**Multiply has higher priority **some calculators will do in correct order but not all DO NOT rely on your calculator!

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1 1 Chemistry 047 Math in Chem Math in Chemistry: A. Multiplication/Division symbols B. Order of operations C. Ratio D. proportion E. Scientific notation F. Unit conversions G. Dimensional analysis H. Derived quantities I. Experimental uncertainty, accuracy, and precision J. Sig figs K. Reading scales L. Density and rearranging any equation A. Multiplying or dividing: symbols 2x *3 8/ Multiplying 2 (2)(3) 2 8 Dividing B. Order of operations Highest *Bracket priority *exponent *division/multiplication all x and, L to R Lowest *addition/subtraction all + and -, L to R If only 1 type of operation proceed L to R If greater than 1 operation must do in order of priority i.e., x 3 if you solve it in order it would =24 but it is =14 NOT 24. Solve mult first, then addition **Multiply has higher priority **some calculators will do in correct order but not all DO NOT rely on your calculator! 1) Brackets highest priority, evaluate operation inside bracket first i.e., 2(3 + 4) 2(7) = 14 Nested brackets: work from inside out; ( ) [ ] { } i.e., {4 [ 3+ ( 7 2 )]} 2 {4 [ 3+ 5 ]} 2 {4 [ 8 ]} 2 {32} 2 =30 Sometimes brackets all ( ) so pay attention to working from inside out! P a g e 1 11

2 2 Chemistry 047 Math in Chem 2) Exponents 10 2 exponent multiply the base by itself, the exponent # of times base i.e., 10 2 = 10 x 10 = = 1x1x1x1x1x1x1 = 1 a 2 = a. a 3 2 = 3 x 3 = = 2 x 2 x 2 x 2 x 2 = 32 C. Ratio - A ratio is a comparison of 2 numbers - Units therefore must be identical Written as 5 to 3 word form 5:3 -colon form in chemistry 5 fraction form 3 if units are not identical convert ie, 5 mins to 1 hour both are time units but not identical so 5 mins to 60 mins = 5:60 min or 5 60 Then simplify ratios should always be in simplest form 5 60 = 1 12 but always retain the 2 numbers in ratio form i.e., 6 = 2 do not simplify as far as D. Proportion Compares 2 ratios Idea of proportion used to calculate an unknown # from related pairs 5 moles i.e., 10 moles = 15 grams N grams N = 30 To calculate: cross multiply and divide 15 x 10 N = = 30 5 Get N by itself what is on the bottom comes up and multiplies, what is on the top goes down and divides How can you tell if a word problem can be solved by proportions? If you have 3 numbers and want to calculate a 4 th And pairs of #s are related i.e., If 45 grams of CaCO3 reacts to form 1.2 litres of CO2, how many grams will produce 0.40 litres? P a g e 2 11

3 3 Chemistry 047 Math in Chem Can set up the proportion: 45 g First # here N g = 1.2 L 0.40 L placement of 1 st # directs others or, even better and for future problem solving do in this order: First # here 45 g = N g 1.2 L 0.40 L Start with what you are given followed by what you want Solve by isolating N 45 g X 0.40 L = N 1.2 L 15 g = N E. Scientific notation Scientists use numbers that are sometimes very large or very small and it is very awkward to write it out...the age of the earth is 4,500,000 years old (4 and a half billion) but in order to figure this out we have to count how many zeroes also leaves room for error with the really big or really small numbers Another example Avogadro s number ,214,150,000,000,000,000,000 Obviously for a scientist who needs to deal with lots of very big or very small numbers this is not logical so we use what is called: Scientific Notation: The age of the earth in scientific notation is: 4.5 x 10 6 instead of 4,500,000 With a calculator, you can calculate it and it will work out to be that number How to do it Move the decimal point in the original number so that it is located after the first non zero digit: moves 3 decimal places to the left 2. Multiply by 10 raised to the power of how many decimal places you moved X the sign on the exponent indicates direction the decimal moved Moved right - negative exponent Moved left - positive exponent Avogadro s number? 602,214,150,000,000,000,000,000 or x More Examples: 1) x ) 4,500,000, x 10 9 What about small numbers? 3) x 10-4 Try these: a) 1200 b) 6,600,000 c) d) Answers: a) x 10 3 b) 6.6 x 10 6 c) 4.68 x 10-2 d) 3.0 x 10-5 Try the handout to practice Scientific Notation (Under Extra Worksheets on the website) ***VERY IMPORTANT: You need to know how to use your EXP button on your calculator to do calculations with scientific notation!! P a g e 3 11

