Henry R. Kang (1/2010)

General Chemistry

Lecture 6

Graphing

Henry R. Kang (7/2008)

Outlines

• Logarithm and Exponent Definitions and rules

• Graphing Graphing rules Line drawing

• Linear Regression Deviation of equations Goodness of data fitting Draw the least-square line

Henry R. Kang (7/2008)

Logarithm

and

Exponent

Henry R. Kang (7/2008)

Definition of Exponent

• Exponent is expressed as the power of a base value. Y = BX.

B is the base and X is the power.

Any positive number greater than 1 can be used as the base.

The power X can be a number or a more complex expression such as X = a/4 or X = (a+b)/2.

Commonly used bases are 10 and e = 2.7182818… (e is an irrational number).

Computer uses binary system; the base is 2.

Henry R. Kang (7/2008)

Examples of Exponent

• Positive exponents B1 = B; B2 = B×B; B3 = B×B×B; Bn = B×B - - - B×B (B multiples itself n times );

21 = 2; 22 = 4; 23 = 8; 81 = 8; 82 = 64; 83 = 512; 101 = 10; 102 = 100; 103 = 1000; 10n = 10 - - - 0 (n zeros);

• Negative exponents B–1 = 1/B; B–2 = 1/(B×B); B–3 = 1/(B×B×B); B–n = 1/(B×B - - - B×B) (The inverse of B multiples itself n times );

2–1 = 1/2; 2–2 = 1/4; 2–3 = 1/8; 8–1 = 1/8; 8–2 = 1/64; 8–3 = 1/512; 10–1 = 1/10 = 0.1; 10–2 = 1/100 = 0.01; 10–3 = 1/1000 = 0.001; 10–n = 1/10 - - - 0 = 0.0 - - - 01 (n-1 zeros between 1 and decimal point);

• Zero exponent B0 = 1; 20 = 1; e0 = 1; 80 = 1; 100 = 1

Henry R. Kang (7/2008)

Rules of Exponent

• Multiplication Ba × Bb = Ba+b

2a × 2b = 2a+b 24 × 22 = 26 = 64 10a × 10b = 10a+b 104 × 107 = 1011

• Division Ba Bb = Ba-b

2a 2b = 2a-b 24 207 = 2-3 = 1/8 10a 10b = 10a-b 104 107 = 10-3 = 0.001

• Exponent (Ba)b = Ba×b

(2a)b = 2a×b (23)2 = 106 = 64 (10a)b = 10a×b (104)7 = 1028

• Fraction B1/a = aB

21/a = a2 21/3 = 32 101/a = a10 101/2 = 10

Henry R. Kang (7/2008)

Definition of Logarithm

• Logarithm is the inverse of the exponent.

• log(base)Y = X.

Two popular bases are 10 and e.

• Common (or Briggsian) logarithm uses the base value of 10. log10Y = X or log Y = X.

Some values of common logarithm: log(1) = 0; log(10) = 1; log(100) = 2; etc.

• Natural (or hyperbolic) logarithm uses the base value of e = 2.7182818… logeY = X or lnY = X.

• Common and natural logarithms can be inter-converted. lnY = 2.30259 × logY or logY = lnY / 2.30259.

Henry R. Kang (7/2008)

Rules of Logarithms• The following rules apply to both common and natural

logarithms:

• log(A×B) = logA + logB. log(5×7) = log(5) + log(7) = 0.698970004 + 0.84509804 = 1.544068044 log(35) = 1.544068044

• log(A/B) = logA – logB. log(5/7) = log(5) – log(7) = 0.698970004 - 0.84509804 = -0.146128036 log(5/7) = log(0.714285714) = -0.146128036

• log(An) = n logA. log(73) = 3 log(7) = 2.53529412 log(73) = log(343) = 2.53529412

• log(A-n) = -n logA.

• log(A1/n) = (1/n) logA.

