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.

Mathematics in Chemistry
Lab 1
Outline
• Mathematics in Chemistry
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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
Accuracy and Precision
Statistics
Serial Dilutions
Direct Dilutions
Graphing
Calibration Curves
• MicroLAB™
– The Program
– Reference Sheet
– Pitfalls
Mathematics in Chemistry
• Math is a very important tool, used in all of the
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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 entire semester.
Remember, if you start memorizing rules and formulas
now, you don’t have to do it the night before your
exams!
Units
• Units are very important!
• Units give dimension to numbers.
• They also allow us to use dimensional analysis in
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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.
Rounding
When you have to round to a certain number to obey
significant figure rules, remember to do the following:
For numbers 1 through 4, round down
• For numbers 6 through 9, round up
• For numbers with a terminal 5, round to the closest 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
0.012852 rounded to three significant digits becomes ?
Why is this method statistically more correct?
Digits of Precision and
Significant Figures
• All measurements have some degree of
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uncertainty due to limitations of measuring
devices.
Scientists have come up with a set of rules we
can follow to easily specify the exact amount of
significant figures, without sacrificing the
accuracy of the measuring devices.
Digits of Precision:
Addition and Subtraction
Your answer must contain no more digits after
the decimal point than the number with the
least number of digits after the decimal point.
104.75 + 209.7852 + 1.1 = 315.6
Digits of Precision:
Addition and Subtraction
205.12234
–
72.319
+
4.68
= 137.48334
137.48
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 = ?
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
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?
This one has 6 significant digits.
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 no more digits total
than the number with the least number of
digits total.
5.10 x 6.213 x 5.425 = 172
Significant Figure
Multiplication and Division
205.244
= 76.016
2.7
76
Order of operations
1st: ( ), x2, square roots
2nd: x or /
3rd: + or –
Significant Figure
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:
– 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
• Calculators could be a significant aid in
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performing calculations in scientific notation.
KNOW HOW TO USE YOUR CALCULATOR
– 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 Conversions
• What is the scientific notation equivalent of
0.0432?
1043.50?
• What is the standard decimal notation
equivalent of 3.45 x 103?
6.500 x 10-2?
Scientific Notation
• 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
• 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
• Logarithms might seem strange, but they are
nothing more than another way of representing
exponents.
• logbx = y is the same thing as x = by
• Know how to use your calculator to perform
these functions.
Logarithms
We see logarithms frequently when working with pH
chemistry. If you have a solution of pH 5.2, and you need
to calculate the concentration of hydrogen ions, set the
problem up as follows:
pH = - log [H+]
5.2 = - log [H+]
-5.2 = log [H+]
10-5.2 = 10log [H+]
10-5.2 = [H+]
[H+] = 6.3 x 10-6
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
• Logarithms
log (4.21 x 1010) = 10.6242821  10.624
• Antilogarithms
antilog (- 7.52) = 10-7.52 = 3.01995 x 10-8  3.0 x 10-8
Significant Figures of Equipment
Electronics
• Always report all the digits electronic
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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)
Accuracy
• The accuracy of a measurement represents a
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comparison of the measured value (experimental
value) to the “true” value.
A measure of accuracy is indicated by:
Percent Error =
 Experimental Value  "True Value "

"True Value "


 x 100%

• Tolerances of glassware affect the accuracy of volume
measurements.
Precision
• Precision of a measurement reflects
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reproducibility of an experimental procedure.
Refer to the bull’s eye experiment on page
60.
Graduations on glassware affect the precision
of the glassware in question.
Statistics
• We use statistics in the laboratory in order to validate
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our results.
We evaluate the central tendency of our work by
calculating the mean (related to accuracy) of our
data.
We evaluate the variability in our work by calculating
the standard deviation (s) (related to precision) of our
data.
The relative standard deviation gives us a more
meaningful number than the standard deviation.
Calculation of the Mean
x

