MEASUREMENT ANDCALCULATIONSChapter 2

HOW DO WE ANSWER QUESTIONS INSCIENCE? Scientific Method – A logical approach to

solving problems by: Observing and collecting data Formulating hypotheses Testing hypotheses Formulating theories that are supported by data.

OBSERVING/COLLECTING DATA

Observing – using the senses May give us qualitative (descriptive) or quantitative

(numerical) information

Experimenting – carrying out a procedure to make observations and collect data System – A specific portion of matter in a given

region of space that has been selected for study during an experiment or observation

FORMULATING HYPOTHESES

Hypothesis – A testable statement Serves as a basis for experimentation Often drafted as “if-then” statements

TESTING HYPOTHESES

Controls – experimental conditions that remain constant

Variable – experimental conditions that change

Results here will either prove your hypothesis or cause it to be discarded or modified

THEORIZING

Model A physical representation of a system An explanation of how phenomena occur and how

data/events are related

Theory A broad generalization that explains a body of facts

or phenomena

UNITS OF MEASUREMENT

As part of our data collection, many times we will take measurements

Measurements always contain two parts Number Unit

A number without a unit does not give you any useful information!

Example: Joe is 7. Seven what?

INTERNATIONAL SYSTEM OF UNITS (SI UNITS) Uses seven “base” units:

Meters (m) for length Kilograms (kg) for mass** Kelvin (K) for temperature Moles (mol) for amount of substance Ampere (A) for electric current Candela (cd) for luminous intensity

Mass: the measure of the quantity of matter Weight: the measure of the gravitational pull on

matter

Notice that mass is the only “base” unit that has a prefix!

INTERNATIONAL SYSTEM OF UNITS (SI UNITS) Derived Units

Units made from a combination of two or more SI base units

Examples: Volume

Length x width x height (m3) Liquid volume in liters Handy equivalent: 1 mL = 1 cm3

Density D = 𝑀𝑀

𝑉𝑉

Ratio of mass to volume (g/mL) As the value rises, the objects are “heavier”

DENSITY CALCULATIONSA) If mass is 6.2g and volume is 4.7 mL, what is

the density?

𝐷𝐷 = 𝑀𝑀𝑉𝑉

=6.2g/4.7mL = 1.3 g/mL

B) If the density of a substance is 6.72 g/mL and you have 55.1mL of it, what should the mass be?

M = D•V = (6.72 g/mL) (55.1 mL) = 370. g

DENSITY CALCULATIONS

C) If the density of a substance is 0.824 g/mL and you have 0.451g of it, V = ?

𝑉𝑉 = 𝑀𝑀𝐷𝐷

= (0.451 g) / (0.824 g/mL) = 0.547 mL

INTERNATIONAL SYSTEM OF UNITS (SI UNITS)

Instead of using many units for the same measurement, the SI unit system uses prefixes to change the size of units:

PROBLEM SOLVING/DIMENSIONAL ANALYSIS

Many of the problems you will be asked to solve during this course involve converting units. The method used for this is called the factor-label

method, or dimensional analysis.

In order to convert between units, you must first know how they are related to each other. This relationship is referred to as an equivalency statement. For example: 1 meter = 100 centimeters 60 minutes = 1 hour 3 feet = 1 yard

DIMENSIONAL ANALYSIS

There is an algebraic rule that says a fraction with equivalent values in the numerator and the denominator is equal to one:

= 1

Since the quantities in an equivalency statement are equal, we can make a fraction from them that is equal to one:

= 1

This is a conversion factor – a ratio derived from the equality between two different units

1010

60 minutes1 hour

DIMENSIONAL ANALYSIS

We can use these fractions to set up a problem in which we cancel units to convert values from one unit to another.

3.5 hours = ? minutes1 hour = 60 minutes

NOTICE – since we are technically multiplying by a value of one, we haven’t changed the value of the number, just the unit it is expressed in!

hours 3.5

hour 1minutes 60 minutes 210=

DIMENSIONAL ANALYSIS

What if you do not know a direct relationship between what you want to convert? You can use multiple conversion factors:

380 cm = ? km

380 cm = 0.0038 km

PRACTICE

An island has no currency; instead it has the following exchange rate:

50 bananas = 20 coconuts12 fish = 30 coconuts1 hammock = 100 fish

How many bananas would you need to purchase a hammock?

1 ℎ100 𝑓𝑓

1 ℎ30 𝑐𝑐12 𝑓𝑓

50 𝑏𝑏20 𝑐𝑐

= 625 𝑏𝑏

PRACTICE

It takes a seamstress 4.56 hours to complete a skirt. A boutique in town would like to buy 25 of these skirts. If she worked without stopping, how long would it take her (in minutes) to complete the skirts for the boutique?

