Chapter 1: Let’s Review: The Tools of Quantitative Chemistry

Note 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 is considered to be a prerequisite to this course. Optional Slide

Units of Measure Science predominantly uses the “SI” (System International) system of units, which is similar to the “Metric System”.

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

Temperature Units Temperature is measured in the Fahrenheit, Celsius, and the Kelvin temperature scale.

Temperature Conversion Convert °F to °C TC = 5/9 ( TF – 32.00 ) Convert °C to K TK = TC + 273.15

Length, Volume, and Mass In the metric system: The base unit of length is the meter 1 m ≡ 100 cm 1 inch ≡ 2.54 cm The base unit of volume is the liter 1 L ≡ 1 dm3 1 L ≡ 1000 mL 1 mL ≡ 1 cm3 1 L ≈ 1.057 quart The base unit of mass is the gram 1kg ≡ 1000 g 1 kg ≈ 2.205 lb

Energy Units Energy is confined as the capacity to do work. The SI unit for energy is the joule (J). Energy is also measured in calories (cal) 1 cal = 4.184 J A kcal (kilocalorie) is often written as Cal. 1 Cal = 103 cal

Making Measurements: Precision The precision of a measurement indicates how well several determinations of the same quantity agree.

Making Measurements: Accuracy Accuracy is the agreement of a measurement with the accepted, or true value of the quantity. Accuracy is reflected by the Experimental Error, often reported as the Percent Error: Observed – True Value Percent Error = ———————————— x 100% True Value

Making Measurements: Standard Deviation The Standard Deviation ( s ) of a series of measurements is equal to the square root of the sum of the squares of the deviations from the mean, divided by one less than the number of measurements (n – 1). Measurements are often reported to  the standard deviation to report the precision of a measurement.

Mathematics of Chemistry 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 0.00215 kg = 2.15  103 kg

Exponential or Scientific Notation 1.23  104 Coefficient or Mantissa (this number is 1 and <10 in scientific notation Exponential part Base Exponent

Mathematics of Chemistry Significant figures: The number of digits represented in a number conveys the precision of the number or measurement. A mass measured to  0.1 g is far less precise than a mass measured to  0.0001 g. 1.1 g vs. 1.0001 g (2 sig. fig. vs. 5 sig. fig) In order to be successful in this course, you will need to master the identification and use of significant figures in measurements and calculations!

Counting Significant Figures All non zero numbers are significant All zeros between non zero numbers are significant Leading zeros are NEVER significant. (Leading zeros are the zeros to the left of your first non zero number) 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).

Determining Significant Figures Determine the number of Sig. Figs. in the following numbers 1256 4 sf 1056007 7 sf 0.000345 not trapped by a decimal place. 3 sf 0.00046909 5 sf Instructor Note: Each individual component to this slide may be brought in on unit at a time using the “custom animation” tools in PowerPoint. zeros written explicitly behind the decimal are significant… 1780 3 sf 770.0 4 sf 0.08040 4 sf

Rounding Numbers 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, then leave the last digit alone. If that number is 5 or greater, then increase the previous digit by one.

Rounding Numbers Round the following to 2 significant figures: 1056007 1100000 0.000345 0.00035 Instructor note: Each individual component to this slide may be brought in on unit at a time using the “custom animation” tools in PowerPoint. 1780 1800

Sig. Figures in Calculations Multiplication/Division The number of significant figures in the answer is limited by the factor with the smallest number of significant figures. Addition/Subtraction The number of significant figures in the answer is limited by the least precise number (the number with its last digit at the highest place value). NOTE: Counting numbers ( integers ), such as 12 dimes, will never limit the calculation.

Sig. Figures in Calculations Determine the correct number of sig. figs. in the following calculation, and express the answer in scientific notation. from the calculator: 4 sf 4 sf 2 sf 23.50  0.2001  17 = 1996.501749 10 sf Your calculator knows nothing of sig. fig. !

Sig. Figures in Calculations Determine the correct number of sig. figs. in the following calculation, and express the answer in scientific notation. in sci. notation: 1.996501749  103 Rounding to 2 sf: 2.0  103

Sig. Figures in Calculations Determine the correct number of sig. figs. in the following calculation: 391  12.6 + 156.1456

Sig. Figures in Calculations 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 place. 391  12.6 +156.1456 391 no digits here 12.6 +156.1456

Sig. Figures in Calculations 391 -12.6 +156.1456 one must round to here 534.5456 (answer from calculator) round to here (units place) Answer: 535

Sig. Figures in Calculations Combined Operations: When there are both addition & subtraction and or multiplication & division operations, the correct number of sf is 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) = ???