x
i
N
xi = individual values
N = number of measurements
For significant figures, always keep as many
digits after the decimal point as the original
values. Remember units!
Calculation of the standard
deviation of a set of numbers
s=
x
i
x
2
N1
xi = individual values
x = the average of the individual values
N = number of measurements
For significant digits, report the same digits of
precision as the xi values. The units are the same
as the units for the x values.
Calculation of Relative Standard
Deviation
 s
  x 100%
RSD% =  x 
s = standard deviation of a set of data
x = average of the individual measurements
The calculation itself dictates the number of
significant digits. What would the units be?
Dilutions
Using a solution of known concentration for the
preparation of a solution with a lower
concentration is commonly called dilution.
Solution Preparation from Solids
• Determine the mass of the solid needed.
You will need the following:
– Molar mass of the solid
– Total volume desired
– Final concentration desired
• Calculation:
m = M x MM x V
g = mol/L x g/mol x L
– Remember the precision of your glassware!
Solution Preparation from Solids
Make the solution:
– Weigh out the appropriate mass of solid.
– Place a small volume of distilled water in the
volumetric flask.
– Add the solid to the volumetric flask.
– Add some more distilled water to the flask, stopper,
and invert several times.
– Add distilled water to the calibration line (fill to
volume) using a medicine dropper, stopper, and
invert several times.
Solution Preparation from Liquids
• Determine the volume of stock solution
needed.
You will need the following:
– Concentration of stock solution (M1)
– Desired concentration of diluted solution (M2)
– Desired volume of diluted solution (V2)
• Calculation:
– M1V1 = M2V2
– Remember the precision of your glassware!
Solution Preparation from Liquids
Make the solution:
– Obtain the appropriate volume of stock solution using
a graduated cylinder. (Always add a few mL extra.)
– Place a small volume of distilled water in a
volumetric flask.
– Use the appropriate pipet to transfer the correct
volume of stock solution from the graduated cylinder
to the volumetric flask.
– Add some more distilled water to the flask, stopper,
and invert several times.
– Add distilled water to the calibration line (fill to
volume) using a medicine dropper, stopper, and
invert several times.
Serial Dilution
Serial dilution is a laboratory technique in which
substance concentration is decreased stepwise in
series.
Standard dilution
• Standard dilution is a laboratory technique in
which stock solution is used to prepare a diluted
solution.
Graphing
• Graphing is an important tool used to represent
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experimental outcomes and to set up calibration
curves.
It is a modeling device.
Graphing: Variables
• Having no fixed quantitative value.
– X-variable
– Y-variable
• Graphing in chemistry
– Renamed with a chemistry label
– Paired with a unit most of the time
Graphing: Units
• Give dimension to labels / variables
• Give meaning to numbers
• Essential!
Graphing: Coordinates
• A coordinate set consists of an x-value and y•
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value, plotted as a point on a graph.
X-values: domain (independent variable)
Y-values: range (dependent variable)
Graphing: Axes
• Multiple axes on a graph
• Coordinate sets determine the number of
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axes on a plot
Two dimensional graphs have only two axes
– X-axis
– Y-axis
• Each axis must have a consistent scale
Graphing in Chemistry
• Graph title reflects the:
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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
• Coordinate sets are plotted
– 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
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sets
The equation is used to find unknowns
Graphing: Equations
• 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
Label / Variable
Graphing
•Let’s look at a graphical
representation of the following
data:
Units
[Ni2+], M
Absorbance
0.200
0.041
0.300
0.063
0.400
0.085
0.500
0.101
Graphing
Graph Title
Absorbance
Absorbance vs. [Ni2+], M
0.110
0.100
y = 0.20 x + 0.002
0.090
0.080
0.070
Best-fit Line
0.060
0.050
0.040
0.200 0.250 0.300 0.350 0.400 0.450 0.500
Graph Axis Labels
[Ni 2+], M
Graphing
Always add a title to your graph. On the previous
slide, the title was
Absorbance vs. [Ni2+], M
because absorbance was graphed on the y-axis
and [Ni2+], M was graphed on the x-axis.
(Always y vs. x!)
If a graph title is
Temp F, degrees vs. Temp C degrees
what should be graphed on the x-axis?
Graphing
Always label your axis appropriately. The label for
the y-axis is Absorbance. Absorbance has no
units, so none are listed. The label for the x-axis
