Transcript Mathematics in Chemistry Lab 1 Outline             Mathematics in Chemistry Units Rounding Digits of Precision (Addition and Subtraction) Significant Figures (Multiplication and Division) Order of Operations Mixed Orders Scientific Notation Logarithms and.

Mathematics in
Chemistry
Lab 1
Outline
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Mathematics in Chemistry
Units
Rounding
Digits of Precision (Addition and Subtraction)
Significant Figures (Multiplication and Division)
Order of Operations
Mixed Orders
Scientific Notation
Logarithms and Antilogarithms
Algebraic Equations
Averages
Graphing
Mathematics in Chemistry
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Math is a very important tool, used in all of the sciences to
model results and explain observations.
Chemistry in particular requires a lot of calculations before
even trivial experiments can be performed. In this first
exercise you will be introduced to some of the very basic
calculations you will be required to perform in lab during
the semester.
Remember, if you start memorizing rules and formulas
now, you don’t have to do it the night before your exams!
Units
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Units are very important!
Units give dimension to numbers.
They also allow us to use dimensional analysis in
our calculations.
If a unit belongs next to a number, place it there!!!
Example: 6.23 mL
The unit “mL” indicates to us that our measurement
is a metric system volume and indicates to us the
order of magnitude of that volume.
Common units, equations, and conversions are
given on p. 30 of your lab manual.
Rounding
When you have to round to a certain number, to obey significant figure
rules, remember to do the following:
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For numbers 1 through 4 in the rounding position, round down
For numbers 6 through 9 in the rounding position, round up
For numbers with a terminal 5 in the rounding position, round to the
nearest even number.
0.01255 rounded to three significant digits becomes 0.0126
0.01265 rounded to three significant digits becomes 0.0126
0.01275 rounded to three significant digits becomes 0.0128
0.012851 rounded to three significant digits becomes ?
Why is this method statistically more correct?
Rounding
When you have to round to a certain number, to obey significant figure
rules, remember to do the following:
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For numbers 1 through 4 in the rounding position, round down
For numbers 6 through 9 in the rounding position, round up
For numbers with a terminal 5 in the rounding position, round to the
nearest even number.
0.01255 rounded to three significant digits becomes 0.0126
0.01265 rounded to three significant digits becomes 0.0126
0.01275 rounded to three significant digits becomes 0.0128
0.012851 rounded to three significant digits becomes ? 0.0129
Why is this method statistically more correct?
Digits of Precision and
Significant Figures
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All measurements have some degree of
uncertainty due to limitations of measuring
devices.
Scientists have come up with a set of rules
we can follow to easily specify the exact
digits of precision and amount of significant
figures, without sacrificing the accuracy of the
measuring devices.
Digits of Precision:
Addition and Subtraction
Your answer must contain the same number of
digits after the decimal point as the number
with the least number of digits after the decimal
point.
104.07 + 209.7852 + 1.113 = 314.97
Addition and Subtraction
205.12234
– 72.319
+
4.7
= 137.48334
137.5
Addition of Whole Numbers
When you add or subtract whole numbers, your
answer cannot be more accurate than any of
your individual terms.
20 + 34 + 2400 – 100 = 2400
Limited to the hundreds position
What about:
319 + 870 + 34,650 = ?
Addition of Whole Numbers
When you add or subtract whole numbers, your
answer cannot be more accurate than any of
your individual terms.
20 + 34 + 2400 – 100 = 2400
What about:
319 + 870 + 34,650 = ?
The answer is 35,840
Limited to the tens position.
Significant Figures Rule #1
Numbers with an infinite number of significant
digits do not limit calculations. These numbers are
found in definite relationships, otherwise known as
conversion factors.
100 cm = 1 m
1000 mL = 1 L
Significant Figures Rule #2
All non-zero digits are significant.
1.23 has 3 significant figures
98,832 has 5 significant figures
How many significant digits does 34.21 have?
Significant Figures Rule #2
All non-zero digits are significant.
1.23 has 3 significant figures
98,832 has 5 significant figures
How many significant digits does 34.21 have?
Correct! The answer is 4.
Significant Figures Rule #3
The number of significant figures is
independent of the decimal point.
12.3, 1.23, 0.123 and 0.0123 have 3
significant figures
0.0004381 and 0.4381 have how many
significant figures?
Significant Figures Rule #3
The number of significant figures is
independent of the decimal point.
12.3, 1.23, 0.123 and 0.0123 have 3
significant figures
0.0004381 and 0.4381 have how many
significant figures?
