GC-S006-Graphing

Henry R. Kang (1/2010)
General Chemistry
Lecture 6
Graphing

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

Logarithm
and
Exponent

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.

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

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
= a
√B
 21/a
= a
√2 21/3
= 3
√2
 101/a
= a
√10 101/2
= √10

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.

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.

Graphing

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.

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.

Example – Correct Size
0
200
400
600
800
1000
1200
1400
0 10 20 30 40 50 60 70 80
Tem perature (C)
Pressure(mmH2O)

Example – Incorrect Size
0
200
400
600
800
1000
1200
1400
0 10 20 30 40 50 60 70 80
Temperature (C)
Pressure(mmH2O)

Graphing Rules - Axis
• An axis scale does not need to
start at “zero”
To avoid data points cluster in a
narrow range
Exception
If the extrapolation of the line is
required to the x-axis or y-axis
intercept.

Data Cluster - Incorrect
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20
Tem perature (C)
Volume(mL)

Expand the Scale
0
1
2
3
4
5
6
7
8
9
10
16 16.5 17 17.5 18 18.5 19
Tem perature (C)
Volume(mL)

Extrapolation
0
1
2
3
4
5
6
7
8
9
-10 -5 0 5 10 15 20
X
Y

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 8or
3 60 9 12Not
2
0
4
6
8
10
12
5
10
15
20
25
30
10
20
30
40
50
60
00

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.

Example
0
1
2
3
4
5
6
7
8
9
10
16 16.5 17 17.5 18 18.5 19
Temperature (C)
Volume(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)

Graphing Rules - Resolution
• Ideally, one should be able to read
all significant figures of the data
from its position on the graph
paper
Often, the significant figure of the
data is higher than the resolution of
the graph paper.

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.

Example
0
2
4
6
8
10
12
14
0 5 10 15
X
Y
Series1
Series2
Series3

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.

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
Tem perature (C)
Volume(mL)

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
Tem perature (C)
Volume(mL)

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.

Linear Regression

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

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

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
= m2
xn
2
+ b2
+ yn
2
+ 2mbxn – 2mxnyn – 2byn
• The overall deviation is
 ∑ dn
2
= ∑ (m xn+ b – yn )2
∀ ∑dn
2
can be minimized by taking the partial derivatives
with respect to m and b, respectively.

Linear Regression - Formulas
• Minimize ∑dn
2
by taking the partial derivatives
with respect to m (slope) and b (intercept)
∂(∑ dn
2
)/ ∂m = ∑(2mxn
2
+ 2bxn – 2xnyn) = 0
∂(∑ dn
2
)/ ∂b = ∑(2b + 2mxn – 2yn) = 0
• This set of equations can be solved for m and b.
N (∑xnyn ) – (∑xn) (∑ yn)
N (∑xn
2
) – (∑xn)2
N (∑xn
2
) – (∑xn)2
(∑xn
2
)(∑yn) – (∑xn)(∑xn yn)
m =
b =
where N is the number of data sets

Linear Regression - Example
N (∑xn yn)– (∑xn) (∑ yn)
[N(∑xn
2
) – (∑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 (∑xn
2
) – (∑xn)2
(∑xn
2
)(∑ 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
90.0
r =
(600.)1/2
(14)1/2
0.982
r2
= 0.964

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 (∑xn
2
) – (∑xn)2
]1/2
[N (∑yn
2
) – (∑yn )2
]1/2
r =

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.

GC-S006-Graphing

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