Let ’ s Review: The Tools of Quantitative Chemistry

Jeffrey MackJeffrey MackCalifornia State University, California State University,

SacramentoSacramento

LetLet’’s Review: s Review: The Tools of Quantitative The Tools of Quantitative

ChemistryChemistry

"In physical science the first essential step in the direction of learning any subject is to find principles of numerical reckoning and practicable methods for measuring some quality connected with it.

I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of Science, whatever the matter may be."

Lord Kelvin, "Electrical Units of Measurement", 1883-05-03

The Tools of Quantitative ChemistryThe Tools of Quantitative Chemistry

Note About Math & ChemistryNote About Math & Chemistry

Numbers and mathematics are an inherent and unavoidable part of general chemistry. Students must possess secondary algebra skills and the ability to recognize orders of magnitude quickly with respect to numerical information to assure success in this course.

The material presented in this chapter is considered to be prerequisite to this course.

Science predominantly uses the “SI” (System International) system of units, more commonly known as the “Metric System”.

Units of MeasureUnits of Measure

The base units are modified by a series of prefixes which you will need to memorize.

Units of MeasureUnits of Measure

Temperature is measured in the CelsiusCelsius an the KelvinKelvin temperature scale.

Temperature UnitsTemperature Units

1KT(K) T C 273.15 C

1 C

1K25.0 C 273.15 C 298.2K

1 C

Temperature ConversionTemperature Conversion

The base unit of length in the metric system is the metermeter..

Depending on the object measured, the meter is scaled accordingly.

Length, Volume, and MassLength, Volume, and Mass

Unit conversions: How many picometers are there in 25.4 nm? How many yards?


124

9

2

nm m pm

1m 1 10 pm25.4nm 2.54 10 pm

1 10 nm 1m

m cm in ft yd

10 cm 1in 1ft 1yd25.4m 27.8 yards

1m 2.54cm 12in 3ft

The base unit of volume in the metric system is the literliter. .

1 L = 103 mL 1 mL=1 cm3 1 cm3 = 1 mL


3

3 34 3

L mL cm

10 mL 1cm25.4 L 2.54 10 cm

1 L 1 mL

The base unit of volume in the metric system is the gramgram.

1kg = 103g


119 3

ng g kg

1g 1kg25.4ng 2.54 10 g

1 10 ng 1 10 g

EnergyEnergy is confined as the capacity to do work.The SI unite for energy is the joulejoule (J).

Energy is also measured in calories (cal)1 cal = 4.184J

A kcal (kilocalorie) is often written as Cal. 1 Cal =103 cal

Energy UnitsEnergy Units

The precisionprecision of a measurement indicates how well several determinations of the same quantity agree.

Making Measurements: PrecisionMaking Measurements: Precision

AccuracyAccuracy is the agreement of a measurement with the accepted value of the quantity.

Accuracy is often reflected by Experimental Experimental errorerror.

Making Measurements: AccuracyMaking Measurements: Accuracy

The Standard Deviation Standard Deviation of a series of measurements is equal to the square root of the sum of the squares of the deviations for each measurement from the average divided by one less than the number of measurements (n).

Measurements are often reported to the standard deviation to report the precision of a measurement.

Making Measurements: Making Measurements: Standard DeviationStandard Deviation

Exponential or Scientific Notation:Exponential or Scientific Notation:

Most often in science, numbers are expressed in a format the conveys the order of magnitude.

3285 ft = 3.285 103 ft

0.00215kg = 2.15 103 kg

Mathematics of ChemistryMathematics of Chemistry

1.23 104

Coefficient or Mantissa

(this number is 1 and

<10 in scientific notation

Base Exponent

Exponential part

Exponential or Scientific NotationExponential or Scientific Notation

Significant figures: Significant figures: The number of digits represented in a number conveys the precision of the number or measurement.

A mass measured to 0.1g is far less precise than a mass measured to 0.0001g.

1.1g vs. 1.0001g1.1g vs. 1.0001g(2 sig. figs. vs. 5 sig. figs)

In order to be successful in this course, you will need to master the identification and use of significant figures in measurements and calculations!

Mathematics of ChemistryMathematics of Chemistry

1. All non zero numbers are significant2. All zeros between non zero numbers are

significant3. Leading zeros are NEVER significant.

(Leading zeros are the zeros to the left of your first non zero number)

4. Trailing zeros are significant ONLY if a decimal point is part of the number. (Trailing zeros are the zeros to the right of your last non zero number).

Counting Significant FiguresCounting Significant Figures

Determining Significant FiguresDetermining Significant Figures

Determine the number of Sig. Figs. in the following numbers

4 sf

7 sf

3 sf

5 sf

3 sf

4 sf

4 sf

1256

1056007

0.000345

0.00046909

0.08040

zeros written explicitly behind the decimal are significant…

not trapped by a decimal place.

1780

770.0

1. Find the last digit that is to be kept.2. Check the number immediately to the right:

If that number is less than 5 leave the last digit alone.If that number is 5 or greater increase the previous digit by one.

Rounding NumbersRounding Numbers

11000001056007

0.000345

1780

0.00035

1800

Rounding NumbersRounding Numbers

Round the following to 2 significant figures:

Multiplication/Division

The number of significant figures in the answer is limited by the factor with the smallest numbersmallest number of significant figures.

Addition/Subtraction

The number of significant figures in the answer is limited by the least precise numberleast precise number (the number with its last digit at the highest place value).