4 4 Chemistry 047 Math in Chem In Hebden the following further topics are covered: Unit conversions Derived quantities Density Significant figures Accuracy and precision Reading scales Experimental uncertainty F. Unit Conversions Metric to metric: Scale to remember: Using it the decimal moves in the direction you are going up and down the scale, and the number of places you move determines how far the decimal moves. kilo Hecto deca Base unit deci centi milli Using differences in exponentials to figure out decimal placement i.e., Move from micro to mega 10-6 to decimal points difference - to the left 56 micrograms megagrams or 5. 6 x P a g e 4 11

5 5 Chemistry 047 Math in Chem Using conversion factors: Either use known conversion factors or know how much bigger or smaller something is: Examples: Kilograms is 1000 x larger then grams so conversion factor = 1g/1000kg 1 cm is 100,000x smaller than a km so 1km/100,000cm Temperature conversions and rearranging eqtns Fahrenheit Scale ( o F) Water boils at 212 o F Water freezes at 32 o F Celsius scale ( o C) Water boils at 100 o C Water freezes at 0 o C Kelvin Scale (K) Not based on boiling or freezing based on absolute zero which is the point where no motion of particles Based on the theoretically lowest temp attainable by an object Has no upper limit 0 K = o C Freezing point of water = K K = o C o F = (1.8 x o C) + 32 o C = ( o F 32)/1.8 Do multiple conversions using G. Dimensional Analysis Use this method when possible!! 1) Solve using sequential series of proportions 2) Set up units first to make sure you are cancelling correctly before you put in the #s check to make sure the unit you end up with is the one you want Example: A cyclist is travelling at 54 km/hr. How fast is this in m/sec? Want to go from km/hr m/sec Need to set up conversion factors km to m and hr to sec km x 1000m x 1 hr x 1 min = m/sec hr 1 km 60 min 60 sec Cross off the units to make sure you end up with only the units you want! P a g e 5 11

6 6 Chemistry 047 Math in Chem 54 km x 1000m x 1 hr x 1 min = 5400 m/3600 sec hr 1 km 60 min 60 sec = 15.2 m/sec = 15 m/sec (correct significant figures) you will learn about sig figs near the end of this section. Basic Steps: 1 st unit X conversion factor = 2 nd unit 1) Determine what you want to solve for, and write it down 2) Determine unit conversion factors needed (look it up if needed) 3) Set up problem so units cancel 4) Do operation ensure sig figs are correct 5) Check your answer is it reasonable? H. Derived quantities other examples pg 23 Hebden Derived quantities is a number made by combining 2 or more other values. Thus the importance of using units when solving!! Example: P V 1 atm 22.4 L R = --- = = atm L / (mol K) n T 1 mol 273 K where p=pressure, v=volume, n=moles, T=temp in Kelvin Units of something actually tells us what the formula is for that particular thing in the above example, given that the units are atm L/mol K, that tell us that that IS the formula! I. Experimental uncertainty, accuracy, and precision: In chemistry we make measurements in experiments to understand and learn about physical and chemical changes that occur in whatever substance we are studying. Measurement is always expressed as a number and a unit (a word/symbol): Whenever a measurement is made it is an estimate - measurements are never exact values- it depends on the precision of the instrument and user this will determine the accuracy of the estimate - for example thermometers, scales etc Accuracy = how close a result comes to the true value P a g e 6 11

7 7 Chemistry 047 Math in Chem Precision = the closeness 2 or more measurements are to each other i.e., if you weigh something 4 times and get the same answer every time your answer is very precise this does not mean it is accurate! In a measurement the last digit is considered to be uncertain this is where the variation in the measurement comes in the first digits are considered to be certain. To express max precision number should contain all digits that are known plus 1 digit that is estimated. Precision = reliability of the measured value = Contains all known digits plus one estimated digit Estimating gives uncertainty therefore can only have a limited number of digits to express a measured quantity = significant figures Significant figures are used to indicate the precision of numerical values calculated from measurement. Exact numbers: Exact numbers are counting numbers or defined numbers such as: defined # s 100 cm= 1m, 12 inches in 1 ft etc Count 25 dollars you have exactly 25$ Have no uncertainty therefore they have an infinite # of significant figures. Expressing uncertainty: Generally expressed as ± which implies that the actual value lies somewhere in the range given. i.e.: 42.3 o ±0.2 o implies the value should lie somewhere between 42.1 and 42.5 o J. Significant figures: Because all measurements involve uncertainty using the proper number of significant figures is very important! General rules for Sig Figs: 1. All non zero #s are significant 2. Exact numbers: have infinite # of sig figs have no uncertainty 3. Zeros: Sandwiched - Zeros sandwiched between non zeros are significant i.e, 205 and sig figs 6,109 and sig figs P a g e 7 11

