What are the objectives of Unit 3?

To learn how to critically analyze data using graphs.

To learn how to use graphing techniques to arrive at mathematical equations that describes the relationship between two variables.

To learn to connect the mathematical slope of a line to a physical property

Graphs compensate for shortcomings of data

tables.

Difficult to determine relationship between variables.

Impractical to find equation

The most important purposes of a graph are

Quick and efficient organization of large amounts of data.

Demonstration of relationship between independent and dependent variables.

Calculation of a mathematical equation that describes the relationship in the data.

Scientists who used graphs to make important discoveries.

Galileo’s falling body experiments showed all objects, no matter what they weigh fall at the same rate.

Schwabe’s sunspot data showed that sunspots are at a maximum every 11 years.

Dependent Axis and Independent Axis

In any graph there are at least 2 axes

Vertical axis is called the y-axis Plot the dependent variable on the

y-axis. May refer to it as y.

Horizontal axis is called the x-axis.

Plot the independent

variable here except when time is a variable

Always plot time on the x-axis

May refer to the variable as x

(x,y) is the pair of data. Plotting this point allows you to see the relationship between the variables.

Linear Data – data points form a straight line.

Best Fit LineApproximates the linear trend of the

data. The line passes in between the data

points. The line does not have to pass through

all of the points. Points should be relatively close to

one another. Random error causes about ½ the points to be

above the line and ½ to be below the line. Use a straight edge to approximate.

Can be computed by mathematical equation or computer program like Excel.

Best Fit Curve

Used when data points form a curve.

Should pass between points but must remain smooth.

Types of Linear Relationships

Direct Linear Relationships

As the independent variable increases, the dependent variable increases.

As the independent variable decreases, the dependent variable decreases.

Example of both is ice cream cone - Melting vs. Temperature: Both variables move in the same direction.

Have positive slopes and appear to go uphill.

Equation is y=mx+b

Direct Linear Relationship

Direct Linear Proportion

A direct linear relationship that passes through the origin (0,0).

y-intercept is 0. Equation is y = mx.

Direct Linear Proportion

Indirect Linear Relationships

Indirect means as one variable increases, the other decreases.

Have negative slopes and look like downhill lines.

Indirect linear proportions are extremely unusual in the physical sciences since science deals with real situations.

Indirect Linear Relationship

Distance versus Time

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12 14 16

Time (hr)

Dis

tan

ce

(m

)

Indirect Linear Proportion Data starts at (0,0) and

decreases into the negative region of the graph.

Horizontal Line

1. Indicates there is no measurable relationship between the variables.

2. The independent variable is an irrelevant variable.

Mass and Arc Size in the pendulum lab

Indirect Linear Proportion

Let’s think a minute

What would a vertical line tell you?

The values of the replicates are changing.

A variable has not been controlled. You’ve made a mistake in your

procedure.

Calculating the Equation of a Plotted Line

In Algebra the general equation for a line is y =mx + b

x is the independent variable y is the dependent variable m is the slope b is the y-intercept We will determine a specific

equation for a specific set of data where x and y will be replaced with more descriptive variables.

Calculating the Slope

The slope of a line describes how quickly the dependent variable changes with respect to the independent variable.

Large positive slopes - steep uphill Small positive slopes – gradual uphill Large negative slopes – steep

downhill Small negative slopes – gradual

downhill

To calculate slope

Pick two points on the line (not data points)

Pick points where the line perfectly crosses a corner of the grid. Move from left to right.

Choose points as far apart as possible

Use

12

12

xx

yy

X

Y

run

rise

Significant Figures in Graphing

The significant figures should be determined by the accuracy of the data.

Use the place value of the independent (x) and dependent variables (y) in the data table when selecting points off the graph

Calculating the y- intercept

The y-intercept is the value of the dependent variable at the point where the best fit line crosses the y-axis.

Estimate this value from the graph. Express with units of the dependent

variable. If the y-intercept is not shown on the

graph, plug in values of a point and solve for b in the equation for the line.

Putting it all Together

Change the variables from x and y to more appropriate variable.

2. Example pressure = P, temperature = T

a. clarity b. real situation If there is not a standard

variable, define a variable

Using the Equation

Why use the equation?

So we do not have to redo the experiment each time we need a new data point.

Melissa Hubert Example p 65

Pressure Versus Temperature for a 50L Steel Tank

0

5

10

15

20

0 10 20 30 40 50 60 70 80 90 100

Temperature (˚C)

Pre

ssu

re (

atm

)

(x2, y2)

(x1, y1)

P 66 continued

Select data points (2.0°C, 5.00atm) and (68.0°C,16.00atm)

Calculating Slope:

Slope = 0.167atm/°C Equation of the line is: y = (0.167atm/°C )x + b

Catm

C

atm

CC

atmatmm 167.0

0.66

00.11

0.20.68

00.500.16

Now we need b

y = (0.167 atm/ºC) x + b 5.00 atm = (0.167 atm/ºC) 2.0ºC + b 5.00 atm = 0.34 atm + b b = 5.00 atm – 0.34 atm =

4.66 atm = 4.66 atm Plugging in the value of the y-intercept,

the equation of the best-fit line becomes: y = (0.167 atm/ºC) x + 4.66 atm

Now change the variables

P = (0.167atm/˚C)T + 4.66 atm

Finding Equations PWYR p69

Amount of Energy Released Versus the Starting Mass of the Reactants for the Decomposition of Ammonium Dichromate

0

50

100

150

200

250

300

0 10 20 30 40 50 60 70 80 90 100

Starting Mass of Reactants (g)

Am

ou

nt

of

En

erg

y r

elea

sed

(K

J)

PWYR p69

1. Which variable is being changed by the experimenter? Mass

Explain why you chose this variable? It is the independent variable and is plotted on the x axis.

2. Find the equation of the best-fit line in the graph. (0g,0Kj) and (84g,250Kj)

E = energy

E = (3.0Kj/g) M

P 69 continued 3. How many kilojoules (kJ) of energy would be released if 4,129 grams of

ammonium dichromate decomposed?

E = (3.0Kj/g) M

E = (3.0 Kj/g) 4,129g = 12,387Kj = 12,000Kj

4. If researchers needed to limit the energy released in the chemical reaction to 975 kJ, what is the maximum mass of ammonium dichromate they can use?

975Kj = (3.0 Kj/g) M M=975Kj/3.0 Kj/g = 325 g = 320 g

Derived and Empirical Equations

Derived Equations can be found by graphing data or can be formulated by combining 2 or more already known equations.

C=∏d d=2r where r = radius Substitute 2r for d in the first

equation C=2∏r

Empirical Equations Empirical Equations can be formulated by

observing patterns in nature and creating an equation.

Johanne Kepler observed the patterns of the planets orbiting the sun and concluded they had something in common.

If the periods of their revolutions were squared and divided by their average distances from the sun cubed, the result for all the planets was a constant.

Extrapolation

The extension of a best fit line beyond the plotted points.

Allows you to make predictions. Use it cautiously and make sure the

following conditions are met. The plotted data is accurate. The data points were plotted correctly. The relationship between the 2

variables continues. Small error can result in huge error

over large distances.

Interpolation

To go between data points.

Can have a great deal of confidence that the interpolated data is as good as the experimental data.