Henry R. Kang (7/2008)

Graphing

Henry R. Kang (7/2008)

Advantages of Graphing

• A graph or figure is a very powerful means of delivering information. The information is very compactly

represented. The relationship between parameters is clearly

shown. The general characteristics of the parameters

can be derived.

• A picture is worth a thousand words.

Henry R. Kang (7/2008)

Graphing Rules – Label & Size

• Graph should be neatly presented, easily readable and properly titled. Each axis should be clearly labeled with

the name of the parameter and the unit.

Scales should be selected so that the actual graph covers at least 50% of the space available.

Henry R. Kang (7/2008)

Example – Correct Size

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50 60 70 80

Temperature (C)

Pre

ssu

re (

mm

H2O

)

Henry R. Kang (7/2008)

Example – Incorrect Size

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50 60 70 80

Temperature (C)

Pre

ssu

re (

mm

H2O

)

Henry R. Kang (7/2008)

Graphing Rules - Axis

• An axis scale does not need to start at “zero”To avoid data points cluster in a

narrow rangeException

If the extrapolation of the line is required to the x-axis or y-axis intercept.

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Data Cluster - Incorrect

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20

Temperature (C)

Vo

lum

e (

mL

)

Henry R. Kang (7/2008)

Expand the Scale

0

1

2

3

4

5

6

7

8

9

10

16 16.5 17 17.5 18 18.5 19

Temperature (C)

Vo

lum

e (

mL

)

Henry R. Kang (7/2008)

Extrapolation

0

1

2

3

4

5

6

7

8

9

-10 -5 0 5 10 15 20

X

Y

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Graphing Rules - Divisions

• Scale on the graph paper should have divisions that are easily “divided by the eye”

1, 2, 5, or 10 Not 3, 6, 7, or 11

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20

0 2 4 6 8or3 60 9 12Not

2

0

4

6

8

10

12

5

10

15

20

25

30

10

20

30

40

50

60

00

Henry R. Kang (7/2008)

Graphing Rules – Table of Data

• A table of data may be provided (This rule is not always obeyed)No individual coordinates of data points should appear on the graph.

Henry R. Kang (7/2008)

Example

0

1

2

3

4

5

6

7

8

9

10

16 16.5 17 17.5 18 18.5 19

Temperature (C)

Vo

lum

e (

mL

)

T(°c) V (mL)

17.0 6.58

16.8 5.76

17.8 7.71

18.5 8.84

18.2 8.47

17.5 7.04

16.5 5.25

(17.0, 6.58)

Henry R. Kang (7/2008)

Graphing Rules - Resolution

• Ideally, one should be able to read all significant figures of the data from its position on the graph paperOften, the significant figure of the

data is higher than the resolution of the graph paper.

Henry R. Kang (7/2008)

Graphing Rules – Data Symbols

• A data point should be circled (or other shapes such as square or triangle) surrounding it.

• If more than one data set, each set should have its own shape (or symbol) to represent the data points. You can use different color for different

data set, if color display or print is available.

Henry R. Kang (7/2008)

Example

0

2

4

6

8

10

12

14

0 5 10 15

X

Y

Series1

Series2

Series3

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Graphing Rules - Drawing

• The curve or straight line is drawn smoothly among the points, rather than connecting dots (piece-wise linearization). The curve or line should represent the best average of

the data.Roughly about equal number of points above and below the

curve or line.The curve or line does not have to touch any of the data

points.

Use a clear straightedge for drawing lines. Use French curves for drawing curves.

Henry R. Kang (7/2008)

Incorrect Way of Line Drawing

0

1

2

3

4

5

6

7

8

9

10

16 16.5 17 17.5 18 18.5 19

Temperature (C)

Vo

lum

e (

mL

)

Henry R. Kang (7/2008)

Correct Way of Line Drawing

0

1

2

3

4

5

6

7

8

9

10

16 16.5 17 17.5 18 18.5 19

Temperature (C)

Vo

lum

e (

mL

)

Henry R. Kang (7/2008)

Graphing Rules – Data Points

• Data point should not exist on the axis line.This rule is not always obeyed.You still plot the point if it lies on the axis line.

