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Analyzing Data

Section 2.1Units and Measurements

Section 2.2Scientific Notation and Dimensional Analysis

Section 2.3Uncertainty in Data

Section 2.4Representing Data

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Section 2.1 Units and Measurements

- Define SI base units for time, length, mass, and temperature.

- Explain how adding a prefix changes a unit.
- Compare the derived units for volume and density.

mass: a measurement that reflects the amount of matter an object contains

Section 2.1 Units and Measurements (cont.)

base unit

second

meter

kilogram

kelvin

derived unit

liter

density

Chemists use an internationally recognized system of units to communicate their findings.

Units

- Système Internationale d'Unités (SI) is an internationally agreed upon system of measurements.

- A base unit is a defined unit in a system of measurement that is based on an object or event in the physical world, and is independent of other units.

Units (cont.)

Units (cont.)

Units (cont.)

- The SI base unit of time is the second (s), based on the frequency of radiation given off by a cesium-133 atom.

- The SI base unit for length is the meter (m), the distance light travels in a vacuum in 1/299,792,458th of a second.
- The SI base unit of mass is the kilogram (kg), about 2.2 pounds

Units (cont.)

- The SI base unit of temperature is the kelvin(K).

- Zero kelvin is the point where there is virtually no particle motion or kinetic energy, also known as absolute zero.
- Two other temperature scales are Celsius and Fahrenheit.

Derived Units

- Not all quantities can be measured with SI base units.

- A unit that is defined by a combination of base units is called a derived unit.

Derived Units (cont.)

- Volume is measured in cubic meters (m3), but this is very large. A more convenient measure is the liter, or one cubic decimeter (dm3).

Derived Units (cont.)

- Density is a derived unit, g/cm3, the amount of mass per unit volume.

- The density equation is density = mass/volume.

A

B

C

D

Section 2.1 Assessment

Which of the following is a derived unit?

A.yard

B.second

C.liter

D.kilogram

A

B

C

D

Section 2.1 Assessment

What is the relationship between mass and volume called?

A.density

B.space

C.matter

D.weight

Section 2.2 Scientific Notation and Dimensional Analysis

- Express numbers in scientific notation.

- Convert between units using dimensional analysis.

quantitative data: numerical information describing how much, how little, how big, how tall, how fast, and so on

Section 2.2 Scientific Notation and Dimensional Analysis (cont.)

scientific notation

dimensional analysis

conversion factor

Scientists often express numbers in scientific notation and solve problems using dimensional analysis.

Scientific Notation

- Scientific notation can be used to express any number as a number between 1 and 10 (the coefficient) multiplied by 10 raised to a power (the exponent).

- Count the number of places the decimal point must be moved to give a coefficient between 1 and 10.

Scientific Notation (cont.)

- The number of places moved equals the value of the exponent.

- The exponent is positive when the decimal moves to the left and negative when the decimal moves to the right.

800 = 8.0 102

0.0000343 = 3.43 10–5

Scientific Notation (cont.)

- Addition and subtraction

- Exponents must be the same.
- Rewrite values with the same exponent.
- Add or subtract coefficients.

Scientific Notation (cont.)

- Multiplication and division

- To multiply, multiply the coefficients, then add the exponents.
- To divide, divide the coefficients, then subtract the exponent of the divisor from the exponent of the dividend.

Dimensional Analysis

- Dimensional analysis is a systematic approach to problem solving that uses conversion factors to move, or convert, from one unit to another.

- A conversion factor is a ratio of equivalent values having different units.

Dimensional Analysis (cont.)

- Writing conversion factors

- Conversion factors are derived from equality relationships, such as 1 dozen eggs = 12 eggs.
- Percentages can also be used as conversion factors. They relate the number of parts of one component to 100 total parts.

Dimensional Analysis (cont.)

- Using conversion factors

- A conversion factor must cancel one unit and introduce a new one.

A

B

C

D

Section 2.2 Assessment

What is a systematic approach to problem solving that converts from one unit to another?

A.conversion ratio

B.conversion factor

C.scientific notation

D.dimensional analysis

A

B

C

D

Section 2.2 Assessment

Which of the following expresses 9,640,000 in the correct scientific notation?

A.9.64 104

B.9.64 105

C.9.64 × 106

D.9.64 610

Section 2.3 Uncertainty in Data

- Define and compare accuracy and precision.

- Describe the accuracy of experimental data using error and percent error.
- Apply rules for significant figures to express uncertainty in measured and calculated values.

experiment: a set of controlled observations that test a hypothesis

Section 2.3 Uncertainty in Data (cont.)

accuracy

precision

error

percent error

significant figures

Measurements contain uncertainties that affect how a result is presented.

Accuracy and Precision

- Accuracy refers to how close a measured value is to an accepted value.

- Precision refers to how close a series of measurements are to one another.

Accuracy and Precision (cont.)

- Erroris defined as the difference between and experimental value and an accepted value.

Accuracy and Precision (cont.)

- The error equation is error = experimental value – accepted value.

- Percent errorexpresses error as a percentage of the accepted value.

Significant Figures

- Often, precision is limited by the tools available.

- Significant figures include all known digits plus one estimated digit.

Significant Figures (cont.)

- Rules for significant figures

- Rule 1: Nonzero numbers are always significant.
- Rule 2: Zeros between nonzero numbers are always significant.
- Rule 3: All final zeros to the right of the decimal are significant.
- Rule 4: Placeholder zeros are not significant. To remove placeholder zeros, rewrite the number in scientific notation.
- Rule 5: Counting numbers and defined constants have an infinite number of significant figures.

Rounding Numbers

- Calculators are not aware of significant figures.

- Answers should not have more significant figures than the original data with the fewest figures, and should be rounded.

Rounding Numbers (cont.)