Correct! The answer is 4.
Significant Figures Rule #4
Zeros between non-zero digits are significant.
1.01, 10.1, 0.00101 have 3 significant figures.
How many significant digits are in 10,101?
Significant Figures Rule #4
Zeros between non-zero digits are significant.
1.01, 10.1, 0.00101 have 3 significant figures.
How many significant digits are in 10,101?
The answer is 5!
Significant Figures Rule #5
After the decimal point, zeros to the right of
non-zero digits are significant.
0.00500 has 3 significant figures 0.030 has 2
significant figures.
How many significant figures are in 34.1800?
Significant Figures Rule #5
After the decimal point, zeros to the right of
non-zero digits are significant.
0.00500 has 3 significant figures 0.030 has 2
significant figures.
How many significant figures are in 34.1800?
The answer is 6. Right again.
Significant Figures Rule #6
If there is no decimal point present, zeros to
the right of non-zero digits are not significant.
3000, 50000, 20 all have only 1 significant
figure
How many significant figures are in
32,000,000?
Significant Figures Rule #6
If there is no decimal point present, zeros to
the right of non-zero digits are not significant.
3000, 50000, 20 all have only 1 significant
figure
How many significant figures are in
32,000,000?
The answer is 2!
Significant Figures Rule #7
Zeros to the left of non-zero digits are never
significant.
0.0001, 0.002, 0.3 all have only 1 significant figure
How many significant figures are in 0.0231?
How many significant figures are in 0.02310?
Significant Figures Rule #7
Zeros to the left of non-zero digits are never
significant.
0.0001, 0.002, 0.3 all have only 1 significant figure
How many significant figures are in 0.0231?
This one has 3 significant digits.
How many significant figures are in 0.02310?
This one has 4 significant digits.
Significant Figures:
Multiplication and Division
Your answer must contain the same number of
significant digits as the number with the least
number of significant digits.
5.10 x 6.213 x 5.425 = 172
Significant Figures:
Multiplication and Division
205.244
= 76.016
2.7
76
Order of operations
1st: ( ), x2, square roots
2nd: x or /
3rd: + or –
Significant Figures:
Mixed Orders
29.104
 (21.009 x 0.0032)  1.42
34.2
23
(21.009 x 0.0032)  0.067
29.104
99
 0.850
34.2
99
23
0.850  0.067  1.42  2.20
Scientific Notation
The three main items required for numbers to
be represented in scientific notation are:
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the correct number of significant figures
one non-zero digit before the decimal point, and
the rest of the significant figures after the
decimal point
this number must be multiplied by 10 raised to
some exponential power
123 becomes 1.23 x 102
This number has three significant digits
Scientific Notation
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Calculators could be a significant aid in
performing calculations in scientific notation.
KNOW HOW TO USE YOUR CALCULATOR
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Does your calculator retain or suppress zeros in
its display?
In converting between scientific and decimal
notation, the number of significant digits don’t
change.
Scientific Notation
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What is the scientific notation equivalent of
0.0432?
1043.50?
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What is the standard decimal notation
equivalent of 3.45 x 103?
6.500 x 10-2?
Scientific Notation
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What is the scientific notation equivalent of
0.0432?
The answer is 4.32 x 10-2
1043.50?
The answer is 1.04350 x 103
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What is the standard decimal notation
equivalent of 3.45 x 103?
This is 3450
6.500 x 10-2?
This is 0.06500
Scientific Notation
Calculations
Addition:
(4.22 x 105) + (3.97 x 106)
= (4.22 x 105) + (39.7 x 105)
= (4.22 + 39.7) x 105
= 43.9 x 105
= 4.39 x 106
Know how to perform these types of
calculations on your calculator!
Scientific Notation
Calculations
Subtraction:
(4.22 x 105)
- (3.97 x 106)
= (4.22 x 105) - (39.7 x 105)
= (4.22 – 39.7) x 105
= -35.5 x 105
= -3.55 x 106
Know how to perform these types of
calculations on your calculator!
Scientific Notation
Calculations
Multiplication:
(4.22 x 105) x (3.97 x 106)
= (4.22 x 3.97) x 10(5+6)
= 16.8 x 1011
= 1.68 x 1012
Know how to perform these types of
calculations on your calculator!
Scientific Notation
Calculations
Division:
(4.22 x 105) / (3.97 x 106)
= (4.22 / 3.97) x 10(5-6)
= 1.06 x 10-1
Know how to perform these types of
calculations on your calculator!
Logarithms
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Logarithms might seem strange, but they are
nothing more than another way of
representing exponents.
