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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

2x3 8÷2 2*3 8/2

2 . 3 Multiplying 8

2 Dividing

(2)(3) 2 8

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., 2 + 4 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 !

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2 Chemistry 047 – Math in Chem

2) Exponents 102 exponent multiply the base by itself, the exponent # of times base

i.e., 102 = 10 x 10 = 100

17 = 1x1x1x1x1x1x1 = 1 a2 = a . a 32 = 3 x 3 = 9 25 = 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

3 fraction form

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

3=

2

1 do not simplify as far as 2

D. Proportion

Compares 2 ratios Idea of proportion used to calculate an unknown # from related pairs

i.e., 5 𝑚𝑜𝑙𝑒𝑠

10 𝑚𝑜𝑙𝑒𝑠=

15 𝑔𝑟𝑎𝑚𝑠

𝑁 𝑔𝑟𝑎𝑚𝑠

N = 30 To calculate: cross multiply and divide

N = 15 x 10

5 = 30

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 4th 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?

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3 Chemistry 047 – Math in Chem

Can set up the proportion:

First # here 45 𝑔

𝑁 𝑔=

1.2 𝐿

0.40 𝐿 placement of 1st # directs others

or, even better and for future problem solving do in this order:

First # here 45 𝑔

1.2 𝐿=

𝑁 𝑔

0.40 𝐿

Start with what you are given followed by what you want Solve by isolating N –

45 𝑔 X 0.40 𝐿

1.2 𝐿= 𝑁

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 4,540,000,000 years old (approx. 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... 602,214,150,000,000,000,000,000 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: 4.54 x 109 instead of 4,540,000,000 With a calculator, you can calculate it and it will work out to be that number – How to do it...

1. Move the decimal point in the original number so that it is located after the first non zero digit: 2468 2.468 moves 3 decimal places to the left

2. Multiply by 10 raised to the power of how many decimal places you moved... 2.468 X 103

3. 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 6.022 x 1023 More Examples: 1) 5283 5.283 x 103 2) 4,500,000,000 4.5 x 109 What about small numbers? 3) 0.000123 1.23 x 10 -4

Try these: a) 1200 b) 6,600,000 c) 0.0468 d) 0.00003 Answers: a) 1.200 x 103 b) 6.6 x 106 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!!

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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 1000 100 10 1 0.1 0.01 0.001

Using differences in exponentials to figure out decimal placement…

i.e., Move from micro

to mega – 10-6 to 106 12 decimal points difference - to the left 56 micrograms megagrams – 0.000000000056 or 5. 6 x 10-11

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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 (oF) Water boils at 212 oF Water freezes at 32 oF

Celsius scale (oC) Water boils at 100 oC Water freezes at 0 oC

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 = -273.15 0 oC Freezing point of water = 273.15 K

K = oC + 273.15

oF = (1.8 x oC) + 32

oC = (oF – 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

ℎ𝑟x

1000𝑚

1 𝑘𝑚x

1 ℎ𝑟

60 𝑚𝑖𝑛x

1 𝑚𝑖𝑛

60 𝑠𝑒𝑐 = m/sec

Cross off the units to make sure you end up with only the units you want!

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54 𝑘𝑚

ℎ𝑟x

1000𝑚

1 𝑘𝑚x

1 ℎ𝑟

60 𝑚𝑖𝑛x

1 𝑚𝑖𝑛

60 𝑠𝑒𝑐 = 5400 m/3600 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: 1st unit X conversion factor = 2nd 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 = --- = ------------ = 0.08205 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

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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.: 42.3o±0.2o implies the value should lie somewhere between 42.1 and 42.5o

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 2.05 - 3 sig figs

6,109 and 61.09 - 4 sig figs

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Zeros right -If decimal - Zeros to the right of non zero numbers are always significant if there is a decimal point in the #.

3.00 – 3 sig figs 25.160 – 5 sig figs 0.500 – 3 sig figs

Zeros right -if no decimal pt in # - zeros to the right are NOT significant

590 - 2 sig figs

1000 – 1 sig fig

Zeros Left– Not! All zeros to the left of non zero #s are NOT significant

0.0025 – 2 sig figs

0.0108 – 3 sig figs

Examples:

a) 4.5 inches = 2 sig figs

b) 3.025 ft = 4

c) 125.0 m = 4

d) 0.001 mile = 1

e) 25.0 g = 3

f) 12.20 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 102

Rounding off numbers:

Example: 2.3 ÷ 4.215 = 0.5456702 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: 0.5456702

2. 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!

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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) = 438.38 – now change to correct sig figs:

2.3 has least with 2 sig figs so answer must have 2 sig figs:

Round to make 438.38 have 2 sig figs = 440 or 4.4 x 102

Example: (13.59)(6.3)/12 = 7.13475

6.3 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: 125.17 + 129 + 52.2 306.37

# with least precision is 129 (least decimal places) so answer will be 306 ex: 1587 -120 1467 but 120 has least precision =1470 or 1.47 x 103

Do ex: 1.039 – 1.020

1.039 = 0.018286814

Need to do subtraction 1st 1.039

-1.020

0.019

Now eqtn is 0.019

1.039 and 0.019 has 2 sig figs - final answer = 0.018 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)

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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 upper surface of a liquid in a tube or container.

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/cm3 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

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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(=20oC at 1 atmosphere) for liquids and solids and 0 oC and 1

atmosphere pressure for gases.

When comparing density of solids and liquids: water at 4 oC has a density of 1.000 –

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 1.293 g/L at 0

oC - 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…