is [Ni2+] and the units M. What does “M” stand
for?
If an axis is labeled with Temp F, degrees
which one is the unit?
Graphing
If your data points look like they fall on a line, be
sure to add a “linear” calibration curve to them. If
they don’t appear linear, DO NOT add a linear line.
When you add a calibration curve, an equation
results. This equation describes the line and can
help you solve unknown values. The equation on
our graph was: y = 0.20 x + 0.002
Calibration Curves
• A calibration curve gives you a graphical
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representation of an instrument’s response to a
particular analyte.
If we were to declare your 1992 Ford Escort an
“instrument” and the gas it uses an “analyte,” we
could construct a calibration curve for:
Distance Driven, miles vs. Gas Consumed, gallons
Calibration Curves
• The measurements that are made are all
made:
– with the same vehicle
– using the same set of tires
– driving under similar road and environmental
conditions
– using the same type of gas
Data Table of Standards
Distance Driven, miles
Gas Consumed, gallons
25.3
1.0
49.2
2.0
73.9
3.0
98.2
4.0
122.8
5.0
Calibration Curve
Distance Driven, miles vs. Gas Consumed, gallons
Distance Driven, miles
140.0
y = 24 x + 0.7
R2 = 1
120.0
100.0
80.0
60.0
40.0
20.0
0.0
0.0
1.0
2.0
3.0
4.0
5.0
Gas Consumed, gallons
Notice that the average gas consumption of this vehicle is 24 mpg!
Unknowns
• When we talk about unknown “analytes,” we are referring to an
unknown measurement, not an unknown identity.
• If we were to analyze three unknowns related to our previous
example…we are still talking about gas, the unknown
measurement refers to either the gallons of gas consumed, or
the distance driven.
• No matter which unknown we are trying to determine, our
analysis must be made under the same conditions as
previously, in other words, unknown measurements should be
made:
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with the same vehicle
using the same set of tires
driving under similar road and environmental conditions
using the same type of gas
Data Table of Unknowns
Distance Driven, miles
Gas Consumed, gallons
41.5
?
82.1
?
103.6
?
Calibration Equation
• We can use the previously determined
•
calibration equation to determine how many
gallons of gas it would take to drive the number
of miles indicated on the previous slide.
Since Distance Driven was plotted on the y-axis
and Gas Consumed was plotted on the x-axis,
the new equation becomes:
Distance = 24 (Gas Consumed) + 0.7
Calibration Equation
Distance = 24 (Gas Consumed) + 0.7
Let’s solve this equation for Gas Consumed:
Gas Consumed = (Distance – 0.7) / 24
Let’s solve for our unknowns:
Solving Unknowns
Gas Consumed = (Distance – 0.7) / 24
Gas Consumed = (41.5 – 0.7) / 24
= 1.7 gallons
Gas Consumed = (82.1 – 0.7) / 24
= 3.4 gallons
Gas Consumed = (103.6 – 0.7) / 24
= 4.3 gallons
Laboratory Computer Etiquette
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Do not surf the web
Do not check e-mail unrelated to this course
Do not print materials unrelated to this course
Do not connect a USB mass storage drive
No social networking!!!
Do not open any attachments, unless directly
from your lab Blackboard shell or lab instructor.
You may access Blackboard from your lab
computer once given permission.
MicroLAB™
• MicroLAB™ is a computerized system that allows us to
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collect experimental data in real-time.
MicroLAB™ also has a spreadsheet program with
several data analysis features.
Your lab manual has the instructions for this part of
Lab 1.
Double check your work onscreen before printing out
anything.
MicroLAB™ does not give you the correct number of
significant figures for statistics. Use your data set to
determine these.
Don’t save any of your files!
Important…
• Always type a “0” before a decimal point for
•
•
numbers smaller than 1, e.g. 0.123.
Label columns correctly the first time. If you don’t,
you will have to delete and redo them, which will
result in a loss of data.
If you are told to label a column [Fe], M the “[Fe]”
refers to the label and the “M” the unit. If there
are no units present, then the particular variable in
question does not have any units.
Also Important…
• Do not give two columns the same label.
• Do not label a column for which you will need to
•
create a formula.
The digits of precision you set your column
properties to should reflect the digits of precision
of your tabulated data.
More Important…
• Select “Accept Data” often! Always select
•
•
“Accept Data” right before looking up column
statistics!
Column statistics can be accessed by right
clicking on a column and selecting “Column
Statistics.”
When asked to look at a graph to determine
certain values, always reference the spreadsheet
for the exact values instead of visually
estimating from the graph.