4.56 h = 1 s60 min = 1 h

25 𝑠𝑠4.56 ℎ

1 𝑠𝑠60 𝑚𝑚𝑚𝑚𝑚𝑚

1 ℎ= 6,840 𝑚𝑚𝑚𝑚𝑚𝑚

PRACTICE

A store sells its own blend of perfume by volume. A girl wants to refill her 0.015L bottle with her favorite fragrance, which costs $1.75 for each milliliter. How much will it cost to fill the bottle?

1 mL = 1.75 d1 L = 1000 mL

0.015 𝐿𝐿1000 𝑚𝑚𝐿𝐿

1 𝐿𝐿1.75 𝑑𝑑1 𝑚𝑚𝐿𝐿

= 26.25 d

PRACTICE

How long (in minutes) will it take to run 1.45 km if the average speed of a runner is 1.85 m/s?

1.85 m = 1 s1000 m = 1 km60 s = 1 min

1.45 𝑘𝑘𝑚𝑚1000 𝑚𝑚

1 𝑘𝑘𝑚𝑚1 𝑠𝑠

1.85 𝑚𝑚1 𝑚𝑚𝑚𝑚𝑚𝑚60 𝑠𝑠

= 13.1 𝑚𝑚𝑚𝑚𝑚𝑚

PRACTICE

Maggie has a job at the local convenience store. She gets paid $7.50/hour for her work. If she wants to purchase a new iPod that costs $199.00, how many SECONDS of work will she have to do in order to purchase the iPod? (Neglect the fact that taxes will be taken out of her pay, and that tax will be charged on her purchase)

95,520 seconds

PRACTICE

Convert 85.6 deciliters into microliters.

1 L = 10 dL1 L = 1,000,000 µL

85.6 𝑑𝑑𝐿𝐿1 𝐿𝐿

10 𝑑𝑑𝐿𝐿1,000,000 𝜇𝜇𝐿𝐿

1 𝐿𝐿= 8,560,000 𝜇𝜇𝐿𝐿

RELIABILITY IN MEASUREMENTS

When we make measurements, we must know that they are reliable. There are two benchmarks for reliability:

Precision – How close multiple measurements are to one another.

Accuracy – How close a single measurement is to the accepted value.

ERASER/DARTBOARD ANALOGY: If I aim at

something and I hit my intended target once, that is accuracy.

If I hit the same thing each time, regardless of where I have aimed, that is precision.

To have bothprecision and accuracy, you must hit your intended target each time you throw.

PERCENT ERROR

In chemistry, we often want to make a comparison of how close we have come to an accepted value, also known as percent error.

𝑃𝑃𝑃𝑃𝑃𝑃𝑐𝑐𝑃𝑃𝑚𝑚𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑒𝑒𝑃𝑃 = 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 − 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑒𝑒𝑎𝑎𝑎𝑎𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑎𝑎𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑒𝑒𝑎𝑎𝑎𝑎𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑎𝑎

𝑥𝑥 100

If percent error is negative, the experimental value is LESS THAN the accepted value.

If percent error is positive, the experimental value is MORE THAN the accepted value.

PERCENT ERROR

A 100.00g mixture is 85.00g sulfur and 15.00g iron. Using a magnet, you recover 13.75g of iron. What is your percent error?

% error = (13.75 g-15.00 g) x 100%15.00 g

= - 8.33% error (rounded to sig figs)

TAKING MEASUREMENTS

Any time we take a measurement, we must use some type of instrument Temperature – thermometers Mass – balance (scale) Length – ruler Volume – graduated cylinder

When taking a measurement with a digital instrument, the instrument will estimate your final digit for you.

When taking a measurement with a manual instrument, we must include all of the digits we know for certain, plus one estimated digit. We call this rounding to significant figures.

TAKING MEASUREMENTS

How long would the block below be?

2.15 cm

TAKING MEASUREMENTS

How many milliliters in this graduated cylinder? We measure to the

BOTTOM of the meniscus!

43.0 mL

TAKING MEASUREMENTS

What is the length of this screw to sig figs? 5.10 cm

TAKING MEASUREMENTS

What about now? 5.1 cm

TAKING MEASUREMENTS

And now? 5 cm

TAKING MEASUREMENTS

Measure the Celsius temperature from this thermometer: 8.0 °C

TAKING MEASUREMENTS

How about this one? -7 °C

PRACTICE Measure each to the correct number of significant figures:

1.92 cm

52.8 mL

1.0 °C

SIGNIFICANT FIGURES IN MEASUREMENTS

Whenever we take a measurement, we must always include the correct number of significant figures, based upon our instrument. Remember, include all digits you know FOR SURE

plus one estimated digit!