Sig. Figures in Calculations Example: Complete the following mathematical operation and report the value with the correct # of sig. fig. (26.05 + 32.1)  (0.0032 + 7.7) = ??? 2nd determine the correct # of sf in the denominator (bottom) 1st determine the correct # of sf in the numerator (top) The result will be limited by the least # of sf (division rule)

Sig. Figures in Calculations 3 sf 26.05 + 32.1 = 58.150 7.7032 0.0032 + 7.7 2 sf The result may only have 2 sf

Sig. Figures in Calculations Carry all of the digits through the calculation and round at the end. 3 sig fig 58.150 7.7032 2 sig fig = 7.5488 = 7.5 Round to here 2 sf

Problem Solving and Chemical Arithmetic Dimensional Analysis: Dimensional analysis converts one unit to another by using a conversion factor (CF). The resulting quantity is equivalent to the original quantity, it differs only by the units. = unit (2) unit (1)  conversion factor

Problem Solving and Chemical Arithmetic Dimensional Analysis: Dimensional analysis converts one unit to another by using a conversion factor (CF). Conversion factors come from equalities: 1 m = 100 cm 1 m 100 cm or

Problem Solving and Chemical Arithmetic Dimensional Analysis: Dimensional analysis converts one unit to another by using a conversion factor (CF). Wanted Units CF = ————————— Given Units

Examples of Conversion Factors Exact Conversion Factors: CF in the same system of units, and CF determined by definition 1 m ≡ 100 cm 1 inch ≡ 2.54 cm 1 mile ≡ 5280 feet Use of exact CF’s will not affect the significant figures in a calculation, because exact CF’s contain an infinite number of SF’s

Examples of Conversion Factors Inexact Conversion Factors: CF that relate quantities in different systems of units 1.000 kg = 2.205 lb SI units British Std. (4 sig. figs.) Use of inexact CF’s will affect significant figures.

Problem Solving and Chemical Arithmetic Problem solving in chemistry requires “critical thinking skills”. Most questions go beyond basic knowledge and comprehension. 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 – try

Problem Solving and Chemical Arithmetic Before starting a problem, devise a “Strategy 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.

Problem Solving and Chemical Arithmetic Most importantly, before you start... PUT YOUR CALCULATOR DOWN! Your calculator wont help you until you are ready to solve the problem based on your strategy map.

Problem Solving and Chemical Arithmetic Example: How many meters are there in 125 miles? First: Outline of the conversion: miles  ft  in  cm  m Each arrow indicates the use of a conversion factor.

Problem Solving and Chemical Arithmetic Example: How many meters are there in 125 miles? Second: Setup the problem using Dimensional Analysis: m Miles ft  in  cm  =

Problem Solving and Chemical Arithmetic Example: How many meters are there in 125 miles? Third: Check your sf and cancel out units. m = miles  ft  in  cm  3 sf exact / / / / / / / /

Problem Solving and Chemical Arithmetic Example: How many meters are there in 125 miles? Fourth: Now use your calculator: m / = miles  ft  in  cm  3 sf exact Carry though all digits, round at end 2.01168 x 105 m Round to: 2.01 x 105 (3 sf)

Problem Solving and Chemical Arithmetic Example: How many square feet are in 25.4 cm2 ? Map out your conversion: ft2 / 2.73403  10-2 ft2 = cm2  in2  Round to: 2.73  10-2 ft2 (3 sf) exact 3 sf

Problem Solving and Chemical Arithmetic Example: How many cubic feet are in 25.4 cm3 ? Map out your conversion: cm3  in3  ft3 / / = 8.96993  10-4 ft3 / / 3 sf exact exact Round to: 8.97  10-4 ft3 (3 sf)

Problem Solving and Chemical Arithmetic Example: What volume in cubic feet would 0.851 g of air occupy if it’s density is 1.29 g / L ? Map out your conversion: ft3 L  in3  cm3  g  / 3 sf exact

Problem Solving and Chemical Arithmetic Example: How many picometers are in 25.4 nm? How many yards are in 25.4 m?

Problem Solving and Chemical Arithmetic Unit conversions: How many cm3 are there in 25.4 L ? 1 L = 103 mL 1 mL = 1 cm3

Problem Solving and Chemical Arithmetic Unit conversions: How many grams are there in 5.67 pounds ? 1 kg ≈ 2.205 lb 1 kg ≡ 1000 g lb → kg → g 1.000 kg 1000 g 5.67 lb x ———— x ——— = 2.57 x 103 g 2.205 lb 1 kg