mathematics and graphing in chemistry lab 1. outline mathematics in chemistry units rounding digits...
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Mathematics and Graphing 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 Antilogarithms Algebraic Equations Averages Graphing
Mathematics in Chemistry
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 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
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:
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 0.01260.01265 rounded to three significant digits becomes 0.01260.01275 rounded to three significant digits becomes 0.01280.012851 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:
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 0.01260.01265 rounded to three significant digits becomes 0.01260.01275 rounded to three significant digits becomes 0.01280.012851 rounded to three significant digits becomes ? 0.0129
Why is this method statistically more correct?
Digits of Precision and Significant Figures 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 NumbersWhen you add or subtract whole numbers, your answer cannot be more accurate than any of your individual terms.
20 + 34 + 2400 – 100 = 2400Limited to the hundreds position
What about:319 + 870 + 34,650 = ?
Addition of Whole NumbersWhen 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 figures98,832 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
= 76.016 76205.244
2.7
Order of operations
1st: ( ), x2, square roots
2nd: x or /
3rd: + or –
Significant Figures: Mixed Orders
23
99
99 23
29.104(21.009 x 0.0032) 1.42
34.2
(21.009 x 0.0032) 0.067
29.1040.850
34.2
0.850 0.067 1.42 2.20
Scientific NotationThe 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
performing calculations in scientific notation.
KNOW HOW TO USE YOUR CALCULATORDoes 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
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 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 EquipmentElectronics
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)
Graphing
Graphing is an important tool used to represent experimental outcomes and to set up calibration curves.
It is a modeling device.
Graphing: Variables
Having no fixed quantitative value.X-variableY-variable
Graphing in chemistryRenamed with a chemistry labelPaired 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-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
axes on a plot Two dimensional graphs have only two
axesX-axisY-axis
Each axis must have a consistent scale
Graphing in Chemistry
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
Coordinate sets are plotted x-variable matching the x-value on the x-axisy-variable matching the y-value on the y-axisA 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
Of the form y = mx + bm = slope of the graphb = y-intercept of the graphx = any x-value from the graphy = corresponding y-coordinate
Your manual has sample calculations to derive the slope and y-intercept from two coordinate sets
Graphing
Let’s look at the following data:
[Ni2+], M Absorbance
0.200 0.041
0.300 0.063
0.400 0.085
0.500 0.101
Graphing
y = 0.20 x + 0.002
0.0400.0500.0600.0700.0800.0900.1000.110
0.200 0.250 0.300 0.350 0.400 0.450 0.500
Ab
sorb
an
ce
[Ni2+], M
Absorbance vs. [Ni2+], M
GraphingWhat 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?