How do you know which digits are significant if you did not take the measurement yourself?

IDENTIFICATION OF SIGNIFICANT FIGURES

IDENTIFICATION OF SIGNIFICANT FIGURES

Identify the number of significant figures in each of the following measurements:

465 m 327.0 L 0.062 s 1000 g 6802 mL

34214

IDENTIFICATION OF SIGNIFICANT FIGURES

Identify the number of significant figures in:

0.02103 g 2.0100 mL 1900 cm 2.4050 L 0.0002010650 g

45257

IDENTIFICATION OF SIGNIFICANT FIGURES

How about these ones?

1001.10 m 26.050 s 0.008010 g 7200 cm 0.0710 mL 0.02001 L

654234

SIGNIFICANT FIGURES IN CALCULATIONS

If we multiply 7.7m • 5.4m in our calculator, we get 41.58 m2. Can a calculated value be more precise than our

measured values? No, so we must round our answer manually to make

it consistent with our measurements.

The way that we round depends upon the operation we have performed.

ADDITION/SUBTRACTION

Round to the same number of decimal places as the measurement with the least number of decimal places. An easy way to do this is to set up your problem in a

column and draw a vertical line next to the last column that has a digit for each entry:

12.52 m349.0 m+ 8.24 m369.76 m

In the correct number of sig figs, the answer is 369.8 m.

1. Line up the decimal points 2. Cut off at least precise measurement with a line 3. Look to the right of the line to round up or down

ADDITION/SUBTRACTION

Let’s try a subtraction:

74.626 m-28.34 m46.286 m

Your rounded answer is 46.29 m

ADDITION/SUBTRACTION

Practice: 61.2 m + 9.35 m + 8.6 m

9.44 L – 2.11 L

1.36 g + 10.17 g

34.61 m – 17.3 m

79.2 m

7.33 L

11.53 g

17.3 m

79.15 m=

17.31 m=

MULTIPLICATION/DIVISION

Round off answer to the least number of sig figs used in any measurement in the calculation.

**The position of the decimal point has NOTHING to do with the rounding process!!

7.55m •0.34m (3 sf) (2 sf)

2.4526 g ÷ 8.4(5 sf) (2 sf)

= 2.567 m2

= 2.6 m2

= 0.291976 g = 0.29 g

MULTIPLICATION/DIVISION

Practice: 8.3 m • 2.22 m

8432 g ÷ 12.5

5.4 cm • 3.21cm • 1.871 cm

0.053 L ÷ 0.255

18.426 m2 ⇒ 18 m2

674.56 g ⇒ 675 g

32.431914 cm3 ⇒ 32 cm3

0.20784313 L ⇒ 0.21 L

SIG FIGS AND CONVERSION FACTORS

It is important to note that conversion factors are generally considered to be exact, so they will NOT affect the certainty of your answer. This means that when you multiply by a conversion

factor, the number of digits in the factor will NOT change or limit the number of significant figures that your answer should be given in!

Example:

4.608 𝑚𝑚100 𝑐𝑐𝑚𝑚

1 𝑚𝑚= 460.8 𝑐𝑐𝑚𝑚

MIXED OPERATIONS PRACTICE

8.7 g + 15.43 g + 19 g

4.32 cm • 1.7 cm

853.2 L – 627.443 L

639.45 g / 0.058

43.13 g ⇒ 43 g

7.344 cm2 ⇒ 7.3 cm2

225.757 L ⇒ 225.8 L

11,025g ⇒ 11,000 g

WARM UP

Perform each operation and round to the appropriate number of significant figures:

32.5m * 0.021m

145.6231 cL + 26.4 cL

38.742 kg ÷ 0.421

69.5214 s – 28.348569 s

=0.68 m2

=172.0 cL

=92.0 kg

=41.1728 s

0.6825 m2

172.0231 cL

92.0237597 kg

41.172831 s

MIXED OPERATIONS PRACTICE

A. 6.25 x 104 + 3.98 x 106

B. 5.98 x 107 * 6.82 x 105

C. 4.37 x 104 – 6.34 x 103

D. 5.23 x 106 / 4.92 x 10-3

4.04 x 106

4.08 x 1013

3.74 x 104

1.06 x 109

SCIENTIFIC NOTATION

In chemistry, we often use very small or very large numbers: 1 atom = 0.000 000 000 000 000 000 000 327 g

In order to avoid writing out all these numbers, we use scientific notation. Scientific Notation – a number written as the product

of a coefficient and a power of ten. Example: 540,000 is written as 5.4 x 105

5.4 is the coefficient, 105 is the power of ten, or an exponent

WRITING NUMBERS IN SCIENTIFIC NOTATION The coefficient is always a number greater than

or equal to one AND less than ten: (1≤ coefficient <10)