NOTE: counted numbers like 10 dimes never limit calculations.

Sig. Figures in CalculationsSig. Figures in Calculations

Determine the correct number of sig. figs. in the following calculation, express the answer in scientific notation.

23.50 0.2001 174 sf 4 sf 2 sf

= 1996.501749 10 sf

Your calculator knows nothing of sig. figs. !!!Your calculator knows nothing of sig. figs. !!!

from the calculator:


Determine the correct number of sig. figs. in the following calculation, express the answer in scientific notation.

in sci. notation: 1.996501749 103

Rounding to 2 sf: 2.0 103


Determine the correct number of sig. figs. in the following calculation:

391 12.6 + 156.1456


To determine the correct decimal to round to, align the numbers at the decimal place:

One must round the calculation to the least significant decimal.

391 12.6 +156.1456

39112.6

+156.1456

no digits hereno digits here


one must round to here391-12.6

+156.1456

534.5456 (answer from calculator)

round to here (units place)

Answer: 535


Combined Operations:Combined Operations: When there are both addition & subtraction and or multiplication & division operations, the correct number of sf must be determined by examination of each step.

Example: Complete the following math mathematical operation and report the value with the correct # of sig. figs.

(26.05 + 32.1) (0.0032 + 7.7) = ???


Example: Complete the following math mathematical operation and report the value with the correct # of sig. figs.

(26.05 + 32.1) (0.0032 + 7.7) = ???

1st determine the correct # of sf in the numerator (top)

2nd determine the correct # of sf in the denominator (bottom)

The result will be limited by the least # of sf (division rule)


26.05

+ 32.1

0.0032

+ 7.7

3 sf

2 sfThe result

may only have 2 sf

=58.150

7.7032


2 sig figs!

3 sig figs

7.7032

58.150

= 7.5488 = 7.5

2 sfRound to here

Carry all of the digits through the calculation and round at the end.


Dimensional Analysis:Dimensional analysis converts one unit to another by using conversion factors (CFconversion factors (CF’’s)s)..

The resulting quantity is equivalent to the original quantity, it differs only by the units.

= unit (2)unit (1) conversion factorconversion factor

Problem Solving and Problem Solving and Chemical ArithmeticChemical Arithmetic

Dimensional Analysis:Dimensional Analysis:Dimensional analysis converts one unit to another by using conversion factors (CFconversion factors (CF’’s)s)..

Conversion factors come from equalities:

1 m = 100 cm

1 m

100 cmor

1 m

100 cm


Exact Conversion FactorsExact Conversion Factors:: Those in the same system of units

1 m = 100 cm

Use of exact CF’s will not affect the significant figures in a calculation.

Examples of Conversion FactorsExamples of Conversion Factors

1.000 kg = 2.205 lb

SI unitsSI units British Std.British Std.

Use of inexact CF’s will affect significant figures.

(4 sig. figs.)

Inexact Conversion FactorsInexact Conversion Factors:: CF’s that relate quantities in different systems of units

Examples of Conversion FactorsExamples of Conversion Factors

• Problem solving in chemistry requires “critical thinking skills”.

• Most questions go beyond basic knowledge and comprehension. (Who is buried in Grant’s tomb?)

• You must first have a plan to solve a problem before you plug in numbers.

• You must evaluate the result to see if it makes sense. (units, order of magnitude)

• You must also practice to become proficient because...

Chem – is – tryChem – is – try


• Before starting a problem, devise a “Strategy MapStrategy Map”.

• Use this to collect the information given to work your way through the problem.

• Solve the problem using Dimensional Analysis.

• Check to see that you have the correct units along the way.


Most importantly, before you start...

PUT YOUR CALCULATOR DOWN!PUT YOUR CALCULATOR DOWN!

Your calculator wont help you until you are ready to solve the problem based on your strategy map.


ExampleExample: How many meters are there in 125 miles?

First: Outline of the conversion:



First: Outline of the conversion:

m miles ft in cm

Each arrow indicates the use of a conversion factor.



2.54 cm

1 in =

Second: Setup the problem using Dimensional Analysis:

m miles ft in cm



Third: Check your sig. figs. & cancel out units.

m

2.54 cm

1 in =

miles ft in cm

3 sf exact exact exact3 sf


//

/ // /

/ /


Fourth: Now use your calculator. :

m

2.54 cm

1 in/

//

//

//

/ =

miles ft in cm


Carry though all digits, round at end



2.54 cm

1 in

//

/

2.01168 105

=

or 2.01 105 m (3 sf)


Lastly: Check your answer for sig. figs & magnitude.

m

//

miles ft in cm

///


ExampleExample: How many square feet are there in 25.4 cm2?

Map out your conversion:

ft2

//// 2.73403 10-2

ft2=

cm2 in2

or 2.73 10-2 ft2 (3 sf)

3 sf exact exact


ExampleExample: How many cubic feet are there in 25.4 cm3?


ft3

//// 8.96993 10-4

ft3=

cm3 in3

or 8.97 10-4 ft3 (3 sf)

3 sf exact exact


ExampleExample: What volume in cubic feet would 0.851 grams of air occupy if the density is 1.29 g/L?


ft3 L in3 cm3 g

/

3 sf

3 sf3 sf 3 sf exact3 sf

//

//

//

/


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Let ’ s Review: The Tools of Quantitative Chemistry