbs”

• Other examples 0.5 [ ] 0.50 [ ] 0.050 [ ] 

• Sig. figs apply to scientific notation as well 9.7 x 10 2 = 970 [ ]1.20 x 10 -4 = .000120 [ ]

Calculating with Measurements ( Sig Fig Math )• Rounding Rules • XY ---------------------> Y

When Y > 5, increase X by 1 When Y < 5, don’t change X When Y = 5,

• If X is odd, increase X by 1• If X is even, then don’t change X

 • Ex1: round to 3 sig figs 35.27 =

87.24 = 95.25 = 95.15 =

• Note - the “5” rule only applies to a “dead even” 5 - if any digit other than 0 follows a 5 to be rounded, then the number gets rounded up without regard to the previous digit.

• Ex2: round to 3 sig figs 35.250000000000000000000000001 =

Calculating rules:1. Multiplying or dividing – round results to the smaller # of

sig. figs in the original problem. 

• Ex1: 3.10 cm Ex2: 7.9312 g

4.102 cm / 0.98 m

x 8.13124 cm

Calculating rules:2. Adding or subtracting - round to the last common

decimal place on the right. • Ex1: 21.52 Ex2: 73.01234 g + 3.1? - 73.014?? g  • Note - exact conversion factors do not limit the # of sig

figs - the final answer should always end with the # of sig figs that started the problem

 ex. convert 7866 cm to m

2.1 Units of Measurement• Factor Label Method (Dimensional Analysis)• A method of problem solving that treats units like algebraic

factors• Rules

1. Put the known quantity over the number 1.2. On the bottom of the next term, put the unit on top of the previous term.3. On top of the current term put a unit that you are trying to get to.4. On the top and bottom of the current term, put in numbers in order to create equality.5. If the unit on top is the unit of your final answer, multiply/divide and cancel units. If not, return to step # 2.6. As far as sig figs are concerned, end with what you start with!

2.1 Units of Measurement• Factor Label Method (Dimensional Analysis)• Ex1 - convert 26 inches to feet 

• Ex2 - convert 1.8 years to seconds

 

• Ex3 - convert 2.50 ft to cm if 1 inch = 2.54 cm  

 

2.1 Units of Measurement• Factor Label Method (Dimensional Analysis)• Ex4 - convert 75 cm to Hm

 

 • Ex5 - convert 150 g to ug  

 • Ex6 - convert 0.75 L to cm3 (1 cm3 = 1 mL)

 

2.1 Units of Measurement• Density – ratio of mass to volume• The common density units are:

g/cm3 for solids g/ml for liquids g/L for gases

• Formula is D = m/v • Density is

a) a characteristic b) and intensive property c) temperature dependent

• Two ways to find volume in density problems:1. Water displacement 2. Volume formula

 • Note: the density is the same no matter what is the size or shape of

the sample.

2.1 Units of Measurement• Ex1: Find the density of an object with

m= 10g and v=2 cm3

 • Ex2: A cube of lead 3.00 cm on a side has a mass of

305.0 g. What is the density of lead?   First, calculate it’s volume:

Next, calculate the density:• Density = mass/volume =

2.1 Units of Measurement• Ex 3: A graduated cylinder contains 25 mL of water.

When a 4.5 g paper clip is dropped into the water, the water level rises to 36 mL. What is the density of the paper clip?

2.3 Using Scientific Measurement• Precision vs. Accuracy Precision Accuracy• Reproducibility• Check by repeating measurements• A function of the instrument• Poor precision results from poor

technique

• Correctness – closeness to the true value

• Check by using a different method• A function of the user• Poor accuracy results from

procedural or equipment flaws

2.3 Using Scientific Measurement• Precision vs. Accuracy

good precision & good accuracy

poor accuracy but good precision

good accuracy but poor precision

poor precision & poor accuracy

2.3 Using Scientific Measurement• Percent Error - experiments don’t always give true results -

error is pretty much a given • Observed value (experimental value) - data found in an

experiment• True value (accepted value, theoretical value) - data that is

generally accepted as true • Percent (%) error = (experimental – true) x 100

true value • +/- shows show the direction of the error - values are either too

high or too low• Note: some texts teach that percent error should be treated as

absolute value - I say you should use +/- in order to show direction of error and better analyze your experiments.

• Ex1: 66 Co is the answer in your experiment 65 CO is the theoretical value

2.3 Using Scientific Measurement• Two important points:

• Uncertainty in Measurement• making a measurement usually involves comparison with a unit or

a scale of units

When making a measurement, include all readable digits and 1 estimated digit

always read between the lines! the digit read between the lines is always uncertain

2.3 Using Scientific Measurement• Uncertainty in Measurement

• when measuring, include all readable digits and 1 estimated digit

• if the measurement is exactly half way between lines record it as 0.5

• if it is a little over, record .7 or .8• if it is a little under, record .2 or .3• You would read this as 18.0 mL and not

18.5 mL.

2.4 Solving Quantitative Problems

1. Analyze – read carefully, list data with units, draw a picture

2. Plan – list conversion factors, show that units will work

3. Compute – use a calculator and use significant figures

4. Evaluate – does the answer “seem right” ?

2.4 Solving Quantitative Problems• Proportional relationships• Directly proportional – examples – density (mass of

water vs. volume of water), grades vs. freedom

In this example, we would say that, “volume of water is directly proportional to mass of water.” We can write it as V α m

2.4 Solving Quantitative Problems• Proportional relationships• Inversely proportional – examples – speed vs. time, more

accidents = less drivingAnother chemistry example, as the pressure of a gas increases,

the volume decresases:Volume ofgas (mL)

Pressureof gas (atm)

25.2 0.971

28.3 0.865

32.4 0.755

39.6 0.618

43.1 0.568

47.8 0.512

52.6 0.465

When two variables are related this way, they are said to be inversely proportional.We can write it as P 1/α V