To get the power of ten, count how many spaces you must move the decimal point to get the coefficient between one and ten:• Numbers > 10 have positive exponents (move decimal

point to left)• Numbers < 1 have negative exponents (move decimal

point to right)

“Regular” numbers can be written in scientific notation as well: 5 = 5 x 100

WRITING NUMBERS IN SCIENTIFIC NOTATION

Place the following into scientific notation:

245,000

0.000 039 5

32,000,000

0.000 000 006 5

2.45 x 105

3.95 x 10-5

3.2 x 107

6.5 x 10-9

PUTTING NUMBERS INTO SCIENTIFICNOTATION

Try some:A. 84,300B. 0.000 006 2C. 68,200,000,000D. 0.000 000 000 52E. 849,000,000,000,000F. 0.002 34

A. 8.43 x 104

B. 6.2 x 10-6

C. 6.82 x 1010

D. 5.2 x 10-10

E. 8.49 x 1014

F. 2.34 x 10-3

WRITING NUMBERS IN SCIENTIFIC NOTATION

Take the following out of scientific notation:

6.25 x 105

2.98 x 10-4

3.57 x 107

8.74 x 10-6

625,000

0.000 298

35,700,000

0.000 008 74

TAKING NUMBERS OUT OF SCIENTIFICNOTATION

Try some:A. 5.48 x 105

B. 9.3 x 10-4

C. 3.8 x 108

D. 7.1 x 10-7

E. 2.62 x 103

F. 8.3 x 10-9

A. 548,000B. 0.000 93C. 380,000,000D. 0.000 000 71E. 2,620F. 0.000 000 008 3

OPERATIONS USING SCIENTIFIC NOTATION You MUST be able to perform these operations

manually (without your calculator!)

You will NOT need to worry about keeping track of Sig Figs here; just round the final coefficient to three (3) sig figs

If your answer does not follow our standard form for standard scientific notation, you can adjust it: If you make the value of the coefficient bigger, the

exponent must get smaller. If you make the value of the coefficient smaller, the

exponent must get bigger.

MULTIPLICATION

To multiply: Multiply the coefficients and add the exponents

(3.00 x 104) ● (5.00 x 102) =(3.00●5.00) x 10 (4+2)

=15.0 x 10 6= 1.50 x 107

DIVISION

To divide: Divide the coefficients and subtract the exponents (bottom from top)

(6.00 x 104) ÷ (2.00 x 102) = (6.00÷2.00) x 10 (4-2)

= 3.00 x 10 2

MULTIPLICATION/DIVISION PRACTICE

Practice:6.8 x 103 ● 4.54 x 106

9.2 x 10-3 ÷ 6.3 x 106

2.0 x 10-1 ● 8.5 x 105

2.4 x 106 ÷ 5.49 x 10-9

30.9 x 109

1.46 x 10-9

17.0 x 104

0.437 x 1015

= 3.09 x 1010

= 1.70 x 105

= 4.37 x 1014

ADDITION/SUBTRACTION

To add or subtract, you must first adjust the decimal point so that the exponents in each number are the same. Then, just add/subtract the coefficients.

ADDITION/SUBTRACTION

(3.00 x 104) + (2.00 x 102)

= (3.00 x 104) + (0.02 x 104) = 3.02 x 104

(3.00 x 104) - (2.00 x 102)

= (3.00 x 104) - (0.02 x 104) = 2.98 x 104

= (300. x 102) + (2.00 x 102)= 302 x 102

= 3.02 x 104

= (300. x 102) - (2.00 x 102) = 298 x 102

= 2.98 x 104

ADDITION/SUBTRACTION PRACTICE

7.2 x 105 – 6.4 x 103

4.5 x 102 + 1.6 x 104

4.9 x 106 – 8.2 x 104

6.3 x 109 + 4.1 x 1011

7.13 x 105

1.65 x 104

4.82 x 106

4.16 x 1011

PROBLEM SOLVING

There are four steps generally followed when one wants to solve a problem:

Analyze – Read the problem, identify known and unknown quantities

Plan – Identify equations that might be helpful, find a way to solve

Compute – Follow your plan to solve the problem Evaluate – Make sure your answer makes sense,

check your work

PROPORTIONS Direct Proportions

Dividing one quantity by the other gives a constant value. As “y” increases/decreases, so must “x” so that “k” stays

constant

𝑦𝑦𝑥𝑥

= 𝑘𝑘

Inverse Proportions Product of two quantities are constant values. As “y” increases/decreases, “x” must do the oppositeso that

“k” stays constant 𝑥𝑥𝑥𝑥 = 𝑘𝑘

In both of these situations, k is a proportionality constant!