- Rules for rounding

- Rule 1: If the digit to the right of the last significant figure is less than 5, do not change the last significant figure.
- Rule 2: If the digit to the right of the last significant figure is greater than 5, round up to the last significant figure.
- Rule 3: If the digits to the right of the last significant figure are a 5 followed by a nonzero digit, round up to the last significant figure.

Rounding Numbers (cont.)

- Rules for rounding (cont.)

- Rule 4: If the digits to the right of the last significant figure are a 5 followed by a 0 or no other number at all, look at the last significant figure. If it is odd, round it up; if it is even, do not round up.

Rounding Numbers (cont.)

- Addition and subtraction

- Round numbers so all numbers have the same number of digits to the right of the decimal.

- Multiplication and division

- Round the answer to the same number of significant figures as the original measurement with the fewest significant figures.

A

B

C

D

Section 2.3 Assessment

Determine the number of significant figures in the following: 8,200, 723.0, and 0.01.

A.4, 4, and 3

B.4, 3, and 3

C.2, 3, and 1

D.2, 4, and 1

A

B

C

D

Section 2.3 Assessment

A substance has an accepted density of 2.00 g/L. You measured the density as 1.80 g/L. What is the percent error?

A.0.20 g/L

B.–0.20 g/L

C.0.10 g/L

D.0.90 g/L

Section 2.4 Representing Data

- Create graphics to reveal patterns in data.

independent variable: the variable that is changed during an experiment

- Interpret graphs.

graph

Graphs visually depict data, making it easier to see patterns and trends.

Graphing

- A graphis a visual display of data that makes trends easier to see than in a table.

Graphing (cont.)

- A circle graph, or pie chart, has wedges that visually represent percentages of a fixed whole.

Graphing (cont.)

- Bar graphs are often used to show how a quantity varies across categories.

Graphing (cont.)

- On line graphs, independent variables are plotted on the x-axis and dependent variables are plotted on the y-axis.

Graphing (cont.)

- If a line through the points is straight, the relationship is linear and can be analyzed further by examining the slope.

Interpreting Graphs

- Interpolation is reading and estimating values falling between points on the graph.

- Extrapolation is estimating values outside the points by extending the line.

Interpreting Graphs (cont.)

- This graph shows important ozone measurements and helps the viewer visualize a trend from two different time periods.

A

B

C

D

Section 2.4 Assessment

____ variables are plotted on the ____-axis in a line graph.

A.independent, x

B.independent, y

C.dependent, x

D.dependent, z

A

B

C

D

Section 2.4 Assessment

What kind of graph shows how quantities vary across categories?

A.pie charts

B.line graphs

C.Venn diagrams

D.bar graphs

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Section 2.1 Units and Measurements

Key Concepts

- SI measurement units allow scientists to report data to other scientists.

- Adding prefixes to SI units extends the range of possible measurements.
- To convert to Kelvin temperature, add 273 to the Celsius temperature. K = °C + 273
- Volume and density have derived units. Density, which is a ratio of mass to volume, can be used to identify an unknown sample of matter.

Section 2.2 Scientific Notation and Dimensional Analysis

Key Concepts

- A number expressed in scientific notation is written as a coefficient between 1 and 10 multiplied by 10 raised to a power.

- To add or subtract numbers in scientific notation, the numbers must have the same exponent.
- To multiply or divide numbers in scientific notation, multiply or divide the coefficients and then add or subtract the exponents, respectively.
- Dimensional analysis uses conversion factors to solve problems.

Section 2.3 Uncertainty in Data

Key Concepts

- An accurate measurement is close to the accepted value. A set of precise measurements shows little variation.

- The measurement device determines the degree of precision possible.
- Error is the difference between the measured value and the accepted value. Percent error gives the percent deviation from the accepted value.
- error = experimental value – accepted value

Section 2.3 Uncertainty in Data (cont.)

Key Concepts

- The number of significant figures reflects the precision of reported data.

- Calculations should be rounded to the correct number of significant figures.

Section 2.4 Representing Data

Key Concepts

- Circle graphs show parts of a whole. Bar graphs show how a factor varies with time, location, or temperature.

- Independent (x-axis) variables and dependent (y-axis) variables can be related in a linear or a nonlinear manner. The slope of a straight line is defined as rise/run, or ∆y/∆x.

- Because line graph data are considered continuous, you can interpolate between data points or extrapolate beyond them.

A

B

C

D

Which of the following is the SI derived unit of volume?

A.gallon

B.quart

C.m3

D.kilogram

A

B

C

D

Which prefix means 1/10th?

A.deci-

B.hemi-

C.kilo-

D.centi-

A

B

C

D

Divide 6.0 109 by 1.5 103.

A.4.0 106

B.4.5 103

C.4.0 103

D.4.5 106

A

B

C

D

Round the following to 3 significant figures 2.3450.

A.2.35

B.2.345

C.2.34

D.2.40

A

B

C

D

The rise divided by the run on a line graph is the ____.

A.x-axis

B.slope

C.y-axis

D.y-intercept

A

B

C

D

Which is NOT an SI base unit?

A.meter

B.second

C.liter

D.kelvin

A

B

C

D

Which value is NOT equivalent to the others?

A.800 m

B.0.8 km

C.80 dm

D.8.0 x 105 cm

A

B

C

D

Find the solution with the correct number of significant figures:25 0.25

A.6.25

B.6.2

C.6.3

D.6.250

A

B

C

D

How many significant figures are there in 0.0000245010 meters?

A.4

B.5

C.6

D.11

A

B

C

D

Which is NOT a quantitative measurement of a liquid?

A.color

B.volume

C.mass

D.density

Click on an image to enlarge.

Table 2.2 SI Prefixes

Figure 2.10 Accuracy and Precision

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