8 8 Chemistry 047 Math in Chem Zeros right -If decimal - Zeros to the right of non zero numbers are always significant if there is a decimal point in the # sig figs sig figs sig figs Zeros right -if no decimal pt in # - zeros to the right are NOT significant sig figs sig fig Zeros Left Not! All zeros to the left of non zero #s are NOT significant sig figs sig figs Examples: a) 4.5 inches = 2 sig figs b) ft = 4 c) m = 4 d) mile = 1 e) 25.0 g = 3 f) L = 4 g) 100,000 people = 1 h) 205 birds = 3 Can show number of sig figs by using scientific notation: 590 has 2 sig figs can write it like: 5.9 x 10 2 Rounding off numbers: Example: = answer has more # s then justified after division based on the precision of the initial measured values: Answer MUST have the same precision or # of sig figs as the least sig figs in the initial measured values therefore must round the number to the correct sig figs: Procedure for rounding #s: 1. Underline last digit you want to round to i.e, if you need 2 sig figs in the # such as above: Look at digit to rt of underlined digit 3. If digit is 4 or lower, underlined digit stays the same 4. If digit is 5 or higher (as in this example), the underlined digit is rounded up by 1 drop all digits to the rt of the underline if right of decimal i.e, 0.55 Significant figures in calculations Answer cannot be more precise then the least precise measurement! P a g e 8 11

9 9 Chemistry 047 Math in Chem Multiplication or division Answer must contain the same sig figs as the measurement with the least # of sig figs: Example: (190.6)(2.3) = now change to correct sig figs: 2.3 has least with 2 sig figs so answer must have 2 sig figs: Round to make have 2 sig figs = 440 or 4.4 x 10 2 Example: (13.59)(6.3)/12 = and 12 have 2 sig figs Therefore: 7.1 Addition or subtraction Answer must be expressed to the same precision as the least precise measurement -this means that answer must be rounded to the same # of decimal places as the value with the fewest decimal places: ex: # with least precision is 129 (least decimal places) so answer will be 306 ex: but 120 has least precision =1470 or 1.47 x 10 3 Do ex: = Need to do subtraction 1 st Now eqtn is and has 2 sig figs - final answer = or 1.8 x10-2 Rules for Sig Figs in Calculations: Multiply or divide = answer has same # of sig figs as measurement with the least sig figs Add or subtract = answer has same # of sig figs as the least precise measurement (which = the one with the least decimal places) P a g e 9 11

10 10 Chemistry 047 Math in Chem K. How to read scales: Your ability to read a scale properly will affect the accuracy of your answer the courseness of the scale will determine precision. Look at the numbered divisions and determine the number of unnumbered subdivisions and calculate the value of the intervals. These will be certain numbers. The uncertainty comes in when reading where the pointer lies on or between these lines. When reading a scale ALWAYS read it at eye level!! Meniscus = the curved or container. upper surface of a liquid in a tube L. Density (and how to rearrange equations to solve for unknown): Ratio of mass of substance to its volume Considered a physical characteristic of a substance Density = mass/volume or mass per unit volume Can be expressed as gram/cm 3 or g/ml or g/l and can refer to density of solid, liquid or gas **density is dependent on temp Density changes especially of gases or liquids with changes in temp - Therefore must specify at what temp the object is P a g e 10 11

11 11 Chemistry 047 Math in Chem Also dependant on atmospheric pressure (especially gases) So unless otherwise stated, densities are reported at STP = Standard temperature and pressure(=20 o C at 1 atmosphere) for liquids and solids and 0 o C and 1 atmosphere pressure for gases. When comparing density of solids and liquids: water at 4 o C has a density of at this temperature water is at its maximum density (i.e. 1 L of water would weigh about 1 kg)- The maximum density of water is used as a standard of comparison for expressing the density of other liquids and solids. When comparing densities of gases we use a reference of air which is g/l at 0 o C - not covered in this level Going back to the section on derived quantities the units tell us what the formula is for density: units are g/l, therefore, formula is: D= g L which means D= mass volume Once you know one version of the formula, you can solve for any part of it that is unknown by rearranging the equation. This is more easily explained in class P a g e 11 11