Mathematics in Chemistry Lab 1. Outline Mathematics in Chemistry Mathematics in Chemistry –Units...

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 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 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.

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

0.01260.01265 rounded to three significant digits becomes

0.01260.01275 rounded to three significant digits becomes

0.01280.012852 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

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 figures98,832 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?

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

= 76.016 76205.244

2.7

Order of operations

1st: ( ), x2, square roots2nd: x or /3rd: + or –

Significant Figure 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 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 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 + 0.0009If 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 comparison of the measured value (experimental value) to the “true” value.

• A measure of accuracy is indicated by:

Percent Error =

• Tolerances of glassware affect the accuracy of volume measurements.

Experimental Value "True Value"x 100%

"True Value"

Precision

• Precision of a measurement reflects 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 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

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!

ixx

N

Calculation of the standard deviation of a set of numbers

s =

xi = individual values = the average of the individual valuesN = 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.

2

ix x

N 1

x

Calculation of Relative Standard Deviation

RSD% =

s = standard deviation of a set of data = average of the individual measurements

The calculation itself dictates the number of significant digits. What would the units be?

sx 100%

x

x

DilutionsUsing 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 Vg = 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 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-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

axes– X-axis– Y-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-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

• Of the form y = mx + b• 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 a graphical representation of the following data:

[Ni2+], M Absorbance

0.200 0.041

0.300 0.063

0.400 0.085

0.500 0.101

Label / Variable

Units

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

Graph Title

Graph Axis Labels

Best-fit Line

GraphingAlways 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, degreeswhich 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 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

y = 24 x + 0.7R2 = 1

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

0.0 1.0 2.0 3.0 4.0 5.0

Dis

tan

ce

Dri

ve

n, m

ile

s

Gas Consumed, gallons

Distance Driven, miles vs. 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:– 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• 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 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.