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logbx = y is the same thing as x = by
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Know how to use your calculator to perform
these functions.
Logs and Antilogs
To enter log 100 on your calculator:
 Press: log  1  0  0  Enter
or
 Press: 1  0  0  log for reverse entry
To enter the antilog 2 on your calculator
 Press: 2nd  log  2  Enter
or
 Press: 2  2nd  log for reverse entry
Did you notice anything?
Significant Figure Rules
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Logarithms
log (4.21 x 1010) = 10.6242821  10.624
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Antilogarithms
antilog (- 7.52) = 10-7.52 = 3.01995 x 10-8  3.0 x 10-8
Significant Figures of Equipment
Electronics
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Always report all the digits electronic
equipment gives you.
When calibrating a probe, the digits of
precision of your calibration values
determine the digits of precision of the
output of the data.
Algebraic Equations
It is important to understand how to manipulate
algebraic equations to determine unknowns
and to interpolate and extrapolate data.
Don’t forget about significant figures.
For y = 1.0783 x + 0.0009
If x = 0.021, find y (answer = 0.024)
If y = 4.3, find x (answer = 4.0)
Finding Averages
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To find the average (algebraic mean) of a set
of data, simply add the data and divide by the
number of data points.
9.98 mL, 10.00 mL, 9.99 mL, 9.97 mL
9.98mL  10.00mL  9.99mL  9.97mL 39.94
Average 
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 9.99 mL
4
4
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What is the average of 23.3 g + 25.6 g + 24.9 g?
Finding Averages
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To find the average (algebraic mean) of a set of
data, simply add the data and divide by the number
of data points.
9.98 mL, 10.00 mL, 9.99 mL, 9.97 mL
9.98mL  10.00mL  9.99mL  9.97mL 39.94
Average 
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 9.99 mL
4
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What is the average of 23.3 g + 25.6 g + 24.9 g?
That’s right… it is 24.6 g. Remember units!
Graphing
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Graphing is an important tool used to
represent experimental outcomes and to
set up calibration curves.
It is a modeling device.
Graphing: Variables
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Having no fixed quantitative value.
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x-variable
y-variable
Graphing in chemistry
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Renamed with a chemistry label
Paired with a unit most of the time
Graphing: Units
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Give dimension to labels / variables
Give meaning to numbers
Essential!
Graphing: Coordinates
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A coordinate set consists of an x-value and yvalue, plotted as a point on a graph.
Plotted x-values: domain (independent variable)
Plotted y-values: range (dependent variable)
Graphing: Axes
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Multiple axes on a graph
Coordinate sets determine the number of
axes on a plot
Two dimensional graphs have only two axes
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x-axis
y-axis
Each axis must have a consistent scale
Graphing in Chemistry
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Graph title reflects the:
Dependent vs. Independent variables
x-axis – labeled appropriately with variable
and unit
y-axis – labeled appropriately with variable
and unit
Each axis has a consistent scale
Graphing in Chemistry
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Coordinate sets are plotted
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x-variable matching the x-value on the x-axis
y-variable matching the y-value on the y-axis
A single point results
A line is drawn through all the points
An equation is derived from two coordinate
sets
The equation is used to find unknowns
Graphing: Equations
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Of the form y = mx + b
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m = slope of the graph
b = y-intercept of the graph
x = any x-value from the graph
y = corresponding y-coordinate
Graphing
Let’s look at the following
data and the resulting plot
on the next slide.
[Ni2+], M
Absorbance
0.200
0.041
0.300
0.063
0.400
0.085
0.500
0.101
Graphing
Absorbance
Absorbance vs. [Ni2+], M
0.110
0.100
y = 0.20 x + 0.002
0.090
0.080
0.070
0.060
0.050
0.040
0.200 0.250 0.300 0.350 0.400 0.450 0.500
[Ni 2+], M
Graphing
a)
b)
c)
d)
e)
f)
g)
h)
What is the title of the previous graph?
Which variable is plotted on the x-axis?
What is the unit of that variable?
Which variable is plotted on the y-axis?
What is the unit of that variable?
What is the equation of the regression line?
What is the slope of the equation? And unit?
What is the y-intercept of the equation? And
unit?
Graphing
a)
b)
c)
d)
e)
f)
g)
h)
Absorbance vs. [Ni2+], M
[Ni2+]
M (molarity or mol/L)
Absorbance
No unit / dimensionless
y = 0.20 x + 0.002
Slope = 0.20; Unit = 1 / M
y-intercept = 0.002; Unit = none