Henry R. Kang (7/2008)

Linear Regression

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Linear Regression - Equation

• Linear regression is used to find the straight line that fits the data best.

• General equation for a line is Y = m X + b X is the independent variable, Y is the dependent variable, m is the slope of the line, and b is the y-axis intercept

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Linear Regression - Deviation

• Let yn = the observed values

• and ýn = the calculated value from the linear equation

• The deviation dn = ýn – yn (n = 1, 2, 3, - - -, N)

N is the number of data sets.

• The best result is obtained by minimizing the deviation (or the square of the deviation)

dn2 = (y1–ý1)2 + (y2–ý2)2 + - - - + (yN–ýN)

Henry R. Kang (7/2008)

Linear Regression – Minimize Deviation

• Calculate ýn from the linear equation, we have

ýn = m xn + b

• The deviation becomes dn = ýn – yn = m xn + b – yn

• The square of deviation is dn

2 = (m xn + b – yn )2

= m2xn2 + b2 + yn

2 + 2mbxn – 2mxnyn – 2byn

• The overall deviation is dn

2 = (m xn + b – yn )2

dn2 can be minimized by taking the partial derivatives

with respect to m and b, respectively.

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Linear Regression - Formulas

• Minimize dn2 by taking the partial derivatives

with respect to m (slope) and b (intercept)

( dn2)/ m = (2mxn

2 + 2bxn – 2xnyn) = 0

( dn2)/ b = (2b + 2mxn – 2yn) = 0

• This set of equations can be solved for m and b.N (xnyn ) – (xn) ( yn)

N (xn2) – (xn)2

N (xn2) – (xn)2

(xn2)(yn) – (xn)(xn yn)

m =

b =

where N is the number of data sets

Henry R. Kang (7/2008)

Linear Regression - Example

N (xn yn) – (xn) ( yn)

[N(xn2) – (xn)2]1/2 [N(yn

2) – (yn )2]1/2r =

x(C)

y(liter)

xy x2 y2

1 0.0 20.0 0.0 0.0 400.

2 10.0 22.0 220. 100. 484.

3 20.0 23.0 460. 400. 529.

Sum 30.0(xn)

65.0( yn)

680.(xn yn)

500.(xn

2)1413(yn

2)

b =

m =N (xn

2) – (xn)2

N (xn yn) – (xn) ( yn)= 3×680. – 30.0×65.0

3×500. – 30.02= 2040. – 1950.

1500. – 900.= 90.0

600.= 0.150

N (xn2) – (xn)2

(xn2)( yn) – (xn)(xn yn)

= 500.×65.0 – 30.0×680.

3×500. – 30.02= 12100.

600.= 20.2

=

=3×680. – 30.0×65.0

(3×500. – 30.02)1/2 (3×1413 – 65.02)1/2

90.0r =(600.)1/2 (14)1/2

0.982

r2 = 0.964

Henry R. Kang (7/2008)

Linear Regression - Goodness

• The “goodness” of the data fitting is expressed by the regression coefficient r2

• If r2 = 1, perfect fit• If r2 > 0.95, excellent fit• If r2 > 0.90, good fit• If r2 > 0.80, reasonable fit• If r2 = 0, completely unrelated

N (xn yn) – (xn) ( yn)

[N (xn2) – (xn)2]1/2 [N (yn

2) – (yn )2]1/2r =

Henry R. Kang (7/2008)

Draw the Least-Square Line

• Once the slope m and intercept b are calculated from a set of data (x and y), the best line can be drawn to fit the data.

• A line, y = mx + b, is defined by two points.

• The least computational cost to find these two points is Set x = 0, then y = b, giving the first point (0, b)

Set y = 0, then x = -b/m, giving the second point (-b/m, 0)

Put these two points in the graph, then draw a straight line connecting these two points.

This line is the least-square line to fit the given data set (x and y) best with a minimum error between the calculated and measured y values.